Subject: sci.physics Frequently Asked Questions (Part 1 of 4)
From: [email-masked]
Date: 13 Oct 1995 14:40:48 GMT
Newsgroups: sci.physics,sci.physics.research,sci.physics.cond-matter,sci.physics.particle,alt.sci.physics.new-theories,sci.answers,alt.answers,news.answers

Posted-By: auto-faq 3.1.1.2
Archive-name: physics-faq/part1

--------------------------------------------------------------------------------
              FREQUENTLY ASKED QUESTIONS ON SCI.PHYSICS - Part 1/4
--------------------------------------------------------------------------------

        This Frequently Asked Questions List is posted monthly to the
USENET newsgroups sci.physics, sci.physics.cond-matter,
sci.physics.research, sci.physics.particle, and
alt.sci.physics.new-theories in an attempt to provide good answers to
frequently asked questions and other reference material which is worth
preserving. If you have corrections or answers to other frequently
asked questions that you would like included in this posting, send
E-mail to [email-masked] (Michael Weiss).  The originator and
original maintainer of this FAQ was Scott I. Chase.

        This document, as a collection, is Copyright (c) 1994 by Scott
I. Chase.  The individual articles are Copyright (c) 1994/5 by the
individual authors listed.  All rights are reserved.  Permission to
use, copy and distribute this unmodified document by any means and for
any purpose EXCEPT PROFIT PURPOSES is hereby granted, provided that
both the above Copyright notice and this permission notice appear in
all copies of the FAQ itself.  Reproducing this FAQ by any means,
included, but not limited to, printing, copying existing prints,
publishing by electronic or other means, implies full agreement to the
above non-profit-use clause, unless upon explicit prior written
permission of the authors.

        This FAQ is provided by the authors "as is," with all its faults.
Any express or implied warranties, including, but not limited to, any 
implied warranties of merchantability, accuracy, or fitness for any 
particular purpose, are disclaimed.  If you use the information in this
document, in any way, you do so at your own risk.

        This document is probably out of date if you are reading it
more than 30 days after the date which appears in the header.  You can
get it by FTP from rtfm.mit.edu or one of its mirror sites:

   North America: ftp.uu.net                     /usenet/news.answers
                  mirrors.aol.com                /pub/rtfm/usenet
                  ftp.seas.gwu.edu               /pub/rtfm
                  rtfm.mit.edu                   /pub/usenet/news.answers
   Europe:        ftp.uni-paderborn.de           /pub/FAQ
                  ftp.Germany.EU.net             /pub/newsarchive/news.answers
                  ftp.sunet.se                   /pub/usenet
   Asia:          nctuccca.edu.tw                /USENET/FAQ
                  hwarang.postech.ac.kr          /pub/usenet/news.answers
                  ftp.hk.super.net               /mirror/faqs

Look for the files

   physics-faq/part1
   physics-faq/part2
   physics-faq/part3
   physics-faq/part4

        If you are a new reader of the Physics newsgroups, please read 
item #1, below.   If you do not wish to read the FAQ at all, add 
"Frequently Asked Questions" to your .KILL file.  

        A listing of new items can be found above the subject index,
so that you can quickly identify new subjects of interest.  To locate
old items which have been updated since the last posting, look for the
stars (*) in the subject index, which indicate new material (other
than minor corrections).

        Items which have been submitted by a single individual are 
attributed to the original author.  All other contributors have been thanked
privately. 

New Item: #25. Can You See the Lorentz Contraction?

Index of Subjects
-----------------

FAQ 1/4 - Administriva and Reference

 1. An Introduction to the Physics Newsgroups on USENET
 2. The Care and Feeding of Kill Files
 3. Accessing and Using Online Physics Resources
 4. A Physics Booklist - Recommendations from the Net
 5. The Nobel Prize for Physics

FAQ 2/4 - Cosmology and Astrophysics

 6. Gravitational Radiation
 7. Is Energy Conserved in General Relativity?
 8. Olbers' Paradox
 9. What is Dark Matter?
10. Some Frequently Asked Questions About Black Holes
11. The Solar Neutrino Problem
12. The Expanding Universe

FAQ 3/4 - General Physics

13. Apparent Superluminal Velocity of Galaxies
14. Hot Water Freezes Faster than Cold!
15. Why are Golf Balls Dimpled?
16. How to Change Nuclear Decay Rates
17. What is a Dippy Bird, and how is it used?
18. Below Absolute Zero - What Does Negative Temperature Mean?
19. Which Way Will my Bathtub Drain?
20. Why do Mirrors Reverse Left and Right?
21. Why Do Stars Twinkle While Planets Do Not? 
22. Time Travel - Fact or Fiction?
23. Open Questions

FAQ 4/4 - Particles, Special Relativity and Quantum Mechanics

24. Special Relativistic Paradoxes and Puzzles
    (a) The Barn and the Pole
    (b) The Twin Paradox
    (c) The Superluminal Scissors
*25. Can You See the Lorentz-Fitzgerald Contraction?
26. Tachyons
27. The Particle Zoo
28. Does Antimatter Fall Up or Down?
29. What is the Mass of a Photon?
30. Baryogenesis - Why Are There More Protons Than Antiprotons?
31. The EPR Paradox and Bell's Inequality Principle
32. Some Frequently Asked Questions About Virtual Particles 

********************************************************************************
Item 1.                                         updated 10-APR-1994 by SIC
                                                original by Scott I. Chase

An Introduction to the Physics Newsgroups on USENET
---------------------------------------------------

        The USENET hierarchy contains a number of newsgroups dedicated
to the discussion of physics and physics-related topics.  These
include sci.physics, sci.physics.research, sci.physics.cond-matter,
sci.physics.particle and alt.sci.physics.new-theories, to all of which
this general physics FAQ is cross-posted.  Some of the more narrowly
focussed physics newsgroups have their own FAQs, which can, of course,
be found in the appropriate newsgroups.

        Sci.Physics is an unmoderated newsgroup dedicated to the
discussion of physics, news from the physics community, and
physics-related social issues.  Sci.Physics.Research is a moderated
newgroup designed to offer an environment with less traffic and more
opportunity for discussion of serious topics in physics among experts
and beginners alike.  The current moderators of sci.physics.research
are John Baez ([email-masked]), William Johnson
([email-masked]), Cameron Randale (Dale) Bass
([email-masked]), and Lee Sawyer
([email-masked]).  Sci.physics.cond-matter is an unmoderated
newsgroup dedicated to the discussion of the physics of condensed
matter.  Sci.physics.particle is an unmoderated newsgroup dedicated to
the discussion of all aspects of particle physics by people with all
levels of expertise.  Alt.sci.physics.new-theories is an open forum
for discussion of any topics related to conventional or unconventional
physics.  In this context, "unconventional physics" includes any ideas
on physical science, whether or not they are widely accepted by the
mainstream physics community.

        People from a wide variety of non-physics backgrounds, as well
as students and experts in all areas of physics participate in the ongoing
discussions on these newsgroups.  Professors, industrial scientists, 
graduate students, etc., are all on hand to bring physics expertise to 
bear on almost any question.   But the only requirement for participation 
is interest in physics, so feel free to post -- but before you do, please 
do the following: 

(1) Read this posting, a.k.a., the FAQ.  It contains good answers,
contributed by the readership,  to some of the most frequently asked
questions. 

(2) Understand "netiquette."  If you are not sure what this means,
subscribe to news.announce.newusers and read the excellent discussion of
proper net behavior that is posted there periodically.  

(3) Be aware that there is another newsgroup dedicated to the discussion of
"alternative" physics.  It is alt.sci.physics.new-theories, and is the
appropriate forum for discussion of physics ideas which are not widely
accepted by the physics community.  Sci.Physics is not the group for such
discussions.  A quick look at articles posted to both groups will make the
distinction apparent. 

(4) Read the responses already posted in the thread to which you want to
contribute.  If a good answer is already posted, or the point you wanted
to make has already been made, let it be.  Old questions have probably been
thoroughly discussed by the time you get there -- save bandwidth by posting
only new information.  Post to as narrow a geographic region as is 
appropriate.  If your comments are directed at only one person, try E-mail.

(5) Get the facts right!  Opinions may differ, but facts should not.  It is
very tempting for new participants to jump in with quick answers to physics
questions posed to the group.  But it is very easy to end up feeling silly
when people barrage you with corrections.  So before you give us all a
physics lesson you'll regret -- look it up. 

(6) Don't post textbook problems in the hope that someone will do your 
homework for you.  Do your own homework; it's good for you.   On the other 
hand, questions, even about elementary physics, are always welcome.  So 
if you want to discuss the physics which is relevent to your homework,
feel free to do so.  Be warned that you may still have plenty of 
work to do, trying to figure out which of the many answers you get are 
correct.

(7) Be prepared for heated discussion.  People have strong opinions about
the issues, and discussions can get a little "loud" at times. Don't take it
personally if someone seems to always jump all over everything you say. 
Everyone was jumping all over everybody long before you got there!  You
can keep the discussion at a low boil by trying to stick to the facts. 
Clearly separate facts from opinion -- don't let people think you are
confusing your opinions with scientific truth.  And keep the focus of
discussion on the ideas, not the people who post them. 

(8) Tolerate everyone.  People of many different points of view, and widely
varying educational backgrounds from around the world participate in this
newsgroup.  Respect for others will be returned in kind.  Personal
criticism is usually not welcome. 

********************************************************************************
Item 2.

The Care and Feeding of Kill Files              updated 28-SEP-1993 by SIC
----------------------------------              original by Scott I. Chase

        With most newsreaders, it is possible for you to selectively ignore
articles with certain title words, or by a certain author.  This feature is
implemented as a "kill file," which contains instructions to your
newsreader about how to filter out unwanted articles.  The exact details on
how to specify articles you want to ignore varies from program to program,
so you should check the documentation for your particular newsreader. Some
examples are given below for a few common newsreaders.  If your newsreader
does not support kill files, you may want to consider upgrading to one that
does.  Some of the more popular newsreaders that support kill files are rn,
trn, nn, xrn, gnews, and gnus. 

        Let's say that you wish to `kill' all posts made by a certain user.
Using the `rn' or `trn' newsreader, you would type a [CTRL]-K while in read
mode to begin editing the kill file, and then type the following: 

     /From: [email-masked]/h:j

This will look for articles that come with "From: [email-masked]" in
the header, junk them, and then display the subject lines of titles that
just got zapped. 

To kill articles by Subject titles, you would type something like this:

     /: *The Big Bang Never Happened/:j
     /: *Space Potatoes Have Inertia/:j

When finished, save the kill file in the normal manner for the editor
you're using. 

In trn 3.0 and higher you can use the faster command

     /username@sitename\.com/f:j

to kill all of username's postings. In trn change the 'j' to ',' to kill
all the replies as well.  Note the '\' to escape the '.'. This is needed in
any search string in a kill file (although they usually work if you
forget). Also in [t]rn you can simply hit K to automatically killfile the
current subject without directly editing the file. 

        For the `nn' newsreader, type a capital K when viewing the contents
of a newsgroup.  nn will then ask you a few questions on whether it is a
Subject or a Name, duration of time that the posts are to be killed, etc. 
Simply answer the questions accordingly. 

        There's a lot more to it, of course, when you become proficient.
You can kill all articles cross-posted to specific groups, for example, or
kill any article with a particular name or phrase appearing anywhere in the
header.  A good primer is in the "rn KILL file FAQ" which appears
periodically in news.answers.  You should also check the man pages for your
particular newsreader. 

********************************************************************************
Item 3.                                       slightly updated 1-AUG-1995 by MW
                                              updated 5-DEC-1994 by SIC
                                              original by Scott I. Chase
Accessing and Using Online Physics Resources
--------------------------------------------

(I) Physical Constants

These are available on the Web, at URL

    http://physics.nist.gov/PhysRefData/codata86/codata86.html

(II) Particle Physics Databases

        The Full Listings of the Review of Particle Properties (RPP), as 
well as other particle physics databases, are accessible on-line.  Here is 
a summary of the major ones, as described in the RPP:

(A) SLAC Databases

PARTICLES   - Full listings of the RPP
HEP         - Guide to particle physics preprints, journal articles, reports,
              theses, conference papers, etc.
CONF        - Listing of past and future conferences in particle physics
HEPNAMES    - E-mail addresses of many HEP people
INST        - Addresses of HEP institutions
DATAGUIDE   - Adjunct to HEP, indexes papers
REACTIONS   - Numerical data on reactions (cross-sections, polarizations, etc)
EXPERIMENTS - Guide to current and past experiments

Anyone with a SLAC account can access these databases.  Alternately, most
of us can access them via QSPIRES.  You can access QSPIRES via BITNET with
the 'send' command ('tell','bsend', or other system-specific command) or by
using E-mail.  For example, send QSPIRES@SLACVM FIND TITLE Z0 will get you
a search of HEP for all papers which reference the Z0 in the title.  By
E-mail, you would send the one line message "FIND TITLE Z0" with a blank
subject line to [email-masked] or [email-masked].
QSPIRES is free.  Help can be obtained by mailing "HELP" to QSPIRES.

For more detailed information, see the RPP, p.I.12, or contact: Louise
Addis ([email-masked]) or Harvey Galic ([email-masked]).

(B) CERN Databases on ALICE

LIB         - Library catalogue of books, preprints, reports, etc.
PREP        - Subset of LIB containing preprints, CERN publications, and 
              conference papers.
CONF        - Subset of LIB containing upcoming and past conferences since 1986
DIR         - Directory of Research Institutes in HEP, with addresses, fax,
              telex, e-mail addresses, and info on research programs

ALICE can be accessed via DECNET or INTERNET.  It runs on the CERN library's
VXLIB, alias ALICE.CERN.CH (IP# 128.141.201.44).  Use Username ALICE (no 
password required.)  Remote users with no access to the CERN Ethernet can
use QALICE, similar to QSPIRES.  Send E-mail to [email-masked], put
the query in the subject field and leave the message field blank.  For 
more information, send the subject "HELP" to QALICE or contact CERN 
Scientific Information Service, CERN, CH-1211 Geneva 23, Switzerland,
or E-mail [email-masked].

Regular weekly or monthly searches of the CERN databases can be arranged
according to a personal search profile.  Contact David Dallman, CERN SIS
(address above) or E-mail [email-masked].

DIR is available in Filemaker PRO format for Macintosh.  Contact Wolfgang
Simon ([email-masked]).

(C) Particle Data Group Online Service

        The Particle Data Group is maintaining a new user-friendly computer
database of the Full Listings from the Review of Particle Properties. Users
may query by paper, particle, mass range, quantum numbers, or detector and
can select specific properties or classes of properties like masses or
decay parameters. All other relevant information (e.g. footnotes and
references) is included. Complete instructions are available online. 

        The last complete update of the RPP database was a copy of the Full
Listings from the Review of Particle Properties which was published as
Physical Review D45, Part 2 (1 June 1992). A subsequent update made on 27
April 1993 was complete for unstable mesons, less complete for the W, Z, D
mesons, and stable baryons, and otherwise was unchanged from the 1992
version. 

DECNET access: SET HOST MUSE or SET HOST 42062
TCP/IP access: TELNET MUSE.LBL.GOV or TELNET 131.243.48.11
Login to: PDG_PUBLIC with password HEPDATA.

Contact: Gary S. Wagman, (510)486-6610.  Email: ([email-masked]).

(D) Other Databases

Durham-RAL and Serpukhov both maintain large databases containing Particle
Properties, reaction data, experiments, E-mail ID's, cross-section
compilations (CS), etc.  Except for the Serpukhov CS, these databases
overlap SPIRES at SLAC considerably, though they are not the same and may
be more up-to-date.  For details, see the RPP, p.I.14, or contact:
For Durham-RAL, Mike Whalley ([email-masked],[email-masked]) or 
Dick Roberts ([email-masked]).  For Serpukhov, contact Sergey Alekhin 
([email-masked]) or Vladimir Exhela ([email-masked]). 

(III) Online Preprint Sources

There are a number of online sources of preprints:

[email-masked] (algebraic geometry)
[email-masked]           (astrophysics)
[email-masked]           (condensed matter)
[email-masked]           (functional analysis)
[email-masked]             (e-mail address database)
[email-masked]            (computational and lattice physics)
[email-masked]                 (high energy physics phenomenological)
[email-masked]                 (high energy physics theoretical)
[email-masked]                 (high energy physics experimental)
[email-masked]              (liquid crystals, optical materials)
[email-masked]                  (general relativity, quantum cosmology)
[email-masked],               (nuclear physics theory)
[email-masked]               (nonlinear science)

Note that babbage.sissa.it also mirrors hep-ph, hep-th and gr-qc.

        To get things if you know the preprint number, send a message to 
the appropriate address with subject header "get (preprint number)" and 
no message body. If you *don't* know the preprint number, or want to get 
preprints regularly, or want other information, send a message with 
subject header "help" and no message body. 

On the Web, some of these preprint archive databases are accessible at 
url http://xxx.lanl.gov/.

The following GOPHER servers which are concerned with physics are currently 
running on the Internet.  They mainly provide a full-text indexed archive 
to the preprint mailing lists:

xyz.lanl.gov, port 70                (LANL Nonlinear Sciences)
mentor.lanl.gov,70                   ('traditional' preprint lists)
babbage.sissa.it,70                  ('traditional' preprint lists)
physinfo.uni-augsburg.de,70          (all lists, but only abstracts) 

(IV) Mailing Lists

In addition to the preprint services already described, there are
several mailing lists that allow one to regularly receive material via
email.  To get a long list of many of them, send mail to
[email-masked] with the following command in the text (not the
subject) of your message:

  LISTS global

To subscribe, send mail to [email-masked] with the following
command in the text (not the subject) of your message:

  SUBSCRIBE <listname> <your-first-name> <your-last-name>

where <listname> is the name of the list.  Example: 

  SUBSCRIBE PHYSICS Isaac Newton

Here are a few of the physics-related lists:

ACC-PHYS        Preprint server for Accelerator Physics        
ALPHA-L         L3 Alpha physics block analysis diagram group
ASTRO-PH        Preprint server for Astrophysics
FUSION          Redistribution of sci.physics.fusion
OPTICS-L        Optics Newsletter
PHYS-L          Forum for Physics Teachers
PHYS-STU        Physics Student Discussion List
PHYSHARE        Sharing resources for high school physics
PHYSIC-L        Physics List
PHYSICS         Physics Discussion
POLYMER         Polymer-related discussions and announcements
POLYMERP        Polymer Physics discussions
SPACE           sci.space.tech Digest
SUP-COND        SuperConductivity List
WKSPHYS         Workshop Physics List

The AIP runs several mailing lists.  The server is "[email-masked]".
Leave the subject line blank, and send text of "help" and "longindex"
on separate lines for a general help file and description of the
mailing lists.  Three mailing lists are

   physnews     a digest of physics news items arising from physics
                meetings, physics journals, newspapers and magazines,
                and other news sources.  Physics News Update appears
                approximately once a week.

    pen         summarizes information on resources, national
                initiatives, outreach programs, grants, professional
                development opportunities, and publications related to
                physics and science education.  It is issued twice a
                month.

    fyi         summarizes science policy developments in Washington
                affecting the physics and astronomy community. It is
                issued between two and five times every week.

To add yourself to a mailing list, send the command
                add <address> <listname>
in the text of a message to the server.  Example: add [email-masked] fyi

(V) The World Wide Web

        There is a wealth of information, on all sorts of topics, available 
on the World Wide Web [WWW], a distributed HyperText system (a network of 
documents connected by links which can be activated electronically). 
Subject matter includes some physics areas such as High Energy Physics,
Astrophysics abstracts, and Space Science, but also includes such diverse
subjects as bioscience, music, and the law.

* How to get to the Web

        If you have no clue what WWW is, you can go over the Internet with
telnet to info.cern.ch (no login required) which brings you to the WWW 
Home Page at CERN. You are now using the simple line mode browser. To move 
around the Web, enter the number given after an item. 

* Browsing the Web

        If you have a WWW browser up and running, you can move around
more easily. The by far nicest way of "browsing" through WWW uses the 
X-Terminal based tool "XMosaic". Binaries for many platforms (ready for use) 
and sources are available via anonymous FTP from ftp.ncsa.uiuc.edu in directory 
Web/xmosaic.  The general FTP repository for browser software is info.cern.ch
(including a hypertext browser/editor for NeXTStep 3.0)

* For Further Information

        For questions related to WWW, try consulting the WWW-FAQ: Its most 
recent version is available via anonymous FTP on rtfm.mit.edu in 
/pub/usenet/news.answers/www-faq , or on WWW at 
http://www.vuw.ac.nz:80/non-local/gnat/www-faq.html

        The official contact (in fact the midwife of the World Wide Web) 
is Tim Berners-Lee, [email-masked]. For general matters on WWW, try 
[email-masked] or Robert Cailliau (responsible for the "physics" 
content of the Web, [email-masked]).

* Finding stuff on the Web

        The URL http://www.yahoo.com is one good starting place for
locating information; for example, 
http://www.yahoo.com/Reference/Scientific_Constants will get you to
a list of scientific constants.

(V) Other Archive Sites 

 http://pdg.lbl.gov/cpep/adventure.html

This page is part of the Contemporary Physics Education Project.

(A)     FreeHEP

        The FreeHEP collection of software, useful to high energy physicists
is available on the Web as  

  http://heplibw3.slac.stanford.edu:80/FIND/FHMAIN.HTML

or anonymous ftp to freehep.scri.fsu.edu.  This is high-energy oriented but 
has much which is useful to other fields also.  Contact Saul Youssef
([email-masked]) for more information.

(B)     AIP Archives

        An archive of the electronic newsletters of the American Institute 
of Physics is now available on nic.hep.net.  The three publications are 
"For Your Information", "The Physics News Update" written by Dr. Phil Schewe, 
and "What's New" written by Dr. Robert Park".

FYI is archived as [ANON_FTP.AIP-FYI.199*]AIPFYI-nnn-mmmddyyyy.TXT.
PNU is archived as [ANON_FTP.PHYSICS-NEWS.199*]PHYSICS-NEWS-yyyy-mm-dd.TXT. 
WN  is archived as [ANON_FTP.WHATS-NEW.199*]WHATS-NEW-yyyy-mm-dd.TXT

In each case, the last issue received is always available as: latest.txt. 

(C)
        There is an FTP archive site of preprints and programs
for nonlinear dynamics, signal processing, and related subjects on node
lyapunov.ucsd.edu (132.239.86.10) at the Institute for Nonlinear Science,
UCSD.  Just login anonymously, using your host id as your password. Contact
Matt Kennel ([email-masked]) for more information.

(VI) Physics Education Online

(A) Mailing Lists

PHYS-L           PHYS-L@UWF        Forum for Physics Teachers
PHYS-STU         PHYS-STU@UWF      Physics Student Discussion List

(B) On the Web

The Computers in Physics Education Committee  of the AAPT has endorsed a 
project to have a site which would point to the all the known physics 
education resources on the net. Alan Cairns has agreed to maintain the site 
until the AAPT is convinced to put some funding into maintaining it. The 
current URL is:

         http://www.halcyon.com/cairns/physics.html

This project is still in its infancy - anyone with an interest in physics 
education is invited to take a look. Your submissions will allow the site to 
grow into a mature resource.  Contact: [email-masked].

********************************************************************************
Item 4.                                         original Vijay D. Fafat
                                                updated 28-JUL-1994 by SIC

A Physics Booklist - Recommendations from the Net
-------------------------------------------------

This article is a complilation of books recommended by sci.physics
participants as the 'standard' or 'classic' texts on a wide variety of 
topics of general interest to physicists and physics students.  As a 
guide to finding the right book for you, many of the comments from the 
contributors have been retained.

This document is still under construction.  Many entries are incomplete,
and many good books are not yet listed.  Please feel free to contribute 
to this project. Contact [email-masked], who will compile the
information for future updates.  

The formatting and organization of this article will also be reviewed 
and improved in future updates.  This is the first try, and it shows.
Please bear with us.

Subject Index
-------------

You can find books in the area of your choice by searching forward for 
the following keywords:  

General Physics
Classical Mechanics
Classical Electromagnetism
Quantum Mechanics
Statistical Mechanics
Condensed Matter
Special Relativity
Particle Physics
General Relativity
Mathematical Methods
Nuclear Physics
Cosmology
Astronomy
Plasma Physics
Numerical Methods/Simulations
Fluid Dynamics 
Nonlinear Dynamics, Complexity and Chaos
Optics (Classical and Quantum), Lasers
Mathematical Phyiscs
Atomic Phyiscs
Low Temperature Physics, Superconductivity

------------------------------
Subject: General Physics (so even mathematicians can understand it!)

1] M. S. Longair, Theoretical concepts in physics, 1986.
An alternative view of theoretical reasoning in Physics for
final year undergrads.

2] Sommerfeld, Arnold - Lectures on Theoretical Physics
Sommerfeld is God for mathematical physics.

3] Feynman, R: The Feynman lectures on Physics - 3 vols.
        
4] Walker, Jearle: The Flying Circus of Physics

Note: There is the entire Landau and Lifshitz series. They have volumes
on classical mechanics, classical field theory, E&M, QM, QFT, Statistical
Physics, and more. Very good series that spans entire graduate level
curriculum.

5] The New physics / edited by Paul Davies.
This is one *big* book to go through and takes time to look through
topics as diverse as general relativity, astrophysics, particle
theory, quantum mechanics, chaos and nonlinearity, low temperature physics
and phase transitions.  Nevertheless, this is one excellent book of
recent (1989) physics articles, written by several
physicists/astrophysicists.

6] QED : The strange theory of light and matter / Richard P.
Feynman.
One need no longer be confused by this beautiful theory.
Richard Feynman gives an exposition that is once again and by itself
a beautiful explanation of the theory of photon-matter interactions.

------------------------------
Subject: Classical Mechanics

1] Goldstein, Herbert "Classical Mechanics", 2nd ed, 1980.
intermediate to advanced; excellent bibliography

2] Introductory: The Feyman Lectures, vol 1.

3] Symon, Keith - Mechanics, 3rd ed., 1971
undergrad level.

4] Corbin, H and Stehle, P - Classical Mechanics, 2nd ed., 1960

5] V.I. Arnold, Mathematical methods of classical mechanics, translated
by K. Vogtmann and A. Weinstein, 2nd ed., 1989. 
The appendices are somewhat more advanced and cover all sorts of 
nifty topics. Deals with Geometrical aspects of classical mechanics

6] Resnick, R and Halliday, D - Physics, vol 1, 4th Ed., 1993
Excellent introduction without much calculus. Lots of problems and
review questions.

7] Marion, J & Thornton, "Classical Dynamics of Particles and Systems", 
2nd ed., 1970.
Undergrad level. A useful intro to classical dynamics.  Not as advanced 
as Goldstein, but with real worked-out examples.

8] Fetter, A and Walecka, J: Theoretical mechanics of particles and continua.
graduate level text, a little less impressive than Goldstein (and sometimes 
a little less obtuse)

9] Many-Particle Physics, G. Mahan

10] Fetter & Walecka: Theoretical Mechanics of Particles and Continua.

------------------------------
Subject: Classical Electromagnetism

1] Jackson, J. D. "Classical Electrodynamics", 2nd ed., 1975
intermediate to advanced.

2] a] Edward Purcell, Berkely Physics Series Vol 2.
You can't beat this for the intelligent, reasonably sophisticated 
beginning physics student. He tells you on the very first page
about the experimental proof of how charge does not vary with
speed.

   b] Chen, Min, Berkeley Physics problems with solutions.

3] Reitz, J, Milford, F and Christy, R: Foundations of Electromagnetic Theory
3rd ed., 1979
Undergraduate level. Pretty difficult to learn from at first, but good 
reference, for  some calculations involving stacks of thin films and their 
reflectance and transmission properties, for eg.  It's a good, rigorous text
as far as it goes, which is pretty far, but not all the way.  For example, 
they have a great section on optical properties of a single thin film 
between two dielectric semi-infinate media, but no generalization to stacks 
of films.

4] Feynman, R: Feynman Lectures, vol 2

5] Lorrain, P & Corson D: Electromagnetism, Principles and Applications, 1979

6] Resnick, R and Halliday, D: Physics, vol 2, 4th ed., 1993

7] Igor Irodov, Problems in Physics.
Excellent and extensive collection of EM problems for undergrads. 

8] Smythe, William: Static and Dynamic Electricity, 3rd ed., 1968
For the extreme masochists. Some of the most hair-raising EM 
problems you'll ever see. Definitely not for the weak-of-heart.

9] Landau, Lifschitz, and Pitaevskii, "Electrodynamics of Continuous Media,"
2nd ed., 1984
same level as Jackson and with lots of material not in Jackson.

10] Marion, J and Heald, M: "Classical Electromagnetic Radiation," 2nd ed.,
1980  undergraduate or low-level graduate level

------------------------------
Subject: Quantum Mechanics  

1] Cohen-Tannoudji, "Quantum Mechanics I & II", 1977.
introductory to intermediate.

2] Liboff - Introductory Quantum Mechanics, 2nd ed., 1992
elementary level. Makes a few mistakes.

3] Sakurai, J - Modern Quantum Mechanics, 1985

4] Sakurai, J - Advanced Quantum Mechanics, 1967
Good as an introduction to the very basic beginnings of quantum field 
theory, except that it has the unfortunate feature of using 'imaginary time'
to make Minkowski space look Euclidean. 

5] Wheeler, J and Zurek, W (eds.)  Quantum Theory and Measurement, 1983
On the philosophical end. People who want to know about interpretations 
of quantum mechanics should definitely look at this collection of 
relevant articles.  

6] DeWitt, C and Neill Graham: The Many Worlds Interpretation of Quantum 
Mechanics
Philosophical. Collection of articles.

7] Everett, H: "Theory of the Universal Wavefunction" 
An exposition which has some gems on thermodynamics and probability.
Worth reading for this alone.

8] Bjorken, J and Drell, S - Relativistic Quantum Mechanics/ Relativistic
Quantum Fields
(for comments, see under Particle Physics)

9] Ryder, Lewis - Quantum Field Theory, 1984

10] Guidry, M - Gauge Field Theories :  an introduction with applications,
1991      

11] Messiah, A: Quantum Mechanics, 1961

12] Dirac, Paul:        
    a] Principles of QM, 4th ed., 1958
    b] Lectures in QM, 1964
    c] Lectures on Quantum Field Theory, 1966

13] Itzykson, C and Zuber, J: Quantum Field Theory, 1980
Very advanced level.

14] Slater, J: "Quantum theory: Address, essays, lectures.
Good follow on to Schiff.
        
      note: Schiff, Bjorken and Drell, Fetter and Walecka, and Slater 
      are all volumes in "International Series in pure and Applied Physics"
      published by McGraw Hill. 

15] Pierre Ramond, Field Theory: A Modern Primer, 2nd edition.
Volume 74 in the FiP series. The so-called "revised printing" is a  
must, as they must've rushed the first printing of the 2nd edition,  
and it's full of inexcusable mistakes.

16] Feynman, R: Lectures - vol III :
A non-traditional approach. A good place to get an intuitive feel
for QM, if one already knows the traditional approach.

&&&&&&& 17] Heitler & London, "Quantum theory of molecules"??

18] Bell: Speakable and Unspeakable in Quantum Mechanics, 1987
An excellent collection of essays on the philosophical aspects of QM.

19] Milonni: The quantum vacuum: an introduction to quantum electrodynamics
                1994.

20] Holland: The Quantum Theory of Motion
A good bet for strong foundation in QM.

21] John Von Neumann: Mathematical foundations of quantum mechanics, 1955.  
For the more mathematical side of quantum theory, especially for
those who are going to be arguing about measurement theory.

22] Schiff, Leonard, L: Quantum Mechanics, 3rd ed., 1968
A little old. Not much emphasis on airy-fairy things like many worlds
or excessive angst over Heisenberg UP. Straight up QM for people
who want to do calculations. Introductory gradauate level. Mostly
Schrod. eqn. Spin included, but only in an adjunct to Schrod. Not
much emphasis on things like Dirac eqn., etc.

23] "Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles" by
Eisberg and Resnick, 2nd ed., 1985. 
This is a basic intro. to QM, and it is excellent for
undergrads. It is not thorough with math, but fills in a lot
of the  intuitive stuff that most textbooks do not present.

24] Elementary Quantum Mechanics, David Saxon
It's a decent undergraduate (senior level) text.

25] Intermediate Quantum Mechanics, Bethe and Jackiw

------------------------------
Subject: Statistical Mechanics

1] David Chandler, "Introduction to Modern Statistical Mechanics", 1987
                
2] Kittel & Kroemer: Statistical Thermodynamics.
Best of a bad lot.

3] Reif, F : Principles of statistical and thermal physics.
the big and little Reif stat mech books. Big Reif is much better than 
Kittel & Kroemer. He uses clear language but avoids the handwaving that 
thermodynamics often gives rise to. More classical than QM oriented.

4] Bloch, Felix: Fundamentals of Statistical Mechanics.

5] Radu Balescu "Statistical Physics"
Graduate Level. Good description of non-equilibrium stat. mech.
but difficult to read. It is all there, but often you don't
realize it until after you have learned it somewhere else.
Nice development in early chapters about parallels between 
classical and quantum Stat. Mech.

6] Huang (grad)
        
The following 6 books deal with modern topics in (mostly) classical 
statistical mechanics, namely, the central notions of linear response theory
(Forster) and critical phenomena (the rest) at level suitable for
beginning graduate students.

7] Thermodynamics, by H. Callen.  

8] Statistical Mechanics, by R. K. Pathria

9] Hydrodynamic Fluctuations, Broken Symmetry, and Correlation
Functions, by D. Forster

10] Introduction to Phase Transitions and Critical Phenomena,
by H. E. Stanley

11] Modern Theory of Critical Phenomena, by S. K. Ma

12] Lectures on Phase Transitions and the Renormalization Group,
by N. Goldenfeld

13] Methods of Quantum Field Theory in Statistical Physics,
Abrikosov, Gorkov, and Dyzaloshinski

------------------------------
Subject: Condensed Matter

1] Charles Kittel, "Introduction to Solid State Physics" (ISSP), introductory

2] Ashcroft and Mermin, "Solid State Physics", interm to advanced

3] Charles Kittel, Quantum Theory of Solids.  
This is from before the days of his ISSP; it is a more advanced book.
At a similar level...

4] Solid State Theory, by W. A. Harrison (a great bargain now that it's
published by Dover)

5] Theory of Solids, by Ziman.

6] Fundamentals of the Theory of Metals, by Abrikosov
Half of the book is on superconductivity.

------------------------------
Subject: Special Relativity

1] Taylor and Wheeler, _Spacetime Physics_
Still the best introduction out there.  

2] "Relativity" :  Einstein's popular exposition.

3] Wolfgang Rindler, Essential Relativity. Springer 1977
With a heavy bias towards astrophysics and therefore on a more 
moderate level formally. Quite strong on intuition.

4] A P French: Special Relativity
A through introductory text.  Good discussion of the twin
paradox, pole and the barn etc.  Plenty of diagrams
illustrating lorentz transformed co-ordinates, giving both an
algebraic and geometrical insight to SR.

------------------------------
Subject: Particle Physics

1] Kerson Huang, Quarks, leptons & gauge fields, World Scientific, 1982.
Good on mathematical aspects of gauge theory and topology.

2] L. B. Okun, Leptons and quarks, translated from Russian by V. I. Kisin,
North-Holland, 1982.

3] T. D. Lee, Particle physics and introduction to field theory.

4] Itzykson: Particle Physics

5] Bjorken & Drell: "Relativistic Quantum Mechanics"
One of the more terse books.  The first volume on 
Relativistic quantum mechanics covers the subject in a blinding 300 
pages.  Very good if you *really* want to know the subject.

6] Francis Halzen & Alan D. Martin, "Quarks & Leptons", beginner to 
intermediate, this is a standard textbook for graduate level courses. 
Good knowledge of quantum mechanics and special relativity is assumed.
A very good introduction to the concepts of particle physics.  Good
examples, but not a lot of Feynman diagram calculation.  For this, 
see Bjorken & Drell.

7] Donald H. Perkins: Introduction to high energy physics
Regarded by many people in the field as the best introductory text at
the undergraduate level. Covers basically everything with almost no
mathematics.

8] Close,Marten, and Sutton: The Particle Explosion.  A popular
exposition of the history of particle physics with terrific photography.

9]  Christine Sutton: Spaceship Neutrino
        A good, historical, largely intuitive introduction to
particle physics, seen from the neutrino viewpoint.

------------------------------
Subject: General Relativity

1] The telephone book, er, that is, MTW, Meisner, Thorne and Wheeler. 
The "bible". W. H. Freeman & Co., San Francisco 1973

2] Robert M. Wald, Space, Time, and Gravity : the Theory of the Big Bang
   and Black Holes.
A good nontechnical introduction, with a nice mix of mathematical 
rigor and comprehensible physics. 

3] Schutz: First Course in General Relativity.

4] Weinberg: Gravitation and Cosmology 
Good reference book, but not a very good read.

5] Hans Ohanian: Gravitation & Spacetime (recently back in print)
For someone who actually wants to learn to work problems, ideal
for self-teaching, and math is introduced as needed, rather than
in a colossal blast.

6] Robert Wald, General Relativity
It's a more advanced textbook than Wald's earlier book, appropriate
for an introductory graduate course in GR.  It strikes just the right
balance, in my opinion, between mathematical rigor and physical
intuition.  It has great mathematics appendices for those who care
about proving theorems carefully, and a good introduction to the
problems behind quantum gravity (although not to their solutions).
I think it's MUCH better than either MTW or Weinberg.

------------------------------
Subject: Mathematical Methods (so that even physicists can understand it!)

1] Morse and Feshbach - Methods of Theoretical Physics (can be hard to
   find)

2] Mathews and Walker, Mathematical Methods for Physicists. 
An absolute joy for those who love math, and very informative even 
for those who don't.

3] Arfken "Mathematical Methods for Physicists" Academic Press
Good introduction at graduate level. Not comprehensive in any
area, but covers many areas widely. Arfken is to math methods
what numerical recipes is to numerical methods -- good intro, but
not the last word.

4] Zwillinger "Handbook of Differential Equations." Academic Press
Kind of like CRC tables but for ODE's and PDE's. Good
reference book when you've got a Diff. Eq. and wnat to find a
solution.

5] Gradshteyn and Ryzhik "Table of Integrals, Series, and Products" Academic
THE book of integrals. Huge, but useful when you need an integral.

------------------------------
Subject: Nuclear Physics

1] Preston and Bhaduri, "Structure of the Nucleus" 

2] Blatt and Weisskopf - Theoretical Nuclear Physics

3] DeShalit and Feshbach - Theoretical Nuclear Physics
This is serious stuff.  Also quite expensive even in paper.  I think 
the hard cover is out of print.  This is volume I (structure).  
Volume II (scattering) is also available. 

4] Satchler: "Direct Nuclear Reactions". 

------------------------------
Subject: Cosmology

1] J. V. Narlikar, Introduction to Cosmology.1983 Jones & Bartlett Publ.
For people with a solid background in physics and higher math, THE 
introductory text, IMHO, because it hits the balance between 
mathematical accuracy (tensor calculus and stuff) and intuitive
clarity/geometrical models very well for grad student level. Of course,
it has flaws but only noticeable by the Real Experts (TM) ...

2] Hawking: Brief History of Time 
Popular Science

3] Weinberg: First Three Minutes 
A very good book.  It's pretty old, but most of the information in it 
is still correct.

4] Timothy Ferris: Coming of Age in the Milky Way.
Popular Science.

5] Kolb and Turner: The Early Universe.
At a more advanced level, a standard reference.  As the title implies, 
K&T cover mostly the strange physics of very early times: it's heavy 
on the particle physics, and skimps on the astrophysics.  There's a 
primer on large-scale structure, which is the most active area of 
cosmological research, but it's really not all that good.

6] Peebles: Principles of Physical Cosmology.
Comprehensive, and on the whole it's quite a good book, but it's 
rather poorly organized.  I find myself jumping back and forth through
the book whenever I want to find anything.

7] "Black Holes and Warped Spacetime", by William J. Kaufmann, III.
This is a great, fairly thorough, though non-mathematical
description  of black holes and spacetime as it relates to
cosmology. I was impressed by how few mistakes Kaufmann makes in
simplifying, while most such books tend to sacrifice accuracy for
simplicity.

8] "Principles of Cosmology and Gravitation", Berry, M. V.
This is very well-written, and useful as an undergrad text.

9] Dennis Overbye: Lonely Hearts of the Cosmos
The unfinished history of converge on Hubble's constant is 
presented, from the perspective of competing astrophysics 
rival teams and institute, along with a lot of background on cosmology 
(a lot on inflation, for instance).  A good insight into the scientific
process.

10] The big bang / Joseph Silk.
I consider Silk's book an absolute must for those who want a quick run
at the current state of big bang cosmology and some of the recent
(1988)issues which have given so many of us lots of problems to solve.

10] Bubbles, voids, and bumps in time : the new cosmology / edited
by James Cornell.
This is quite a nice and relatively short read for some of the
pressing issues (as of 1987-88) in astrophysical cosmology.

11] Structure formation in the universe / T. Padmanabhan.
A no-nonsense book for those who want to calculate some problems
strictly related to the formation of structure in the universe.
The book even comes complete with problems at the end of each chapter.
A bad thing about this book is that there isn't any coverage on
clusters of galaxies and the one really big thing that annoys the
hell outta me is that the bibliography for *each* chapter is all
combined in one big bibliography towards the end of the book which
makes for lots of page flipping.

12] The large-scale structure of the universe / by P. J. E. Peebles.
This is a definitive book for anyone who desires an understanding of
the mathematics required to develop the theory for models of
large scale structure.  The essential techniques in the description
of how mass is able to cluster under gravity from a smooth early
universe are discussed.  While I find it dry in some places, there are
noteworthy sections (e.g. statistical tests, n-point correlation functions,
etc.).

------------------------------
Subject: Astronomy

1] Hannu Karttunen et al. (eds.): Fundamental Astronomy.
The best book covering all of astronomy (also for absolute beginners)
AND still going into a lot of detail for special work for people more 
involved AND presenting excellent graphics and pictures.

2] Pasachoff: Contemporary Astronomy 
Good introductory textbook for the nontechnical reader.  It gives a 
pretty good overview of the important topics, and it has good pictures.

3] Shu, Frank: The physical universe : an introduction to astronomy,

4] Astrophysical formulae : a compendium for the physicist and
astrophysicist / Kenneth R. Lang.
Here is everything you wanted to know (and more!) about astrophysical
formulae on a one-line/one-parargraph/one-shot deal.
Of course, the formulae come complete with references (a tad old, mind
you) but it's a must for everyone who's working in astronomy and
astrophysics. You learn something new everytime you flip through the pages!

------------------------------
Subject: Plasma Physics 

(See Robert Heeter's sci.phys.fusion FAQ for details)

------------------------------
Subject: Numerical Methods/Simulations

1] Johnson and Rees "Numerical Analysis" Addison Wesley
Undergrad. level broad intro.

2] Numerical Recipes in X (X=c,fortran,pascal,etc) Tueklosky and Press

3] Young and Gregory "A survey of Numerical Mathmematics" Dover 2 volumes.
Excellent overview at grad. level. Emphasis toward solution
of elliptic PDE's, but good description of methods to get there
including linear algebra, Matrix techniques, ODE solving methods,
and interpolation theory. Biggest strength is it provides a coherent
framework and structure to attach most commonly used num. methods.
This helps understanding about why to use one method or another. 
2 volumes.

4]Hockney and Eastwood "Computer Simulation Using Particles" Adam Hilger
Good exposition of particle-in-cell (PIC) method and extensions.
Applications to plasmas, astronmy, and solid state are discussed.
Emphasis is on description of algortihms. Some results shown.

5] Birdsall and Langdon "Plasma Physics via Computer Simulations" 
PIC simulation applied to plasmas. Source codes shown. First part
is almost a tutorial on how to do PIC. Second part is like a
series of review articles on different PIC methods.

6] Tajima "Computational Plasma Physics: With Applications to Fusion and 
Astrophysics" Addison Wesley Frontiers in physics Series.
Algorthims described. Emphasis on physics that can be simulated.
Applications limited to plasmas, but subjest areas very broad,
fusion, cosmology, solar astrophysics, magnetospheric physics,
plasma turbulence, general astrophysics.

------------------------------
Subject: Fluid Dynamics

1] Triton "Physical Fluid Dynamics"

2] Batchelor

3] Chandreshekar

------------------------------
Subject: Nonlinear Dynamics, Complexity, and Chaos

There is a FAQ posted regularly to sci.nonlinear. 

1] Prigogine, "Exploring Complexity"
Or any other Prigogine book. If you've read one, you read most of
all of them (A Poincare recurrance maybe?)

2] Guckenheimer and Holmes "Nonlinear Oscillations, Dynamical Systems, and
   Bifurcations of Vector Fields" Springer
Borderline phys/math. Advanced level. Nuts and bolts how to textbook.
No Saganesque visionary thing from the authors. They let the topic
provide all the razz-ma-tazz, which is plenty if you pay attention
and remember the physics that it applies to.

3] Lichtenberg, A. J. and M. A. Lieberman (1982). Regular and Stochastic
Motion. New York, Springer-Verlag.

4] "The Dreams Of Reason" by Heinz Pagels. 
He is a very clear and interesting, captivating writer, and
presents the concepts in a very intuitive way. The level is
popular  science, but it is still useful for physicists who
know  little of complexity.

5] M.Mitchell Waldrop: Complexity.  
A popular intro to the subject of spontaneous orders, complexity 
and so on.  Covers implications for economics, biology etc and not 
just physics.

------------------------------
Subject: Optics (Classical and Quantum), Lasers

1] Born and Wolf
standard reference.

2] Sommerfeld, A: 
For the more classically minded

3] Allen and Eberly's Optical Resonance and Two-Level Atoms.
For quantum optics, the most readable but most limited.

4] Quantum Optics and Electronics (Les Houches summer school 1963-or-4,
but someone has claimed that Gordon and Breach, NY, are going to 
republish it in 1995), edited by DeWitt, Blandin, and Cohen-
Tannoudji, is noteworthy primarily for Glauber's lectures, which 
form the basis of quantum optics as it is known today.

5] Sargent, Scully, & Lamb: Laser Physics

6] Yariv: Quantum Electronics

7] Siegman: Lasers

8] Shen: The Principles of Nonlinear Optics

9] Meystre & Sargent: Elements of Quantum Optics

10] Cohen-Tannoudji, Dupont-Roc, & Grynberg: Photons, Atoms and Atom-Photon
    Interactions.

11] Hecht: Optics
A very good intro optics book (readable by a smart college freshman,
but useful as a reference to the graduate student)

12] "Practical Holography" by Graham Saxby, Prentice Hall: New York; 1988.

This is a very clear and detailed book that is an excellent
introduction to holography for interested undergraduate physics people, as
well as advanced readers, esp. those who are interested in the practical
details of making holograms and the theory behind them.

------------------------------

Subject: Mathematical Physics 
(Lie Algebra, Topology, Knot Theory, Tensors, etc.)

These are books that are sort of talky and fun to read (but still 
substantial - some harder than others). These include things 
mathematicians can read about physics as well as vice versa.  These 
books are different than the "bibles" one must have on hand at all 
times to do mathematical physics.

1] Yvonne Choquet-Bruhat, Cecile DeWitt-Morette, and Margaret
Dillard-Bleick, Analysis, manifolds, and physics (2 volumes)

Something every mathematical physicist should have at her bedside
until she knows it inside and out - but some people say it's not
especially easy to read.

2] Jean Dieudonne, A panorama of pure mathematics, as seen by N. Bourbaki,
   translated by I.G. Macdonald.
Gives the big picture in math.

3] Robert Hermann, Lie groups for physicists, Benjamin-Cummings, 1966.

4] George Mackey, Quantum mechanics from the point of view of the theory 
of group representations, Mathematical Sciences Research Institute,
1984.

5] George Mackey, Unitary group representations in physics, probability,
and number theory.

6] Charles Nash and S. Sen, Topology and geometry for physicists.

7] B. Booss and D.D. Bleecker, Topology and analysis: the Atiyah-Singer
index formula and  gauge-theoretic physics.

8] Bamberg and S. Sternberg, A Course of Mathematics for Students of
   Physics.

9] Bishop & Goldberg: Tensor Analysis on Manifolds.

10] Flanders : Differential Forms with applications to the Physical Sciences.

11] Dodson & Poston Tensor Geometry.

12] von Westenholz: Differential forms in Mathematical Physics.

13] Abraham, Marsden & Ratiu: Manifolds, Tensor Analysis and Applications.

14] M. Nakahara, Topology, Geometry and Physics.

15] Morandi:     The Role of Topology in Classical and Quantum Physics

16] Singer, Thorpe:  Lecture Notes on Elemetary Topology and Geometry

17] L. Kauffman: Knots and Physics, World Scientific, Singapore, 1991.

18] Yang, C and Ge, M: Braid group, Knot Theory & Statistical Mechanics.

19] Kastler, D: C-algebras and their applications to Statistical
    Mechanics and Quantum Field Theory.

20] Courant and Hilbert "Methods of Mathematical Physics" Wiley
Really a math book in disguise. Emphasis on ODE's and PDE's.
Proves existence, etc. Very comprehensive. 2 volumes.

21] Cecille Dewitt: is publishing a book on manifolds that
should be out soon (maybe already is). Very high level, but supposedly
of great importance for anyone needing to set the Feynman path integral
in a firm foundation.

22] Howard Georgi, "Lie Groups for Particle Phyiscs" 
Addison Wesley Frontiers in Physics Series.

23] Synge and Schild 

------------------------------
Subject: Atomic Physics

1] Born and Wolf:
A classic, though a little old. 

------------------------------
Subject: Low Temperature Physics, Superconductivity (high and low Tc), etc.

1] The Theory of Quantum Liquids, by D. Pines and P. Nozieres

2] Superconductivity of Metals and Alloys, P. G. DeGennes
A classic introduction.

3] Theory of Superconductivity, J. R. Schrieffer

4] Superconductivity, M. Tinkham

5] Experimental techniques in low-temperature physics / by Guy K. White.
This is considered by many as a "bible" for those working in
experimental low temperature physics.

Thanks to the contributors who made this compilation possible, including,
but not limited to [email-masked], [email-masked],
[email-masked], [email-masked], p675cen@mpifr-bonn.
mpg.de, [email-masked], [email-masked], 
[email-masked], [email-masked], [email-masked],
[email-masked], [email-masked], [email-masked], 
[email-masked], [email-masked], exunikh
@exu.ericsson.se, [email-masked], [email-masked],
[email-masked], [email-masked], [email-masked], 
[email-masked], [email-masked], [email-masked],
[email-masked], [email-masked], [email-masked],
[email-masked]

********************************************************************************
Item 5.

The Nobel Prize for Physics (1901-1993)         updated 12-OCT-1994 by SIC
---------------------------------------         original by Scott I. Chase

The following is a complete listing of Nobel Prize awards, from the first
award in 1901.  Prizes were not awarded in every year.  The description 
following the names is an abbreviation of the official citation.  

1901    Wilhelm Konrad Roentgen          X-rays
1902    Hendrik Antoon Lorentz          Magnetism in radiation phenomena
        Pieter Zeeman
1903    Antoine Henri Bequerel          Spontaneous radioactivity
        Pierre Curie
        Marie Sklodowska-Curie 
1904    Lord Rayleigh                   Density of gases and 
        (a.k.a. John William Strutt)     discovery of argon
1905    Pilipp Eduard Anton von Lenard  Cathode rays
1906    Joseph John Thomson             Conduction of electricity by gases
1907    Albert Abraham Michelson        Precision meteorological investigations
1908    Gabriel Lippman                 Reproducing colors photographically
                                         based on the phenomenon of interference
1909    Guglielmo Marconi               Wireless telegraphy
        Carl Ferdinand Braun
1910    Johannes Diderik van der Waals  Equation of state of fluids
1911    Wilhelm Wien                    Laws of radiation of heat
1912    Nils Gustaf Dalen               Automatic gas flow regulators 
1913    Heike Kamerlingh Onnes          Matter at low temperature
1914    Max von Laue                    Crystal diffraction of X-rays
1915    William Henry Bragg             X-ray analysis of crystal structure
        William Lawrence Bragg
1917    Charles Glover Barkla           Characteristic X-ray spectra of elements
1918    Max Planck                      Energy quanta
1919    Johannes Stark                  Splitting of spectral lines in E fields
1920    Charles-Edouard Guillaume       Anomalies in nickel steel alloys
1921    Albert Einstein                 Photoelectric Effect
1922    Niels Bohr                      Structure of atoms
1923    Robert Andrew Millikan          Elementary charge of electricity
1924    Karl Manne Georg Siegbahn       X-ray spectroscopy
1925    James Franck                    Impact of an electron upon an atom
        Gustav Hertz
1926    Jean Baptiste Perrin            Sedimentation equilibrium
1927    Arthur Holly Compton            Compton effect
        Charles Thomson Rees Wilson     Invention of the Cloud chamber
1928    Owen Willans Richardson         Thermionic phenomena, Richardson's Law
1929    Prince Louis-Victor de Broglie  Wave nature of electrons
1930    Sir Chandrasekhara Venkata Raman Scattering of light, Raman effect
1932    Werner Heisenberg               Quantum Mechanics 
1933    Erwin Schrodinger               Atomic theory
        Paul Adrien Maurice Dirac
1935    James Chadwick                  The neutron
1936    Victor Franz Hess               Cosmic rays
        Carl D. Anderson                The positron
1937    Clinton Joseph Davisson         Crystal diffraction of electrons
        George Paget Thomson
1938    Enrico Fermi                    New radioactive elements 
1939    Ernest Orlando Lawrence         Invention of the Cyclotron
1943    Otto Stern                      Proton magnetic moment
1944    Isador Isaac Rabi               Magnetic resonance in atomic nuclei
1945    Wolfgang Pauli                  The Exclusion principle
1946    Percy Williams Bridgman         Production of extremely high pressures
1947    Sir Edward Victor Appleton      Physics of the upper atmosphere
1948    Patrick Maynard Stuart Blackett Cosmic ray showers in cloud chambers
1949    Hideki Yukawa                   Prediction of Mesons 
1950    Cecil Frank Powell              Photographic emulsion for meson studies
1951    Sir John Douglas Cockroft       Artificial acceleration of atomic 
        Ernest Thomas Sinton Walton      particles and transmutation of nuclei
1952    Felix Bloch                     Nuclear magnetic precision methods 
        Edward Mills Purcell
1953    Frits Zernike                   Phase-contrast microscope
1954    Max Born                        Fundamental research in QM
        Walther Bothe                   Coincidence counters
1955    Willis Eugene Lamb              Hydrogen fine structure
        Polykarp Kusch                  Electron magnetic moment
1956    William Shockley                Transistors
        John Bardeen
        Walter Houser Brattain
1957    Chen Ning Yang                  Parity violation 
        Tsung Dao Lee
1958    Pavel Aleksejevic Cerenkov      Interpretation of the Cerenkov effect
        Il'ja Mickajlovic Frank
        Igor' Evgen'evic Tamm
1959    Emilio Gino Segre               The Antiproton
        Owen Chamberlain    
1960    Donald Arthur Glaser            The Bubble Chamber
1961    Robert Hofstadter               Electron scattering on nucleons
        Rudolf Ludwig Mossbauer         Resonant absorption of photons
1962    Lev Davidovic Landau            Theory of liquid helium
1963    Eugene P. Wigner                Fundamental symmetry principles
        Maria Goeppert Mayer            Nuclear shell structure
        J. Hans D. Jensen 
1964    Charles H. Townes               Maser-Laser principle
        Nikolai G. Basov
        Alexander M. Prochorov
1965    Sin-Itiro Tomonaga              Quantum electrodynamics
        Julian Schwinger
        Richard P. Feynman
1966    Alfred Kastler                  Study of Hertzian resonance in atoms
1967    Hans Albrecht Bethe             Energy production in stars 
1968    Luis W. Alvarez                 Discovery of many particle resonances
1969    Murray Gell-Mann                Quark model for particle classification
1970    Hannes Alfven                   Magneto-hydrodynamics in plasma physics
        Louis Neel                      Antiferromagnetism and ferromagnetism
1971    Dennis Gabor                    Principles of holography
1972    John Bardeen                    Theory of superconductivity
        Leon N. Cooper
        J. Robert Schrieffer
1973    Leo Esaki                       Tunneling in superconductors
        Ivar Giaever 
        Brian D. Josephson              Super-current through tunnel barriers
1974    Antony Hewish                   Discovery of pulsars
        Sir Martin Ryle                 Pioneering radioastronomy work
1975    Aage Bohr                       Structure of the atomic nucleus
        Ben Mottelson
        James Rainwater
1976    Burton Richter                  Discovery of the J/Psi particle
        Samual Chao Chung Ting
1977    Philip Warren Anderson          Electronic structure of magnetic and 
        Nevill Francis Mott             disordered solids
        John Hasbrouck Van Vleck
1978    Pyotr Kapitsa                   Liquefaction of helium
        Arno A. Penzias                 Cosmic Microwave Background Radiation
        Robert W. Wilson
1979    Sheldon Glashow                 Electroweak Theory, especially
        Steven Weinberg                  weak neutral currents
        Abdus Salam                     
1980    James Cronin                    Discovery of CP violation in the 
        Val Fitch                        asymmetric decay of neutral K-mesons
1981    Kai M. Seigbahn                 High resolution electron spectroscopy
        Nicolaas Bloembergen            Laser spectroscopy
        Arthur L. Schawlow
1982    Kenneth G. Wilson               Critical phenomena in phase transitions 
1983    Subrahmanyan Chandrasekhar      Evolution of stars
        William A. Fowler
1984    Carlo Rubbia                    Discovery of W,Z
        Simon van der Meer              Stochastic cooling for colliders
1985    Klaus von Klitzing              Discovery of quantum Hall effect
1986    Gerd Binning                    Scanning Tunneling Microscopy
        Heinrich Rohrer
        Ernst August Friedrich Ruska    Electron microscopy
1987    Georg Bednorz                   High-temperature superconductivity
        Alex K. Muller 
1988    Leon Max Lederman               Discovery of the muon neutrino leading
        Melvin Schwartz                  to classification of particles in 
        Jack Steinberger                 families
1989    Hans Georg Dehmelt              Penning Trap for charged particles
        Wolfgang Paul                   Paul Trap for charged particles
        Norman F. Ramsey                Control of atomic transitions by the
                                         separated oscillatory fields method
1990    Jerome Isaac Friedman           Deep inelastic scattering experiments
        Henry Way Kendall                leading to the discovery of quarks
        Richard Edward Taylor
1991    Pierre-Gilles de Gennes         Order-disorder transitions in liquid 
                                         crystals and polymers
1992    Georges Charpak                 Multiwire Proportional Chamber
1993    Russell A. Hulse                Discovery of the first binary pulsar
        Joseph H. Taylor                 and subsequent tests of GR
1994    Bertram N. Brockhouse           Neutron scattering experiments 
        Clifford G. Shull                
********************************************************************************
END OF PART 1/4

Article: 5896 of sci.physics.particle
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Subject: sci.physics Frequently Asked Questions (Part 2 of 4)
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Summary: This posting contains a list of Frequently Asked Questions 
	(and their answers) about physics, and should be read by anyone who 
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--------------------------------------------------------------------------------
               FREQUENTLY ASKED QUESTIONS ON SCI.PHYSICS - Part 2/4
--------------------------------------------------------------------------------
Item 6.

Gravitational Radiation                         updated 20-May-1992 by SIC
-----------------------                         original by Scott I. Chase

	Gravitational Radiation is to gravity what light is to
electromagnetism. It is produced when massive bodies accelerate.  You can
accelerate any body so as to produce such radiation, but due to the feeble
strength of gravity, it is entirely undetectable except when produced by
intense astrophysical sources such as supernovae, collisions of black
holes, etc.  These are quite far from us, typically, but they are so
intense that they dwarf all possible laboratory sources of such radiation. 

	Gravitational waves have a polarization pattern that causes objects
to expand in one direction, while contracting in the perpendicular
direction. That is, they have spin two.  This is because gravity waves are
fluctuations in the tensorial metric of space-time. 

	All oscillating radiation fields can be quantized, and in the case
of gravity,  the intermediate boson is called the "graviton" in analogy
with the photon. But quantum gravity is hard, for several reasons: 
        (1) The quantum field theory of gravity is hard, because gauge 
interactions of spin-two fields are not renormalizable.  See Cheng and Li,
Gauge Theory of Elementary Particle Physics (search for "power counting").
        (2) There are conceptual problems - what does it mean to quantize
geometry, or space-time?

	It is possible to quantize weak fluctuations in the gravitational
field.  This gives rise to the spin-2 graviton.  But full quantum gravity
has so far escaped formulation.  It is not likely to look much like the
other quantum field theories.  In addition, there are models of gravity
which include additional bosons with different spins.  Some are the
consequence of non-Einsteinian models, such as Brans-Dicke which has a
spin-0 component. Others are included by hand, to give "fifth force"
components to gravity. For example, if you want to add a weak repulsive
short range component, you will need a massive spin-1 boson.  (Even-spin
bosons always attract.  Odd-spin bosons can attract or repel.)  If
antigravity is real, then this has implications for the boson spectrum as
well. 

	The spin-two polarization provides the method of detection.  Most
experiments to date use a "Weber bar."  This is a cylindrical, very
massive, bar suspended  by fine wire, free to oscillate in response to a
passing graviton.   A high-sensitivity, low noise, capacitive transducer
can turn the oscillations of the bar into an electric signal for analysis. 
So far such searches have failed.  But they are expected to be
insufficiently sensitive for typical  radiation intensity from known types
of sources. 

	A more sensitive technique uses very long baseline laser
interferometry.  This is the principle of LIGO (Laser Interferometric
Gravity wave Observatory).  This is a two-armed detector, with
perpendicular laser beams each travelling several km before meeting to
produce an interference pattern which fluctuates if a gravity wave distorts
the geometry of the detector.  To eliminate noise from seismic effects as
well as human noise sources, two detectors separated by hundreds to
thousands of miles are necessary.  A coincidence measurement then provides
evidence of gravitational radiation.  In order to determine the source of
the signal, a third detector, far from either of the first two, would be
necessary.  Timing differences in the arrival of the signal to the three
detectors would allow triangulation of the angular position in the sky of
the signal. 

	The first stage of LIGO, a two detector setup in the U.S., has been
approved by Congress in 1992.  LIGO researchers have started designing a
prototype detector, and are hoping to enroll another nation, probably in
Europe, to fund and be host to the third detector. 

	The speed of gravitational radiation (C_gw) depends upon the
specific model of Gravitation that you use.  There are quite a few
competing models (all consistent with all experiments to date) including of
course Einstein's but also Brans-Dicke and several families of others.  
All metric models can support gravity waves.  But not all predict radiation
travelling at C_gw = C_em.  (C_em is the speed of electromagnetic waves.)

	There is a class of theories with "prior geometry", in which, as I
understand it, there is an additional metric which does not depend only on
the local matter density.  In such theories, C_gw != C_em in general. 

	However, there is good evidence that C_gw is in fact at least
almost C_em. We observe high energy cosmic rays in the 10^20-10^21 eV
region.  Such particles are travelling at up to (1-10^-18)*C_em.  If C_gw <
C_em, then particles with C_gw < v < C_em will radiate Cerenkov
gravitational radiation into the vacuum, and decelerate from the back
reaction.  So evidence of these very fast cosmic rays is good evidence that
C_gw >= (1-10^-18)*C_em, very close indeed to C_em.  Bottom line: in a
purely Einsteinian universe, C_gw = C_em. However, a class of models not
yet ruled out experimentally does make other predictions.  

	A definitive test would be produced by LIGO in coincidence with
optical measurements of some catastrophic event which generates enough
gravitational radiation to be detected.  Then the "time of flight" of both
gravitons and photons from the source to the Earth could be measured, and
strict direct limits could be set on C_gw. 

	For more information, see Gravitational Radiation (NATO ASI - 
Les Houches 1982), specifically the introductory essay by Kip Thorne.

********************************************************************************
Item 7.

IS ENERGY CONSERVED IN GENERAL RELATIVITY?            original by Michael Weiss
------------------------------------------                    and John Baez
	
	In special cases, yes.  In general--- it depends on what you mean
by "energy", and what you mean by "conserved". 

	In flat spacetime (the backdrop for special relativity) you can
phrase energy conservation in two ways: as a differential equation, or as
an equation involving integrals (gory details below).  The two formulations
are mathematically equivalent.  But when you try to generalize this to
curved spacetimes (the arena for general relativity) this equivalence
breaks down.  The differential form extends with nary a hiccup; not so the
integral form. 
	
	The differential form says, loosely speaking, that no energy is
created in any infinitesimal piece of spacetime.  The integral form says
the same for a finite-sized piece.  (This may remind you of the
"divergence" and "flux" forms of Gauss's law in electrostatics, or the
equation of continuity in fluid dynamics.  Hold on to that thought!) 

	An infinitesimal piece of spacetime "looks flat", while the effects
of curvature become evident in a finite piece.  (The same holds for curved
surfaces in space, of course).  GR relates curvature to gravity.  Now, even
in Newtonian physics, you must include gravitational potential energy to
get energy conservation.  And GR introduces the new phenomenon of
gravitational waves; perhaps these carry energy as well?  Perhaps we need
to include gravitational energy in some fashion, to arrive at a law of
energy conservation for finite pieces of spacetime? 

	Casting about for a mathematical expression of these ideas,
physicists came up with something called an energy pseudo-tensor. (In fact,
several of 'em!)  Now, GR takes pride in treating all coordinate systems
equally.  Mathematicians invented tensors precisely to meet this sort of
demand--- if a tensor equation holds in one coordinate system, it holds in
all.  Pseudo-tensors are not tensors (surprise!), and this alone raises
eyebrows in some circles. In GR, one must always guard against mistaking
artifacts of a particular coordinate system for real physical effects. 
(See the FAQ entry on black holes for some examples.) 

	These pseudo-tensors have some rather strange properties.  If you
choose the "wrong" coordinates, they are non-zero even in flat empty
spacetime.  By another choice of coordinates, they can be made zero at any
chosen point, even in a spacetime full of gravitational radiation.  For
these reasons, most physicists who work in general relativity do not
believe the pseudo-tensors give a good *local* definition of energy
density, although their integrals are sometimes useful as a measure of
total energy. 

	One other complaint about the pseudo-tensors deserves mention. 
Einstein argued that all energy has mass, and all mass acts
gravitationally.  Does "gravitational energy" itself act as a source of
gravity?  Now, the Einstein field equations are 

            G_{mu,nu} = 8pi T_{mu,nu}

	Here G_{mu,nu} is the Einstein curvature tensor, which encodes
information about the curvature of spacetime, and T_{mu,nu} is the
so-called stress-energy tensor, which we will meet again below.  T_{mu,nu}
represents the energy due to matter and electromagnetic fields, but
includes NO contribution from "gravitational energy".  So one can argue
that "gravitational energy" does NOT act as a source of gravity.  On the
other hand, the Einstein field equations are non-linear; this implies that
gravitational waves interact with each other (unlike light waves in
Maxwell's (linear) theory).  So one can argue that "gravitational energy"
IS a source of gravity. 

	In certain special cases, energy conservation works out with fewer
caveats. The two main examples are static spacetimes and asymptotically
flat spacetimes. 

	Let's look at four examples before plunging deeper into the math.
Three examples involve redshift, the other, gravitational radiation. 

(1) Very fast objects emitting light.

	According to *special* relativity, you will see light coming from a
receding object as redshifted.  So if you, and someone moving with the
source, both measure the light's energy, you'll get different answers. 
Note that this has nothing to do with energy conservation per se.  Even in
Newtonian physics, kinetic energy (mv^2/2) depends on the choice of
reference frame.  However, relativity serves up a new twist.  In Newtonian
physics, energy conservation and momentum conservation are two separate
laws.  Special relativity welds them into one law, the conservation of the
*energy-momentum 4-vector*.  To learn the whole scoop on 4-vectors, read a
text on SR, for example Taylor and Wheeler (see refs.)  For our purposes,
it's enough to remark that 4-vectors are vectors in spacetime, which most
people privately picture just like ordinary vectors (unless they have
*very* active imaginations). 

(2) Very massive objects emitting light.

	Light from the Sun appears redshifted to an Earthbound astronomer. 
In quasi-Newtonian terms, we might say that light loses kinetic energy as
it climbs out of the gravitational well of the Sun, but gains potential
energy.  General relativity looks at it differently.  In GR, gravity is
described not by a "potential" but by the "metric" of spacetime.  But "no
problem", as the saying goes.  The Schwarzschild metric describes spacetime
around a massive object, if the object is spherically symmetrical,
uncharged, and "alone in the universe".  The Schwarzschild metric is both
static and asymptotically flat, and energy conservation holds without major
pitfalls.  For further details, consult MTW, chapter 25. 

(3) Gravitational waves.

	A binary pulsar emits gravitational waves, according to GR, and one
expects (innocent word!) that these waves will carry away energy.  So its
orbital period should change.  Einstein derived a formula for the rate of
change (known as the quadrapole formula), and in the centenary of
Einstein's birth, Russell Hulse and Joseph Taylor reported that the binary
pulsar PSR1913+16 bore out Einstein's predictions within a few percent. 
Hulse and Taylor were awarded the Nobel prize in 1993. 

	Despite this success, Einstein's formula remained controversial for
many years, partly because of the subtleties surrounding energy
conservation in GR.  The need to understand this situation better has kept
GR theoreticians busy over the last few years.  Einstein's formula now
seems well-established, both theoretically and observationally. 

(4) Expansion of the universe leading to cosmological redshift.

	The Cosmic Background Radiation (CBR) has red-shifted over billions
of years.  Each photon gets redder and redder.  What happens to this
energy? Cosmologists model the expanding universe with
Friedmann-Robertson-Walker (FRW) spacetimes.  (The familiar "expanding
balloon speckled with galaxies" belongs to this class of models.)  The FRW
spacetimes are neither static nor asymptotically flat.  Those who harbor no
qualms about pseudo-tensors will say that radiant energy becomes
gravitational energy.  Others will say that the energy is simply lost. 

	It's time to look at mathematical fine points.  There are many to
choose from!  The definition of asymptotically flat, for example, calls for
some care (see Stewart); one worries about "boundary conditions at
infinity".  (In fact, both spatial infinity and "null infinity" clamor for
attention--- leading to different kinds of total energy.)  The static case
has close connections with Noether's theorem (see Goldstein or Arnold).  If
the catch-phrase "time translation symmetry implies conservation of energy"
rings a bell (perhaps from quantum mechanics), then you're on the right
track. (Check out "Killing vector" in the index of MTW, Wald, or Sachs and
Wu.) 

	But two issues call for more discussion.  Why does the equivalence
between the two forms of energy conservation break down?  How do the
pseudo-tensors slide around this difficulty? 

	We've seen already that we should be talking about the
energy-momentum 4-vector, not just its time-like component (the energy). 
Let's consider first the case of flat Minkowski spacetime.  Recall that the
notion of "inertial frame" corresponds to a special kind of coordinate
system (Minkowskian coordinates). 

	Pick an inertial reference frame.  Pick a volume V in this frame,
and pick two times t=t_0 and t=t_1.  One formulation of energy-momentum
conservation says that the energy-momentum inside V changes only because of
energy-momentum flowing across the boundary surface (call it S).  It is
"conceptually difficult, mathematically easy" to define a quantity T so
that the captions on the Equation 1 (below) are correct.  (The quoted
phrase comes from Sachs and Wu.) 

  Equation 1:  (valid in flat Minkowski spacetime, when Minkowskian
                coordinates are used) 

                                               t=t_1
       /                  /                    /
       |                  |                    |
       | T dV     -       | T dV       =       | T dt dS
       /                  /                    /
      V,t=t_0           V,t=t_1               t=t_0

   p contained       p contained            p flowing out through
   in volume V    -  in volume V       =    boundary S of V
   at time t_0       at time t_1            during t=t_0 to t=t_1

   (Note: p = energy-momentum 4-vector)

T is called the stress-energy tensor.  You don't need to know what
that means! ---just that you can integrate T, as shown, to get
4-vectors.  Equation 1 may remind you of Gauss's theorem, which deals
with flux across a boundary.  If you look at Equation 1 in the right
4-dimensional frame of mind, you'll discover it really says that the
flux across the boundary of a certain 4-dimensional hypervolume is
zero.  (The hypervolume is swept out by V during the interval t=t_0
to t=t_1.)  MTW, chapter 7, explains this with pictures galore.  (See
also Wheeler.)

	A 4-dimensional analogue to Gauss's theorem shows that Equation 1
is equivalent to: 

  Equation 2:  (valid in flat Minkowski spacetime, with Minkowskian
                coordinates)

       coord_div(T) = sum_mu (partial T/partial x_mu) = 0

We write "coord_div" for the divergence, for we will meet another
divergence in a moment.  Proof?  Quite similar to Gauss's theorem: if
the divergence is zero throughout the hypervolume, then the flux
across the boundary must also be zero.  On the other hand, the flux
out of an infinitesimally small hypervolume turns out to be the
divergence times the measure of the hypervolume.

	Pass now to the general case of any spacetime satisfying Einstein's
field equation.  It is easy to generalize the differential form of
energy-momentum conservation, Equation 2: 

  Equation 3:  (valid in any GR spacetime)

        covariant_div(T) = sum_mu nabla_mu(T) = 0    

                    (where nabla_mu = covariant derivative)

(Side comment: Equation 3 is the correct generalization of Equation 1 for
SR when non-Minkowskian coordinates are used.)

	GR relies heavily on the covariant derivative, because the
covariant derivative of a tensor is a tensor, and as we've seen, GR loves
tensors.  Equation 3 follows from Einstein's field equation (because
something called Bianchi's identity says that covariant_div(G)=0). But
Equation 3 is no longer equivalent to Equation 1! 

	Why not?  Well, the familiar form of Gauss's theorem (from
electrostatics) holds for any spacetime, because essentially you are
summing fluxes over a partition of the volume into infinitesimally small
pieces.  The sum over the faces of one infinitesimal piece is a divergence.
 But the total contribution from an interior face is zero, since what flows
out of one piece flows into its neighbor.  So the integral of the
divergence over the volume equals the flux through the boundary.  "QED". 

	But for the equivalence of Equations 1 and 3, we would need an
extension of Gauss's theorem.  Now the flux through a face is not a scalar,
but a vector (the flux of energy-momentum through the face). The argument
just sketched involves adding these vectors, which are defined at different
points in spacetime.  Such "remote vector comparison" runs into trouble
precisely for curved spacetimes. 

	The mathematician Levi-Civita invented the standard solution to
this problem, and dubbed it "parallel transport".  It's easy to picture
parallel transport: just move the vector along a path, keeping its
direction "as constant as possible".  (Naturally, some non-trivial
mathematics lurks behind the phrase in quotation marks.  But even
pop-science expositions of GR do a good job explaining parallel transport.)
 The parallel transport of a vector depends on the transportation path; for
the canonical example, imagine parallel transporting a vector on a sphere. 
But parallel transportation over an "infinitesimal distance" suffers no
such ambiguity. (It's not hard to see the connection with curvature.) 

	To compute a divergence, we need to compare quantities (here
vectors) on opposite faces.  Using parallel transport for this leads to the
covariant divergence.  This is well-defined, because we're dealing with an
infinitesimal hypervolume.  But to add up fluxes all over a finite-sized
hypervolume (as in the contemplated extension of Gauss's theorem) runs
smack into the dependence on transportation path. So the flux integral is
not well-defined, and we have no analogue for Gauss's theorem. 

	One way to get round this is to pick one coordinate system, and
transport vectors so their *components* stay constant.  Partial derivatives
replace covariant derivatives, and Gauss's theorem is restored.  The energy
pseudo-tensors take this approach (at least some of them do).  If you can
mangle Equation 3 (covariant_div(T) = 0) into the form: 

       coord_div(Theta) = 0

then you can get an "energy conservation law" in integral form.
Einstein was the first to do this; Dirac, Landau and Lifshitz, and
Weinberg all came up with variations on this theme.  We've said
enough already on the pros and cons of this approach.

	We will not delve into definitions of energy in general relativity
such as the Hamiltonian (amusingly, the energy of a closed universe always
works out to zero according to this definition), various kinds of energy
one hopes to obtain by "deparametrizing" Einstein's equations, or
"quasilocal energy".  There's quite a bit to say about this sort of thing! 
Indeed, the issue of energy in general relativity has a lot to do with the
notorious "problem of time" in quantum gravity.... but that's another can
of worms. 

References (vaguely in order of difficulty):

  Clifford Will, "The renaissance of general relativity", in "The New
    Physics" (ed. Paul Davies) gives a semi-technical discussion of the
    controversy over gravitational radiation.
  Wheeler, "A Journey into Gravity and Spacetime".  Wheeler's try at
    a "pop-science" treatment of GR.  Chapters 6 and 7 are a
    tour-de-force: Wheeler tries for a non-technical explanation of
    Cartan's formulation of Einstein's field equation.  It might be
    easier just to read MTW!)
  Taylor and Wheeler, "Spacetime Physics".
  Goldstein, "Classical Mechanics".
  Arnold, "Mathematical Methods in Classical Mechanics".
  Misner, Thorne, and Wheeler (MTW), "Gravitation", chapters 7, 20,
    and 25 
  Wald, "General Relativity", Appendix E.  This has the Hamiltonian
     formalism and a bit about deparametrizing, and chapter 11
     discusses energy in asymptotically flat spacetimes.
  H. A. Buchdahl, "Seventeen Simple Lectures on General Relativity Theory"
    Lecture 15 derives the energy-loss formula for the binary star, and
    criticizes the derivation.
  Sachs and Wu, "General Relativity for Mathematicians", chapter 3
  John Stewart, "Advanced General Relativity".  Chapter 3 ("Asymptopia")
    shows just how careful one has to be in asymptotically flat spacetimes
    to recover energy conservation.  Stewart also discusses the Bondi-Sachs
    mass, another contender for "energy".
  Damour, in "300 Years of Gravitation" (ed. Hawking and Israel). Damour
    heads the "Paris group", which has been active in the theory of
    gravitational radiation.
  Penrose and Rindler, "Spinors and Spacetime", vol II, chapter 9.  The
    Bondi-Sachs mass generalized.
  J. David Brown and James York Jr., "Quasilocal energy in general
    relativity", in "Mathematical Aspects of Classical Field Theory".

********************************************************************************
Item 8.

Olbers' Paradox                                updated: 24-JAN-1993 by SIC
---------------                                original by Scott I. Chase

	Why isn't the night sky as uniformly bright as the surface of the
Sun? If the Universe has infinitely many stars, then it should be.  After
all, if you move the Sun twice as far away from us, we will intercept
one-fourth as many  photons, but the Sun will subtend one-fourth of the
angular area.  So the areal intensity remains constant.  With infinitely
many stars, every angular element of the sky should have a star, and the
entire heavens should be as bright as the sun.  We should have the
impression that we live in the center of a hollow black body whose
temperature is about 6000 degrees Centigrade.   This is Olbers' paradox.  
It can be traced as far back as Kepler in 1610.  It was rediscussed by 
Halley and Cheseaux in the eighteen century, but was not popularized as 
a paradox until Olbers took up the issue in the nineteenth century.

	There are many possible explanations which have been considered. 
Here are a few: 
        (1) There's too much dust to see the distant stars.
        (2) The Universe has only a finite number of stars.
        (3) The distribution of stars is not uniform.  So, for example,
            there could be an infinity of stars, but they hide behind one
            another so that only a finite angular area is subtended by them. 
        (4) The Universe is expanding, so distant stars are red-shifted into
            obscurity.
        (5) The Universe is young.  Distant light hasn't even reached us yet.

	The first explanation is just plain wrong.  In a black body, the
dust will  heat up too.  It does act like a radiation shield, exponentially
damping the  distant starlight.  But you can't put enough dust into the
universe to get rid of enough starlight without also obscuring our own Sun.
So this idea is bad. 

	The premise of the second explanation may technically be correct.
But the number of stars, finite as it might be, is still large enough to 
light up the entire sky, i.e., the total amount of luminous matter  in the 
Universe is too large to allow this escape.  The number of stars is close 
enough to infinite for the purpose of lighting up the sky.  The third 
explanation might be partially correct.  We just don't know.  If the stars 
are distributed fractally, then there could be large patches of empty space, 
and the sky could appear dark except in small areas. 

	But the final two possibilities are are surely each correct and
partly responsible.  There are numerical arguments that suggest that the
effect of the finite age of the Universe is the larger effect.  We live
inside a spherical shell of "Observable Universe" which has radius equal to
the lifetime of the Universe.  Objects more than about 15 billion years
old are too far away for their light ever to reach us. 

	Historically, after Hubble discovered that the Universe was
expanding, but before the Big Bang was firmly established by the discovery
of the cosmic background radiation, Olbers' paradox was presented as proof
of special relativity.  You needed the red-shift (an SR effect) to get rid
of the starlight.  This effect certainly contributes.  But the finite age
of the Universe is the most important effect. 

References:  Ap. J. _367_, 399 (1991). The author, Paul Wesson, is said to
be on a personal crusade to end the confusion surrounding Olbers' paradox. 

_Darkness at Night: A Riddle of the Universe_, Edward Harrison, Harvard
University Press, 1987

********************************************************************************
Item 9.

What is Dark Matter?                            updated 11-MAY-1993 by SIC
--------------------                            original by Scott I. Chase

	The story of dark matter is best divided into two parts.  First we
have the reasons that we know that it exists.  Second is the collection of
possible explanations as to what it is. 

Why the Universe Needs Dark Matter
----------------------------------

	We believe that that the Universe is critically balanced between
being open and closed.  We derive this fact from the observation of the
large scale structure of the Universe.  It requires a certain amount of
matter to accomplish this result.  Call it M. 

	We can estimate the total BARYONIC matter of the universe by
studying Big Bang nucleosynthesis.  This is done by connecting the observed
He/H ratio of the Universe today to the amount of baryonic matter present
during the early hot phase when most of the helium was produced.  Once the 
temperature of the Universe dropped below the neutron-proton mass difference, 
neutrons began decaying into protons.  If the early baryon density was low, 
then it was hard for a proton to find a neutron with which to make helium 
before too many of the neutrons decayed away to account for the amount of 
helium we see today.  So by measuring the He/H ratio today, we can estimate 
the necessary baryon density shortly after the Big Bang, and, consequently, 
the total number of baryons today.  It turns out that you need about 0.05 M 
total baryonic matter to account for the known ratio of light isotopes.  So 
only 1/20 of the total mass of the Universe is baryonic matter.

	Unfortunately, the best estimates of the total mass of everything
that we can see with our telescopes is roughly 0.01 M.  Where is the other
99% of the stuff of the Universe?  Dark Matter!

	So there are two conclusions.  We only see 0.01 M out of 0.05 M 
baryonic matter in the Universe.  The rest must be in baryonic dark matter
halos surrounding galaxies.  And there must be some non-baryonic dark matter 
to account for the remaining 95% of the matter required to give omega, the 
mass of the Universe, in units of critical mass, equal to unity. 

	For those who distrust the conventional Big Bang models, and don't
want to rely upon fancy cosmology to derive the presence of dark matter,
there are other more direct means.   It has been observed in clusters of
galaxies that the motion of galaxies within a cluster suggests that they
are bound by a total gravitational force due to about 5-10 times as much
matter as can be accounted for from luminous matter in said galaxies.  And 
within an individual galaxy, you can measure the rate of rotation of the
stars about the galactic center of rotation.  The resultant "rotation
curve" is simply related to the distribution of matter in the galaxy.  The
outer stars in galaxies seem to rotate too fast for the amount of matter
that we see in the galaxy.  Again, we need about 5 times more matter than
we can see via electromagnetic radiation.  These results can be explained
by assuming that there is a "dark matter halo" surrounding every galaxy. 

What is Dark Matter
-------------------

	This is the open question.  There are many possibilities, and
nobody really knows much about this yet.  Here are a few of the many
published suggestions, which are being currently hunted for by
experimentalists all over the world.  Remember, you need at least one
baryonic candidate and one non-baryonic candidate to make everything
work out, so there there may be more than one correct choice among 
the possibilities given here. 

(1) Normal matter which has so far eluded our gaze, such as 
        (a) dark galaxies
        (b) brown dwarfs
        (c) planetary material (rock, dust, etc.)

(2) Massive Standard Model neutrinos.  If any of the neutrinos are massive,
then this could be the missing mass.  On the other hand, if they are 
too heavy, as the purported 17 KeV neutrino would have been, massive
neutrinos create almost as many problems as they solve in this regard. 

(3) Exotica (See the "Particle Zoo" FAQ entry for some details)

	Massive exotica would provide the missing mass.  For our purposes, 
these fall into two classes: those which have been proposed for other
reasons but happen to solve the dark matter problem, and those which have
been proposed specifically to provide the missing dark matter. 

	Examples of objects in the first class are axions, additional
neutrinos, supersymmetric particles, and a host of others. Their properties
are constrained by the theory which predicts them, but by virtue of their
mass, they solve the dark matter problem if they exist in the correct
abundance. 

	Particles in the second class are generally classed in loose groups. 
Their properties are not specified, but they are merely required to be
massive and have other properties such that they would so far have eluded
discovery in the many experiments which have looked for new particles. 
These include WIMPS (Weakly Interacting Massive Particles), CHAMPS, and a
host of others. 

References:  _Dark Matter in the Universe_ (Jerusalem Winter School for
Theoretical Physics, 1986-7), J.N. Bahcall, T. Piran, & S. Weinberg editors.
_Dark Matter_ (Proceedings of the XXIIIrd Recontre de Moriond) J. Audouze and 
J. Tran Thanh Van. editors.

********************************************************************************
Item 10.

Some Frequently Asked Questions About Black Holes   updated 02-FEB-1995 by MM 
-------------------------------------------------   original by Matt McIrvin

Contents:

1. What is a black hole, really?
2. What happens to you if you fall in?
3. Won't it take forever for you to fall in?  Won't it take forever
   for the black hole to even form? 
4. Will you see the universe end?
5. What about Hawking radiation?  Won't the black hole evaporate 
   before you get there?
6. How does the gravity get out of the black hole?
7. Where did you get that information?

1. What is a black hole, really?

	In 1916, when general relativity was new, Karl Schwarzschild worked
out a useful solution to the Einstein equation describing the evolution of
spacetime geometry.  This solution, a possible shape of spacetime, would
describe the effects of gravity *outside* a spherically symmetric,
uncharged, nonrotating object (and would serve approximately to describe
even slowly rotating objects like the Earth or Sun).  It worked in much the
same way that you can treat the Earth as a point mass for purposes of
Newtonian gravity if all you want to do is describe gravity *outside* the
Earth's surface. 

	What such a solution really looks like is a "metric," which is a
kind of generalization of the Pythagorean formula that gives the length of
a line segment in the plane.  The metric is a formula that may be used to
obtain the "length" of a curve in spacetime.  In the case of a curve
corresponding to the motion of an object as time passes (a "timelike
worldline,") the "length" computed by the metric is actually the elapsed
time experienced by an object with that motion.  The actual formula depends
on the coordinates chosen in which to express things, but it may be
transformed into various coordinate systems without affecting anything
physical, like the spacetime curvature.  Schwarzschild expressed his metric
in terms of coordinates which, at large distances from the object,
resembled spherical coordinates with an extra coordinate t for time. 
Another coordinate, called r, functioned as a radial coordinate at large
distances; out there it just gave the distance to the massive object. 

	Now, at small radii, the solution began to act strangely.  There
was a "singularity" at the center, r=0, where the curvature of spacetime
was infinite.  Surrounding that was a region where the "radial" direction
of decreasing r was actually a direction in *time* rather than in space.
Anything in that region, including light, would be obligated to fall toward
the singularity, to be crushed as tidal forces diverged. This was separated
>from the rest of the universe by a place where Schwarzschild's coordinates
blew up, though nothing was wrong with the curvature of spacetime there. 
(This was called the Schwarzschild radius.  Later, other coordinate systems
were discovered in which the blow-up didn't happen; it was an artifact of
the coordinates, a little like the problem of defining the longitude of the
North Pole.  The physically important thing about the Schwarzschild radius
was not the coordinate problem, but the fact that within it the direction
into the hole became a direction in time.) 

	Nobody really worried about this at the time, because there was no
known object that was dense enough for that inner region to actually be
outside it, so for all known cases, this odd part of the solution would not
apply.  Arthur Stanley Eddington considered the possibility of a dying star
collapsing to such a density,  but rejected it as aesthetically unpleasant
and proposed that some new physics must intervene.  In 1939, Oppenheimer
and Snyder finally took seriously the possibility that stars a few times
more massive than the sun might be doomed to collapse to such a state at
the end of their lives. 

	Once the star gets smaller than the place where Schwarzschild's
coordinates fail (called the Schwarzschild radius for an uncharged,
nonrotating object, or the event horizon) there's no way it can avoid
collapsing further.  It has to collapse all the way to a singularity for
the same reason that you can't keep from moving into the future! Nothing
else that goes into that region afterward can avoid it either, at least in
this simple case.  The event horizon is a point of no return. 

	In 1971 John Archibald Wheeler named such a thing a black hole,
since light could not escape from it.  Astronomers have many candidate
objects they think are probably black holes, on the basis of several kinds
of evidence (typically they are dark objects whose large mass can be
deduced from their gravitational effects on other objects, and which
sometimes emit X-rays, presumably from infalling matter).  But the
properties of black holes I'll talk about here are entirely theoretical.
They're based on general relativity, which is a theory that seems supported
by available evidence. 

2. What happens to you if you fall in?

	Suppose that, possessing a proper spacecraft and a self-destructive
urge, I decide to go black-hole jumping and head for an uncharged,
nonrotating ("Schwarzschild") black hole.  In this and other kinds of hole,
I won't, before I fall in, be able to see anything within the event
horizon.  But there's nothing *locally* special about the event horizon;
when I get there it won't seem like a particularly unusual place, except
that I will see strange optical distortions of the sky around me from all
the bending of light that goes on.  But as soon as I fall through, I'm
doomed.  No bungee will help me, since bungees can't keep Sunday from
turning into Monday.  I have to hit the singularity eventually, and before
I get there there will be enormous tidal forces-- forces due to the
curvature of spacetime-- which will squash me and my spaceship in some
directions and stretch them in another until I look like a piece of
spaghetti.  At the singularity all of present physics is mute as to what
will happen, but I won't care.  I'll be dead. 

	For ordinary black holes of a few solar masses, there are actually
large tidal forces well outside the event horizon, so I probably wouldn't
even make it into the hole alive and unstretched.  For a black hole of 8
solar masses, for instance, the value of r at which tides become fatal is
about 400 km, and the Schwarzschild radius is just 24 km.  But tidal
stresses are proportional to M/r^3.  Therefore the fatal r goes as the cube
root of the mass, whereas the Schwarzschild radius of the black hole is
proportional to the mass.  So for black holes larger than about 1000 solar
masses I could probably fall in alive, and for still larger ones I might
not even notice the tidal forces until I'm through the horizon and doomed. 

3. Won't it take forever for you to fall in?  Won't it take forever
   for the black hole to even form?

	Not in any useful sense.  The time I experience before I hit the
event horizon, and even until I hit the singularity-- the "proper time"
calculated by using Schwarzschild's metric on my worldline -- is finite. 
The same goes for the collapsing star; if I somehow stood on the surface of
the star as it became a black hole, I would experience the star's demise in
a finite time. 

	On my worldline as I fall into the black hole, it turns out that
the Schwarzschild coordinate called t goes to infinity when I go through
the event horizon.  That doesn't correspond to anyone's proper time,
though; it's just a coordinate called t.  In fact, inside the event
horizon, t is actually a *spatial* direction, and the future corresponds
instead to decreasing r.  It's only outside the black hole that t even
points in a direction of increasing time.  In any case, this doesn't
indicate that I take forever to fall in, since the proper time involved is
actually finite. 

	At large distances t *does* approach the proper time of someone who
is at rest with respect to the black hole.  But there isn't any
non-arbitrary sense in which you can call t at smaller r values "the proper
time of a distant observer," since in general relativity there is no
coordinate-independent way to say that two distant events are happening "at
the same time."  The proper time of any observer is only defined locally. 

	A more physical sense in which it might be said that things take
forever to fall in is provided by looking at the paths of emerging light
rays.  The event horizon is what, in relativity parlance, is called a
"lightlike surface"; light rays can remain there.  For an ideal
Schwarzschild hole (which I am considering in this paragraph) the horizon
lasts forever, so the light can stay there without escaping.  (If you
wonder how this is reconciled with the fact that light has to travel at the
constant speed c-- well, the horizon *is* traveling at c! Relative speeds
in GR are also only unambiguously defined locally, and if you're at the
event horizon you are necessarily falling in; it comes at you at the speed
of light.)  Light beams aimed directly outward from just outside the
horizon don't escape to large distances until late values of t.  For
someone at a large distance from the black hole and approximately at rest
with respect to it, the coordinate t does correspond well to proper time. 

	So if you, watching from a safe distance, attempt to witness my
fall into the hole, you'll see me fall more and more slowly as the light
delay increases.  You'll never see me actually *get to* the event horizon.
My watch, to you, will tick more and more slowly, but will never reach the
time that I see as I fall into the black hole.  Notice that this is really
an optical effect caused by the paths of the light rays. 

	This is also true for the dying star itself.  If you attempt to
witness the black hole's formation, you'll see the star collapse more and
more slowly, never precisely reaching the Schwarzschild radius. 

	Now, this led early on to an image of a black hole as a strange
sort of suspended-animation object, a "frozen star" with immobilized
falling debris and gedankenexperiment astronauts hanging above it in
eternally slowing precipitation.  This is, however, not what you'd see. The
reason is that as things get closer to the event horizon, they also get
*dimmer*.  Light from them is redshifted and dimmed, and if one considers
that light is actually made up of discrete photons, the time of escape of
*the last photon* is actually finite, and not very large.  So things would
wink out as they got close, including the dying star, and the name "black
hole" is justified. 

	As an example, take the eight-solar-mass black hole I mentioned
before.  If you start timing from the moment the you see the object half a
Schwarzschild radius away from the event horizon, the light will dim
exponentially from that point on with a characteristic time of about 0.2
milliseconds, and the time of the last photon is about a hundredth of a
second later.  The times scale proportionally to the mass of the black
hole.  If I jump into a black hole, I don't remain visible for long. 

	Also, if I jump in, I won't hit the surface of the "frozen star."
It goes through the event horizon at another point in spacetime from
where/when I do. 

	(Some have pointed out that I really go through the event horizon a
little earlier than a naive calculation would imply.  The reason is that my
addition to the black hole increases its mass, and therefore moves the
event horizon out around me at finite Schwarzschild t coordinate.  This
really doesn't change the situation with regard to whether an external
observer sees me go through, since the event horizon is still lightlike;
light emitted at the event horizon or within it will never escape to large
distances, and light emitted just outside it will take a long time to get
to an observer, timed, say, from when the observer saw me pass the point
half a Schwarzschild radius outside the hole.) 

	All this is not to imply that the black hole can't also be used for
temporal tricks much like the "twin paradox" mentioned elsewhere in this
FAQ.  Suppose that I don't fall into the black hole-- instead, I stop and
wait at a constant r value just outside the event horizon, burning
tremendous amounts of rocket fuel and somehow withstanding the huge
gravitational force that would result.  If I then return home, I'll have
aged less than you.  In this case, general relativity can say something
about the difference in proper time experienced by the two of us, because
our ages can be compared *locally* at the start and end of the journey. 

4. Will you see the universe end?

	If an external observer sees me slow down asymptotically as I fall,
it might seem reasonable that I'd see the universe speed up
asymptotically-- that I'd see the universe end in a spectacular flash as I
went through the horizon.  This isn't the case, though.  What an external
observer sees depends on what light does after I emit it.  What I see,
however, depends on what light does before it gets to me.  And there's no
way that light from future events far away can get to me.  Faraway events
in the arbitrarily distant future never end up on my "past light-cone," the
surface made of light rays that get to me at a given time. 

	That, at least, is the story for an uncharged, nonrotating black
hole.  For charged or rotating holes, the story is different.  Such holes
can contain, in the idealized solutions, "timelike wormholes" which serve
as gateways to otherwise disconnected regions-- effectively, different
universes.  Instead of hitting the singularity, I can go through the
wormhole.  But at the entrance to the wormhole, which acts as a kind of
inner event horizon, an infinite speed-up effect actually does occur.  If I
fall into the wormhole I see the entire history of the universe outside
play itself out to the end.  Even worse, as the picture speeds up the light
gets blueshifted and more energetic, so that as I pass into the wormhole an
"infinite blueshift" happens which fries me with hard radiation.  There is
apparently good reason to believe that the infinite blueshift would imperil
the wormhole itself, replacing it with a singularity no less pernicious
than the one I've managed to miss.  In any case it would render wormhole
travel an undertaking of questionable practicality. 

5. What about Hawking radiation?  Won't the black hole evaporate
   before you get there?

	(First, a caveat: Not a lot is really understood about evaporating
black holes.  The following is largely deduced from information in Wald's
GR text, but what really happens-- especially when the black hole gets very
small-- is unclear.  So take the following with a grain of salt.) 

Short answer:  No, it won't.  This demands some elaboration.

	From thermodynamic arguments Stephen Hawking realized that a black
hole should have a nonzero temperature, and ought therefore to emit
blackbody radiation.  He eventually figured out a quantum- mechanical
mechanism for this.  Suffice it to say that black holes should very, very
slowly lose mass through radiation, a loss which accelerates as the hole
gets smaller and eventually evaporates completely in a burst of radiation. 
This happens in a finite time according to an outside observer. 

	But I just said that an outside observer would *never* observe an
object actually entering the horizon!  If I jump in, will you see the black
hole evaporate out from under me, leaving me intact but marooned in the
very distant future from gravitational time dilation? 

	You won't, and the reason is that the discussion above only applies
to a black hole that is not shrinking to nil from evaporation. Remember
that the apparent slowing of my fall is due to the paths of outgoing light
rays near the event horizon.  If the black hole *does* evaporate, the delay
in escaping light caused by proximity to the event horizon can only last as
long as the event horizon does!  Consider your external view of me as I
fall in. 

	If the black hole is eternal, events happening to me (by my watch)
closer and closer to the time I fall through happen divergingly later
according to you (supposing that your vision is somehow not limited by the
discreteness of photons, or the redshift). 

	If the black hole is mortal, you'll instead see those events happen
closer and closer to the time the black hole evaporates.  Extrapolating,
you would calculate my time of passage through the event horizon as the
exact moment the hole disappears!  (Of course, even if you could see me,
the image would be drowned out by all the radiation from the evaporating
hole.)  I won't experience that cataclysm myself, though; I'll be through
the horizon, leaving only my light behind. As far as I'm concerned, my
grisly fate is unaffected by the evaporation. 

	All of this assumes you can see me at all, of course.  In practice
the time of the last photon would have long been past.  Besides, there's
the brilliant background of Hawking radiation to see through as the hole
shrinks to nothing. 

	(Due to considerations I won't go into here, some physicists think
that the black hole won't disappear completely, that a remnant hole will be
left behind.  Current physics can't really decide the question, any more
than it can decide what really happens at the singularity. If someone ever
figures out quantum gravity, maybe that will provide an answer.) 

6. How does the gravity get out of the black hole?

Purely in terms of general relativity, there is no problem here.  The
gravity doesn't have to get out of the black hole.  General relativity
is a local theory, which means that the field at a certain point in
spacetime is determined entirely by things going on at places that can
communicate with it at speeds less than or equal to c.  If a star 
collapses into a black hole, the gravitational field outside the 
black hole may be calculated entirely from the properties of the star
and its external gravitational field *before* it becomes a black hole.
Just as the light registering late stages in my fall takes longer and
longer to get out to you at a large distance, the gravitational
consequences of events late in the star's collapse take longer and
longer to ripple out to the world at large.  In this sense the black
hole *is* a kind of "frozen star": the gravitational field is a fossil
field.  The same is true of the electromagnetic field that a black
hole may possess.

Often this question is phrased in terms of gravitons, the hypothetical
quanta of spacetime distortion.  If things like gravity correspond to the
exchange of "particles" like gravitons, how can they get out of the
event horizon to do their job?

Gravitons don't exist in general relativity, because GR is not a
quantum theory.  They might be part of a theory of quantum gravity
when it is completely developed, but even then it might not be best to
describe gravitational attraction as produced by virtual gravitons.
See the FAQ on virtual particles for a discussion of this.

Nevertheless, the question in this form is still worth asking, because
black holes *can* have static electric fields, and we know that these
may be described in terms of virtual photons.  So how do the virtual
photons get out of the event horizon?  Well, for one thing, they can
come from the charged matter prior to collapse, just like classical
effects.  In addition, however, virtual particles aren't confined to
the interiors of light cones: they can go faster than light!
Consequently the event horizon, which is really just a surface that
moves at the speed of light, presents no barrier.

I couldn't use these virtual photons after falling into the hole to
communicate with you outside the hole; nor could I escape from the
hole by somehow turning myself into virtual particles.  The reason is
that virtual particles don't carry any *information* outside the light
cone.  See the FAQ on virtual particles for details.

7. Where did you get that information?

	The numbers concerning fatal radii, dimming, and the time of the
last photon came from Misner, Thorne, and Wheeler's _Gravitation_ (San
Francisco: W. H. Freeman & Co., 1973), pp. 860-862 and 872-873. Chapters 32
and 33 (IMHO, the best part of the book) contain nice descriptions of some
of the phenomena I've described. 

	Information about evaporation and wormholes came from Robert Wald's
_General Relativity_ (Chicago: University of Chicago Press, 1984). The
famous conformal diagram of an evaporating hole on page 413 has resolved
several arguments on sci.physics (though its veracity is in question). 

	Steven Weinberg's _Gravitation and Cosmology_ (New York: John Wiley
and Sons, 1972) provided me with the historical dates.  It discusses some
properties of the Schwarzschild solution in chapter 8 and describes
gravitational collapse in chapter 11. 

********************************************************************************
Item 11.

The Solar Neutrino Problem                          original by Bruce Scott
--------------------------			    updated 5-JUN-1994 by SIC

The Short Story:

	Fusion reactions in the core of the Sun produce a huge flux of
neutrinos. These neutrinos can be detected on Earth using large underground
detectors, and the flux measured to see if it agrees with theoretical
calculations based upon our understanding of the workings of the Sun and
the details of the Standard Model (SM) of particle physics. The measured
flux is roughly one-half of the flux expected from theory. The cause of the
deficit is a mystery.  Is our particle physics wrong? Is our model of the
Solar interior wrong?  Are the experiments in error?  This is the "Solar
Neutrino Problem." 

	There are precious few experiments which seem to stand in
disagreement with the SM, which can be studied in the hope of making
breakthroughs in particle physics.  The study of this problem may yield
important new insights which may help us go beyond the Standard Model. 
There are many experiments in progress, so stay tuned. 

The Long Story:

	A middle-aged main-sequence star like the Sun is in a
slowly-evolving equilibrium, in which pressure exerted by the hot gas
balances the self-gravity of the gas mass. Slow evolution results from the
star radiating energy away in the form of light, fusion reactions occurring
in the core heating the gas and replacing the energy lost by radiation, and
slow structural adjustment to compensate the changes in entropy and
composition. 

	We cannot directly observe the center, because the mean-free path
of a photon against absorption or scattering is very short, so short that
the radiation-diffusion time scale is of order 10 million years. But the
main proton-proton reaction (PP1) in the Sun involves emission of a
neutrino: 

        p + p --> D + positron + neutrino(0.26 MeV),

which is directly observable since the cross-section for interaction with
ordinary matter is so small (the 0.26 MeV is the average energy carried
away by the neutrino).  Essentially all the neutrinos make it to the Earth.
Of course, this property also makes it difficult to detect the neutrinos.
The first experiments by Davis and collaborators, involving large tanks of
chloride fluid placed underground, could only detect higher-energy
neutrinos from small side-chains in the solar fusion: 

        PP2:    Be(7) + electron --> Li(7) + neutrino(0.80 MeV),
        PP3:    B(8) --> Be(8) + positron + neutrino(7.2 MeV).

Recently, however, the GALLEX experiment, using a gallium-solution detector
system, has observed the PP1 neutrinos to provide the first unambiguous
confirmation of proton-proton fusion in the Sun.

	There is a "neutrino problem", however, and that is the fact that
every experiment has measured a shortfall of neutrinos. About one- to
two-thirds of the neutrinos expected are observed, depending on
experimental error. In the case of GALLEX, the data read 80 units where 120
are expected, and the discrepancy is about two standard deviations. To
explain the shortfall, one of two things must be the case: (1) either the
temperature at the center is slightly less than we think it is, or (2)
something happens to the neutrinos during their flight over the
150-million-km journey to Earth. A third possibility is that the Sun
undergoes relaxation oscillations in central temperature on a time scale
shorter than 10 Myr, but since no-one has a credible mechanism this
alternative is not seriously entertained. 

(1) The fusion reaction rate is a very strong function of the temperature,
because particles much faster than the thermal average account for most of
it. Reducing the temperature of the standard solar model by 6 per cent
would entirely explain GALLEX; indeed, Bahcall has recently published an
article arguing that there may be no solar neutrino problem at all.
However, the community of solar seismologists, who observe small
oscillations in spectral line strengths due to pressure waves traversing
through the Sun, argue that such a change is not permitted by their
results. 

(2) A mechanism (called MSW, after its authors) has been proposed, by which
the neutrinos self-interact to periodically change flavor between electron,
muon, and tau neutrino types. Here, we would only expect to observe a
fraction of the total, since only electron neutrinos are detected in the
experiments. (The fraction is not exactly 1/3 due to the details of the
theory.) Efforts continue to verify this theory in the laboratory. The MSW
phenomenon, also called "neutrino oscillation", requires that the three
neutrinos have finite and differing mass, which is also still unverified. 

	To use explanation (1) with the Sun in thermal equilibrium
generally requires stretching several independent observations to the
limits of their errors, and in particular the earlier chloride results must
be explained away as unreliable (there was significant scatter in the
earliest ones, casting doubt in some minds on the reliability of the
others).  Further data over longer times will yield better statistics so
that we will better know to what extent there is a problem. Explanation (2)
depends of course on a proposal whose veracity has not been determined.
Until the MSW phenomenon is observed or ruled out in the laboratory, the
matter will remain open. 

	In summary, fusion reactions in the Sun can only be observed
through their neutrino emission. Fewer neutrinos are observed than
expected, by two standard deviations in the best result to date. This can
be explained either by a slightly cooler center than expected or by a
particle-physics mechanism by which neutrinos oscillate between flavors.
The problem is not as severe as the earliest experiments indicated, and
further data with better statistics are needed to settle the matter. 

References:

[0] The main-sequence Sun: D. D. Clayton, Principles of Stellar Evolution
    and Nucleosynthesis, McGraw-Hill, 1968. Still the best text.
[0] Solar neutrino reviews: J. N. Bahcall and M. Pinsonneault, Reviews of
    Modern Physics, vol 64, p 885, 1992; S. Turck-Chieze and I. Lopes,
    Astrophysical Journal, vol 408, p 347, 1993. See also J. N. Bahcall,
    Neutrino Astrophysics (Cambridge, 1989).
[1] Experiments by R. Davis et al: See October 1990 Physics Today, p 17.
[2] The GALLEX team: two articles in Physics Letters B, vol 285, p 376
    and p 390. See August 1992 Physics Today, p 17. Note that 80 "units" 
    correspond to the production of 9 atoms of Ge(71) in 30 tons of
    solution containing 12 tons Ga(71), after three weeks of run time!
[3] Bahcall arguing for new physics: J. N. Bahcall and H. A. Bethe,
    Physical Review D, vol 47, p 1298, 1993; against new physics: J. N. 
    Bahcall et al, "Has a Standard Model Solution to the Solar Neutrino 
    Problem Been Found?", preprint IASSNS-94/13 received at the National
    Radio Astronomy Observatory, 1994.    
[4] The MSW mechanism, after Mikheyev, Smirnov, and Wolfenstein: See the
    second GALLEX paper.
[5] Solar seismology and standard solar models: J. Christensen-Dalsgaard 
    and W. Dappen, Astronomy and Astrophysics Reviews, vol 4, p 267, 1992;
    K. G. Librecht and M. F. Woodard, Science, vol 253, p 152, 1992. See
    also the second GALLEX paper. 

********************************************************************************
Item 12.

The Expanding Universe                          original by Michael Weiss
----------------------				updated 5-DEC-1994 by SIC 

Here are the answers to some commonly asked questions about exactly 
what it means to say that the Universe is expanding.

(1) IF THE UNIVERSE IS EXPANDING, DOES THAT MEAN ATOMS ARE GETTING BIGGER?
IS THE SOLAR SYSTEM EXPANDING?

Mrs. Felix:  Why don't you do your homework?
Allen Felix: The Universe is expanding.  Everything will fall
             apart, and we'll all die.  What's the point?
Mrs. Felix:  We live in Brooklyn.  Brooklyn is not expanding!
             Go do your homework.

                        -from "Annie Hall" by Woody Allen.

Mrs. Felix is right.  Neither Brooklyn, nor its atoms, nor the solar
system, nor even the galaxy, is expanding.  The Universe expands
(according to standard cosmological models) only when averaged over a very
large scale.

	The phrase "expansion of the Universe" refers both to experimental
observation and to theoretical cosmological models.  Lets look at them one
at a time, starting with the observations.

Observation
-----------

	The observation is Hubble's redshift law.  

	In 1929, Hubble reported that the light from distant galaxies is
redshifted.  If you interpret this redshift as a Doppler shift, then the
galaxies are receding according to the law: 

    (velocity of recession) = H * (distance from Earth)

H is called Hubble's constant; Hubble's original value for H was 550
kilometers per second per megaparsec (km/s/Mpc).  Current estimates range
>from 40 to 100 km/s/Mpc.  (Measuring redshift is easy; estimating distance
is hard.  Roughly speaking, astronomers fall into two "camps", some
favoring an H around 80 km/s/Mpc, others an H around 40-55). 

	Hubble's redshift formula does *not* imply that the Earth is in
particularly bad oder in the universe.  The familiar model of the universe
as an expanding balloon speckled with galaxies shows that Hubble's alter
ego on any other galaxy would make the same observation. 

	But astronomical objects in our neck of the woods--- our solar
system, our galaxy, nearby galaxies--- show no such Hubble redshifts.
Nearby stars and galaxies *do* show motion with respect to the Earth
(known as "peculiar velocities"), but this does not look like the
"Hubble flow" that is seen for distant galaxies.  For example, the
Andromeda galaxy shows blueshift instead of redshift.  So the verdict
of observation is: our galaxy is not expanding.

	By the way, Hubble's constant, is not, in spite of its name,
constant in time. In fact, it is decreasing.  Imagine a galaxy D
light-years from the Earth, receding at a velocity V = H*D.  D is
always increasing because of the recession.  But does V increase?  No.
In fact, V is decreasing.  (If you are fond of Newtonian analogies,
you could say that "gravitational attraction" is causing this
deceleration.  But be warned: some general relativists would object
strenuously to this way of speaking.)  So H is going down over time.
But it *is* constant over space, i.e., it is the same number for all
distant objects as we observe them today.

Theory 
------

The theoretical models are, typically, Friedmann-Robertson-Walker (FRW) 
spacetimes.  

	Cosmologists model the universe using "spacetimes", that is to say,
solutions to the field equations of Einstein's theory of general
relativity.  The Russian mathematician Alexander Friedmann discovered an
important class of global solutions in 1923.  The familiar image of the
universe as an expanding balloon speckled with galaxies is a "movie
version" of one of Friedmann's solutions.  Robertson and Walker later
extended Friedmann's work, so you'll find references to
"Friedmann-Robertson-Walker" (FRW) spacetimes in the literature. 

	FRW spacetimes come in a great variety of styles--- expanding,
contracting, flat, curved, open, closed, ....  The "expanding balloon"
picture corresponds to just a few of these. 

	A concept called the metric plays a starring role in general
relativity. The metric encodes a lot of information; the part we care about
(for this FAQ entry) is distances between objects.  In an FRW expanding
universe, the distance between any two "points on the balloon" does
increase over time. However, the FRW model is NOT meant to describe OUR
spacetime accurately on a small scale--- where "small" is interpreted
pretty liberally! 

	You can picture this in a couple of ways.  You may want to think of
the "continuum approximation" in fluid dynamics--- by averaging the motion
of individual molecules over a large enough scale, you obtain a continuous
flow.  (Droplets can condense even as a gas expands.)  Similarly, it is
generally believed that if we average the actual metric of the universe
over a large enough scale, we'll get an FRW spacetime. 

	Or you may want to alter your picture of the "expanding balloon". 
The galaxies are not just painted on, but form part of the substance of the
balloon (poetically speaking), and locally affect its "elasticity". 

	The FRW spacetimes ignore these small-scale variations.  Think of a
uniformly elastic balloon, with the galaxies modelled as mere points.
"Points on the balloon" correspond to a mathematical concept known as a
*comoving geodesic*.  Any two comoving geodesics drift apart over time, in
an expanding FRW spacetime. 

	At the scale of the Solar System, we get a pretty good
approximation to the spacetime metric by using another solution to
Einstein's equations, known as the Schwarzschild metric.  Using evocative
but dubious terminology, we can say this models the gravitational field of
the Sun.  (Dubious because what does "gravitational field" mean in GR, if
it's not just a synonym for "metric"?)  The geodesics in the Schwarzschild
metric do NOT display the "drifting apart" behavior typical of the FRW
comoving geodesics--- or in more familiar terms, the Earth is not drifting
away from the Sun. 

	The "true metric" of the universe is, of course, fantastically
complicated; you can't expect idealized simple solutions (like the FRW and
Schwarzschild metrics) to capture all the complexity.  Our knowledge of the
large-scale structure of the universe is fragmentary and imprecise. 

	In old-fashioned, Newtonian terms, one says that the Solar System
is "gravitationally bound" (ditto the galaxy, the local group).  So the
Solar System is not expanding.  The case for Brooklyn is even clearer: it
is bound by atomic forces, and its atoms do not typically follow geodesics.
So Brooklyn is not expanding.  Now go do your homework. 

References: (My thanks to Jarle Brinchmann, who helped with this list.)

Misner, Thorne, and Wheeler, "Gravitation", chapters 27 and 29.  Page 719
discusses this very question; Box 29.4 outlines the "cosmic distance
ladder" and the difficulty of measuring cosmic distances; Box 29.5 presents
Hubble's work.  MTW refer to Noerdlinger and Petrosian, Ap.J., vol. 168
(1971), pp. 1--9, for an exact mathematical treatment of gravitationally
bound systems in an expanding universe.

M.V.Berry, "Principles of Cosmology and Gravitation".  Chapter 2 discusses
the cosmic distance ladder; chapters 6 and 7 explain FRW spacetimes.

Steven Weinberg, "The First Three Minutes", chapter 2.  A non-technical
treatment.

Hubble's original paper: "A Relation Between Distance And Radial
Velocity Among Extra-Galactic Nebulae", Proc. Natl. Acad. Sci., Vol. 15,
No. 3, pp. 168-173, March 1929.

Sidney van den Bergh, "The cosmic distance scale", Astronomy & Astrophysics
Review 1989 (1) 111-139.

M. Rowan-Robinson, "The Cosmological Distance Ladder", Freeman.

A new method has been devised recently to estimate Hubble's constant, using
gravitational lensing.  The method is described in:

\O Gr\on and Sjur Refsdal, "Gravitational Lenses and the age of the
universe", Eur. J. Phys. 13, 1992 178-183.

S. Refsdal & J. Surdej, Rep. Prog. Phys. 56, 1994 (117-185)

and H is estimated with this method in:

H.Dahle, S.J. Maddox, P.B. Lilje, to appear in ApJ Letters.

Two books may be consulted for what is known (or believed) about the
large-scale structure of the universe: 

P.J.E.Peebles, "An Introduction to Physical Cosmology".
T. Padmanabhan, "Structure Formation in the Universe".

======================================================================

(2) WHAT CAUSES THE HUBBLE REDSHIFT?  ARE THE LIGHT-WAVES "STRETCHED" AS 
THE UNIVERSE EXPANDS, OR IS THE LIGHT DOPPLER-SHIFTED BECAUSE DISTANT 
GALAXIES ARE MOVING AWAY FROM US?

	In a word: yes.  In two sentences: the Doppler-shift explanation is
a linear approximation to the "stretched-light" explanation.  Switching
>from one viewpoint to the other amounts to a change of coordinate systems
in (curved) spacetime. 

	A detailed explanation requires looking at Friedmann-Robertson-Walker 
(FRW) models of spacetime.  The famous "expanding balloon speckled with
galaxies" provides a visual analogy for one of these; like any analogy, it
will mislead you if taken too literally, but handled with caution it can
furnish some insight. 

	Draw a latitude/longitude grid on the balloon.  These define
*co-moving* coordinates.  Imagine a couple of speckles ("galaxies")
imbedded in the rubber surface.  The co-moving coordinates of the speckles
don't change as the balloon expands, but the distance between the speckles
steadily increases.  In co-moving coordinates, we say that the speckles
don't move, but "space itself" stretches between them. 

	A bug starts crawling from one speckle to the other.  A second
after the first bug leaves, his brother follows him.  (Think of the bugs as
two light-pulses, or successive wave-crests in a beam of light.)  Clearly
the separation between the bugs will increase during their journey.  In
co-moving coordinates, light is "stretched" during its journey. 

	Now we switch to a different coordinate system, this one valid only
in a neighborhood (but one large enough to cover both speckles).  Imagine a
clear, flexible, non-stretching patch, attached to the balloon at one
speckle.  The patch clings to the surface of the balloon, which slides
beneath it as the balloon inflates.  (The bugs crawl along *under* the
patch.)  We draw a coordinate grid on the patch.  In the patch coordinates,
the second speckle recedes from the first speckle.  And so in patch
coordinates, we can regard the redshift as a Doppler shift. 

	Is this visually appealing?  I think so.  However, this explanation
glosses over one crucial point: the time coordinate.  FRW spacetimes come
fully-equipped with a specially distinguished time coordinate (called the
co-moving or cosmological time).  For example, a co-moving observer could
set her clock by the average density of surrounding speckles, or by the
temperature of the Cosmic Background Radiation.  (From a purely
mathematical standpoint, the co-moving time coordinate is singled out by a
certain symmetry property.) 

	We have many choices of time-coordinate to go with the
space-coordinates drawn on our patch.  Let's use cosmological time.  Notice
that this is *not* the choice usually made in Special Relativity: though
the two speckles separate rapidly, their cosmological clocks remain
synchronized. Bugs embarking on their journey from the "moving" speckle
appear to crawl "upstream" against flowing space as they head towards the
"home" speckle. The current diminishes as they approach home.  (In other
words, bug-speed is anisotropic in these coordinates.)  These differences
>from the usual SR picture are symptoms of a deeper fact: besides the
obvious "spatial" curvature of the balloon's surface, FRW spacetimes have
"temporal" curvature as well.  Indeed, not all FRW spacetimes exhibit
spatial curvature, but (with one exception) all have temporal curvature. 

	You can work out the magnitude of the redshift using patch
coordinates.  I leave this as an exercise, with a couple of hints. (1)
Since bug-speed is anisotropic far from the home speckle, consider also a
patch attached to the "moving" speckle.  Compute the initial distance
between the bugs (the "wavelength") in both patch coordinate systems, using
the standard *non-relativistic* Doppler formula for a stationary source,
moving receiver.  (2) Now think about how the bug-distance changes as the
bugs journey to the home speckle (this time sticking with home patch
coordinates).  The bug-distance does *not* propagate unchanged.  Consider
instead the analog of the period of a lightwave: the time between
bug-crossings of a grid line on the patch.  This *does* propagate almost
unchanged, *provided* the rate of balloon expansion stays pretty much the
same throughout the bugs' perilous trek.  The final result: the magnitude
of the redshift, computed using Doppler's formula, agrees to first-order
with magnitude computed using the "stretched-light" explanation.  (To the
cognoscenti: the assumptions are that Hx<<1 and (dH/dt)x<<1, where
H(t)=dR(t)/dt, R(t) is the scale factor, t is cosmological time, and x is
the average distance between the "speckles" (co-moving geodesics) during
the course of the journey.) 

	(This long-winded "proof of equivalence" between the Doppler and
"stretched-light" explanations substitutes a paragraph of imagery for a
half-page of calculus.)

	Let me close by emphasizing the word "approximation" from the first
paragraph of this entry.  The Doppler explanation fails for very large
redshifts, for then we must consider how Hubble's "constant" changes over
the course of the journey. 

References:

Misner, Thorne, and Wheeler, "Gravitation", chapter 29.

M.V.Berry, "Principles of Cosmology and Gravitation", chapter 6.

Steven Weinberg, "The First Three Minutes", chapter 2, especially pp. 13
and 30.

********************************************************************************
END OF PART 2/4

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               FREQUENTLY ASKED QUESTIONS ON SCI.PHYSICS - Part 3/4
--------------------------------------------------------------------------------
Item 13.

Apparent Superluminal Velocity of Galaxies      updated 5-DEC-1994 by SIC
------------------------------------------     original by Scott I. Chase

	A distant object can appear to travel faster than the speed of
light relative to us, provided that it has some component of motion towards
us as well as perpendicular to our line of sight.  Say that on Jan. 1 you
make a position measurement of galaxy X.  One month later, you measure it
again. Assuming you know its distance from us by some independent
measurement, you derive its linear speed, and conclude that it is moving
faster than the speed of light. 

	What have you forgotten?  Let's say that on Jan. 1, the object is D
km from us, and that between Jan. 1 and Feb. 1, the object has moved d km
closer to us.  You have assumed that the light you measured on Jan. 1 and
Feb. 1 were emitted exactly one month apart.  Not so.  The first light beam
had further to travel, and was actually emitted (1 + d/c) months before the
second measurement, if we measure c in km/month.  The object has traveled
the given angular distance in more time than you thought.  Similarly, if
the object is moving away from us, the apparent angular velocity will be
too slow, if you do not correct for this effect, which becomes significant
when the object is moving along a line close to our line of sight. 

	Note that most extragalactic objects are moving away from us due to
the Hubble expansion.  So for most objects, you don't get superluminal
apparent velocities.  But the effect is still there, and you need to take
it into account if you want to measure velocities by this technique. 

References: 

Considerations about the Apparent 'Superluminal Expansions' in 
Astrophysics, E. Recami, A. Castellino, G.D. Maccarrone, M. Rodono,
Nuovo Cimento 93B, 119 (1986).

Apparent Superluminal Sources, Comparative Cosmology and the Cosmic 
Distance Scale, Mon. Not. R. Astr. Soc. 242, 423-427 (1990).

********************************************************************************
Item 14.

Hot Water Freezes Faster than Cold!             updated 11-May-1992 by SIC
-----------------------------------             original by Richard M. Mathews

	You put two pails of water outside on a freezing day.  One has hot
water (95 degrees C) and the other has an equal amount of colder water (50
degrees C).  Which freezes first?  The hot water freezes first!  Why?  

	It is commonly argued that the hot water will take some time to
reach the initial temperature of the cold water, and then follow the same
cooling curve.  So it seems at first glance difficult to believe that the
hot water freezes first.  The answer lies mostly in evaporation. The effect
is definitely real and can be duplicated in your own kitchen. 

	Every "proof" that hot water can't freeze faster assumes that the
state of the water can be described by a single number.  Remember that
temperature is a function of position.  There are also other factors
besides temperature, such as motion of the water, gas content, etc. With
these multiple parameters, any argument based on the hot water having to
pass through the initial state of the cold water before reaching the
freezing point will fall apart.  The most important factor is evaporation.

	The cooling of pails without lids is partly Newtonian and partly by
evaporation of the contents.  The proportions depend on the walls and on
temperature.  At sufficiently high temperatures evaporation is more
important.  If equal masses of water are taken at two starting
temperatures, more rapid evaporation from the hotter one may diminish its
mass enough to compensate for the greater temperature range it must cover
to reach freezing.  The mass lost when cooling is by evaporation is not
negligible. In one experiment, water cooling from 100C lost 16% of its mass
by 0C, and lost a further 12% on freezing, for a total loss of 26%. 

	The cooling effect of evaporation is twofold.  First, mass is
carried off so that less needs to be cooled from then on.  Also,
evaporation carries off the hottest molecules, lowering considerably the
average kinetic energy of the molecules remaining. This is why "blowing on
your soup" cools it.  It encourages evaporation by removing the water vapor
above the soup. 

	Thus experiment and theory agree that hot water freezes faster than
cold for sufficiently high starting temperatures, if the cooling is by
evaporation.  Cooling in a wooden pail or barrel is mostly by evaporation. 
In fact, a wooden bucket of water starting at 100C would finish freezing in
90% of the time taken by an equal volume starting at room temperature. The
folklore on this matter may well have started a century or more ago when
wooden pails were usual.  Considerable heat is transferred through the
sides of metal pails, and evaporation no longer dominates the cooling, so
the belief is unlikely to have started from correct observations after
metal pails became common. 

References: 
       "Hot water freezes faster than cold water.  Why does it do so?",
        Jearl Walker in The Amateur Scientist, Scientific American,
        Vol. 237, No. 3, pp 246-257; September, 1977.

       "The Freezing of Hot and Cold Water", G.S. Kell in American
        Journal of Physics, Vol. 37, No. 5, pp 564-565; May, 1969.

********************************************************************************
Item 15.

Why are Golf Balls Dimpled?                     updated 17-NOV-1993 by CDF
---------------------------                     original by Craig DeForest

	The dimples, paradoxically, *do* increase drag slightly.  But they 
also increase `Magnus lift', that peculiar lifting force experienced by 
rotating bodies travelling through a medium.  Contrary to Freshman physics, 
golf balls do not travel in inverted parabolas.  They follow an 'impetus 
trajectory':

                                    *    *       
                              *             *
(golfer)                *                    *
                  *                          * <-- trajectory
 \O/        *                                *
  |   *                                      *
-/ \-T---------------------------------------------------------------ground

	This is because of the combination of drag (which reduces
horizontal speed late in the trajectory) and Magnus lift, which supports
the ball during the initial part of the trajectory, making it relatively
straight.  The trajectory can even curve upwards at first, depending on
conditions!  Here is a cheesy diagram of a golf ball in flight, with some
relevant vectors: 

                             F(magnus)
                             ^
                             |
                F(drag) <--- O -------> V 
                          \     
                           \----> (sense of rotation)

	The Magnus force can be thought of as due to the relative drag on
the air on the top and bottom portions of the golf ball: the top portion is
moving slower relative to the air around it, so there is less drag on the
air that goes over the ball.  The boundary layer is relatively thin, and
air in the not-too-near region moves rapidly relative to the ball.  The
bottom portion moves fast relative to the air around it; there is more drag
on the air passing by the bottom, and the boundary (turbulent) layer is
relatively thick; air in the not-too-near region moves more slowly relative
to the ball. The Bernoulli force produces lift. (Alternatively, one could
say that `the flow lines past the ball are displaced down, so the ball is
pushed up.') 

	The difficulty comes near the transition region between laminar
flow and turbulent flow.  At low speeds, the flow around the ball is
laminar.  As speed is increased, the bottom part tends to go turbulent
*first*.  But turbulent flow can follow a surface much more easily than
laminar flow. 

	As a result, the (laminar) flow lines around the top break away
>from the surface sooner than otherwise, and there is a net displacement
*up* of the flow lines.  The magnus lift goes *negative*. 

	The dimples aid the rapid formation of a turbulent boundary layer
around the golf ball in flight, giving more lift.  Without 'em, the ball
would travel in more of a parabolic trajectory, hitting the ground sooner
(and not coming straight down). 

References: 

Lord Rayleigh, "On the Irregular Flight of a Tennis Ball", _Scientific 
Papers I_, p. 344

Briggs Lyman J., "Effect of Spin and Speed on the Lateral Deflection of 
a Baseball; and the Magnus Effect for Smooth Spheres", Am. J. Phys. _27_, 
589 (1959). [Briggs was trying to explain the mechanism behind the `curve 
ball' in baseball, using specialized apparatus in a wind tunnel at the NBS.  
He stumbled on the reverse effect by accident, because his model `baseball'
had no stitches on it. The stitches on a baseball create turbulence in
flight in much the same way that the dimples on a golf ball do.] 

R. Watts and R. Ferver, "The Lateral Force on a Spinning Sphere" Aerodynamics
of a Curveball", Am. J. Phys. _55_, 40 (1986)

********************************************************************************
Item 16.
                                                updated 9-DEC-1993 by SIC
                                                Original by Bill Johnson
How to Change Nuclear Decay Rates
---------------------------------

"I've had this idea for making radioactive nuclei decay faster/slower than
they normally do.  You do [this, that, and the other thing].  Will this work?" 

Short Answer: Possibly, but probably not usefully.

Long Answer:

	"One of the paradigms of nuclear science since the very early days
of its study has been the general understanding that the half-life, or
decay constant, of a radioactive substance is independent of extranuclear
considerations."  (Emery, cited below.)  Like all paradigms, this one is
subject to some interpretation. Normal decay of radioactive stuff proceeds
via one of four mechanisms: 

	* Emission of an alpha particle -- a helium-4 nucleus -- reducing
	the number of protons and neutrons present in the parent nucleus 
	by two each; 
	* "Beta decay," encompassing several related phenomena in which a
	neutron in the nucleus turns into a proton, or a proton turns into 
	a neutron -- along with some other things including emission of 
	a neutrino.  The "other things", as we shall see, are at the bottom
	of several questions involving perturbation of decay rates; 
	* Emission of one or more gamma rays -- energetic photons -- that
	take a nucleus from an excited state to some other (typically
	ground) state; some of these photons may be replaced by 
	"conversion electrons," of which more shortly; or 
	*Spontaneous fission, in which a sufficiently heavy nucleus simply
	breaks in half.  Most of the discussion about alpha particles will
	also apply to spontaneous fission.

Gamma emission often occurs from the daughter of one of the other decay
modes.  We neglect *very* exotic processes like C-14 emission or double
beta decay in this analysis. 

	"Beta decay" refers most often to a nucleus with a neutron excess,
which decays by converting a neutron into a proton:

         n ----> p + e- + anti-nu(e),

where n means neutron, p means proton, e- means electron, and anti-nu(e) 
means an antineutrino of the electron type.  The type of beta decay which
involves destruction of a proton is not familiar to many people, so
deserves a little elaboration.  Either of two processes may occur when this
kind of decay happens: 

        p ----> n + e+ + nu(e), 

where e+ means positron and nu(e) means electron neutrino; or

        p + e- ----> n + nu(e),

where e- means a negatively charged electron, which is captured from the
neighborhood of the nucleus undergoing decay.  These processes are called
"positron emission" and "electron capture," respectively.  A given nucleus
which has too many protons for stability may undergo beta decay through
either, and typically both, of these reactions. 

	"Conversion electrons" are produced by the process of "internal
conversion," whereby the photon that would normally be emitted in gamma
decay is *virtual* and its energy is absorbed by an atomic electron.  The
absorbed energy is sufficient to unbind the electron from the nucleus
(ignoring a few exceptional cases), and it is ejected from the atom as a
result. 

	Now for the tie-in to decay rates.  Both the electron-capture and
internal conversion phenomena require an electron somewhere close to the
decaying nucleus.  In any normal atom, this requirement is satisfied in
spades: the innermost electrons are in states such that their probability
of being close to the nucleus is both large and insensitive to things in
the environment.  The decay rate depends on the electronic wavefunctions,
i.e, how much of their time the inner electrons spend very near the 
nucleus -- but only very weakly. For most nuclides that decay by electron 
capture or internal conversion, most of the time, the probability of 
grabbing or converting an electron is also insensitive to the environment, 
as the innermost electrons are the ones most likely to get grabbed/converted. 

	However, there are exceptions, the most notable being the
the astrophysically important isotope beryllium-7.  Be-7 decays purely
by electron capture (positron emission being impossible because of
inadequate decay energy) with a half-life of somewhat over 50 days.  It has
been shown that differences in chemical environment result in half-life
variations of the order of 0.2%, and high pressures produce somewhat
similar changes. Other cases where known changes in decay rate occur are
Zr-89 and Sr-85, also electron capturers; Tc-99m ("m" implying an excited
state), which decays by both beta and gamma emission; and various other
"metastable" things that decay by gamma emission with internal conversion. 
With all of these other cases the magnitude of the effect is less than is
typically the case with Be-7. 

	What makes these cases special?  The answer is that one or more
of the usual starting assumptions -- insensitivity of electron wave
function near the nucleus to external forces, or availability of the
innermost electrons for capture/conversion -- are not completely valid. 
Atomic beryllium only has 4 electrons to begin with, so that the "innermost
electrons" are also practically the *outermost* ones and therefore much
more sensitive to chemical effects than usual.  With most of the other
cases, there is so little energy available from the decay (as little as a
few electron volts; compare most radioactive decays, where hundreds or
thousands of *kilo*volts are released), courtesy of accidents of nuclear
structure, that the innermost electrons can't undergo internal conversion. 
Remember that converting an electron requires dumping enough energy into it
to expel it from the atom (more or less); "enough energy," in context, is
typically some tens of keV, so they don't get converted at all in these
cases.  Conversion therefore works only on some of the outer electrons,
which again are more sensitive to the environment. 

	A real anomaly is the beta emitter Re-187.  Its decay energy is
only about 2.6 keV, practically nothing by nuclear standards.  "That this
decay occurs at all is an example of the effects of the atomic environment
on nuclear decay: the bare nucleus Re-187 [i.e., stripped of all orbital
electrons -- MWJ] is stable against beta decay [but not to bound state
beta decay, in which the outgoing electron is captured by the daughter
nucleus into a tightly bound orbital -SIC] and it is the difference of
15 keV in the total electronic binding energy of osmium [to which it decays
-- MWJ] and rhenium ... which makes the decay possible" (Emery).  The
practical significance of this little peculiarity, of course, is low, as
Re-187 already has a half life of over 10^10 years.   

	Alpha decay and spontaneous fission might also be affected by
changes in the electron density near the nucleus, for a different reason. 
These processes occur as a result of penetration of the "Coulomb barrier"
that inhibits emission of charged particles from the nucleus, and their
rate is *very* sensitive to the height of the barrier.  Changes in the
electron density could, in principle, affect the barrier by some tiny
amount.  However, the magnitude of the effect is *very* small, according to
theoretical calculations; for a few alpha emitters, the change has been
estimated to be of the order of 1 part in 10^7 (!) or less, which would be
unmeasurable in view of the fact that the alpha emitters' half lives aren't
known to that degree of accuracy to begin with. 

	All told, the existence of changes in radioactive decay rates due
to the environment of the decaying nuclei is on solid grounds both
experimentally and theoretically.  But the magnitude of the changes is
nothing to get very excited about. 

Reference: The best review article on this subject is now 20 years old: G.
T. Emery, "Perturbation of Nuclear Decay Rates," Annual Review of Nuclear
Science vol. 22, p. 165 (1972).  Papers describing specific experiments are
cited in that article, which contains considerable arcane math but also
gives a reasonable qualitative "feel" for what is involved. 

********************************************************************************
Item 17.
						original by Blair P. Houghton
						([email-masked])

What is a Dippy Bird, and how is it used?
-----------------------------------------

The Anatomy and Habits of a Dippy Bird:

1.  The armature:  The body of the bird is a straight tube attached to two
bulbs, approximately the same size, one at either end.  The tube flows into
the upper bulb, like the neck of a funnel, and extends almost to the bottom
of the lower bulb, like the straw in a soda. 

2.  The pivot:  At about the middle of the tube is clamped a transverse
bar, which allows the apparatus to pivot on a stand (the legs).  The bar is
bent very slightly concave dorsally, to unbalance the bird in the forward
direction (thus discouraging dips to the rear).  The ends of the pivot have
downward protrusions, which hit stops on the stand placed so that the bird
is free to rock when in a vertical position, but can not quite rotate
enough to be horizontal during a dip. 

3.  The wick:  The upper bulb is coated in fuzzy material, and has extended
>from it a beak, made of or covered in the same material. 

4.  The tail.  The tail has no significant external features, except that
it should not be insulated (skin-oil deposited on the bird's glass parts
>from handling will insulate it and can affect its operation). 

5.  The guts:  The bird is partially filled with a somewhat carefully
measured amount of a fluid with suitable lack of viscosity and density and
a low latent heat of evaporation (small d(energy)/d(mass), ld).  For water,
ld is 2250 kJ/kg; for methylene chloride, ld is 406; for mercury, ld is a
wondrous 281; ethyl alcohol has an ld of 880, more than twice that of MC.
Boiling point is not important, here; evaporation and condensation take
place on the surface of a liquid at any temperature.

6.  The frills:  Any hats, eyes, feathers, or liquid coloring have been
added purely for entertainment value. (An anecdote:  as it stood pumping in
the Arizona sun on my kitchen windowsill for several days, the rich,
Kool-Aid red of my bird's motorwater faded to a pale peach.  I have since
retired him to the mantelpiece in the family room). 

7.  Shreddin':  The bird is operated by getting the head wet, taking care
not to make it so wet that it drips down the tube.  (Water on the bottom
bulb will reverse the thermodynamic processes.)  The first cycle will
take somewhat longer than the following cycles.  If you can keep water
where the bird can dip it, the bird will dip for as long as the ambient
humidity remains favorable.

Come on, how does it really work?
---------------------------------

Short answer:  Thermodynamics plus Mechanics.

Medium answer (and essential clues):  Evaporative cooling on the outside;
pV=nRT, evaporation/condensation, and gravity on the inside.

Long answer:

	Initially the system is at equilibrium, with T equal in both
chambers and pV/n in each compensating for the fluid levels.  Evaporation
of water outside the head draws heat from inside it; the vapor inside
condenses, reducing pV/RT.  This imbalances the pressures, so the vapor in
the abdomen pushes down, which pushes fluid up the thorax, which reduces V
in the head.  Since p is decreasing in the abdomen, evaporation occurs,
increasing n, and drawing heat from outside the body. 

	The rising fluid raises the CM above the pivot point; the hips are
slightly concave dorsally, so the bird dips forward.  Tabs on the legs and
the pivot maintain the angle at full dip, for drainage.  The amount of
fluid is set so that at full dip the lower end of the tube is exposed to
the vapor.  (The tube reaches almost to the bottom of the abdomen, like a
straw in a soda, but flows into the head like the neck of a funnel.)  A
bubble of vapor rises in the tube and fluid drains into the abdomen. 

	The rising bubble transfers heat to the head and the falling fluid
releases gravitational potential energy as heat into the rising bubble and
the abdomen.  The CM drops below the pivot point and the bird bobs up.  The
system is thus reset; it's not quite at equilibrium, but is close enough
that the process can repeat this chain of events. 

	The beak acts as a wick, if allowed to dip into a reservoir of
water, to keep the head wet, although it is not necessary for the bird to
drink on every dip. 

Is that all there is to know about dippy birds?
-----------------------------------------------

	Of course not.  Research continues to unravel these unanswered
questions about the amazing dippy-bird:

1.  All of the energy gained by the rising fluid is returned to the system
when the fluid drops; where does this energy go, in what proportions, and
how does this affect the rate at which the bird operates? 

2.  The heat that evaporates the water comes from both the surrounding air
and the inside of the head; but, in what proportion?

3.  Exactly what should the fluid be?  Methylene Chloride is an excellent
candidate, since it's listed in the documentation for recent birds sold by
Edmund Scientific Corp. (trade named Happy Drinking Bird), and because its
latent heat of evaporation (ld) is 406 kJ/kg, compared to 2250 kJ/kg for
water (a 5.5:1 ratio of condensed MC to evaporated water, if all
water-evaporating heat comes from inside the bird).  Ethanol, at 880 kJ/kG,
is only half as efficient.  Mercury would likewise be a good prospective
choice, having an ld of 281 kJ/kG (8:1!), but is expensive and dangerous,
and its density would require careful redesign and greater quality control
in the abdomen and pivot-stops to ensure proper operation at full dip; this
does, however, indicate that the apparatus could be made in miniature,
filled with mercury, and sold through a catalog-store such as The Sharper
Image as a wildly successful yuppie desk-toy (Consider the submission of
this FAQ entry to be prior art for patent purposes). 

4.  Does ambient temperature have an effect on operation aside from the
increase in rate of evaporation of water? I.e., if the temperature and
humidity can be controlled independently such that the rate of evaporation
can be kept constant, what effect does such a change in ambient temperature
and humidity have on the operation of the bird? Is the response transient,
permanent, or composed of both? 

Dippy Bird Tips:
----------------

	They have real trouble working at all in humid climates (like
around the U. of Md., where I owned my first one), but can drive you bats
in dry climates (aside from the constant hammering, it's hard to keep the
water up to a level where the bird can get at it...).  The evaporation of
water from the head depends on the diffusibility of water vapor into the
atmosphere; high partial pressures of water vapor in the atmosphere
translate to low rates of evaporation. 

	If you handle your bird, clean the glass with alcohol or Windex
or Dawn or something; the oil from your hands has a high specific heat,
which damps the transfer of heat, and a low thermal conductivity, which
attenuates the transfer of heat. Once it's clean, grasp the bird only by
the legs or the tube, which are not thermodynamically significant, or
wear rubber gloves, just like a real EMT.

	The hat is there for show; the dippy bird operates okay with or
without it, even though it may reduce the area of evaporation slightly. 
Ditto the feathers and the eyes. 

Bibliography:
-------------

	Chemical data from Gieck, K., _Engineering Formulas_, 3d. Ed.,
McGraw-Hill, 1979, as translated by J. Walters, B. Sc.  

	I've also heard that SciAm had an "Amateur Scientist" column on
this technology a few years ago.  Perhaps someone who understands how a
library works could look up the yr and vol...

	Kool-Aid is a trademark of some huge corporation that makes its
money a farthing at a time...

********************************************************************************
Item 18.

Below Absolute Zero - What Does Negative Temperature Mean?   updated 24-MAR-1993
----------------------------------------------------------   by Scott I. Chase

Questions:  What is negative temperature?  Can you really make a system
which has a temperature below absolute zero?  Can you even give any useful
meaning to the expression 'negative absolute temperature'? 

Answer:  Absolutely. :-)

	Under certain conditions, a closed system *can* be described by a
negative temperature, and, surprisingly, be *hotter* than the same system
at any positive temperature.  This article describes how it all works. 

Step I: What is "Temperature"?
------------------------------

	To get things started, we need a clear definition of "temperature."
Actually various kinds of "temperature" appear in the literature of
physics (e.g., kinetic temperature, color temperature).  The relevant
one here is the one from thermodynamics, in some sense the most
fundamental.

	Our intuitive notion is that two systems in thermal contact
should exchange no heat, on average, if and only if they are at the
same temperature.  Let's call the two systems S1 and S2. The combined
system, treating S1 and S2 together, can be S3.  The important
question, consideration of which will lead us to a useful quantitative
definition of temperature, is "How will the energy of S3 be
distributed between S1 and S2?"  I will briefly explain this below,
but I recommend that you read K&K, referenced below, for a careful,
simple, and thorough explanation of this important and fundamental
result.

	With a total energy E, S has many possible internal states
(microstates).  The atoms of S3 can share the total energy in many ways.
Let's say there are N different states.  Each state corresponds to a
particular division of the total energy in the two subsystems S1 and S2.
Many microstates can correspond to the same division, E1 in S1 and E2 in
S2. A simple counting argument tells you that only one particular division
of the energy, will occur with any significant probability.  It's the one
with the overwhelmingly largest number of microstates for the total system
S3. That number, N(E1,E2) is just the product of the number of states
allowed in each subsystem, N(E1,E2) = N1(E1)*N2(E2), and, since E1 + E2 =
E, N(E1,E2) reaches a maximum when N1*N2 is stationary with respect to
variations of E1 and E2 subject to the total energy constraint. 

	For convenience, physicists prefer to frame the question in terms
of the logarithm of the number of microstates N, and call this the entropy,
S. You can easily see from the above analysis that two systems are in
equilibrium with one another when (dS/dE)_1 = (dS/dE)_2, i.e., the rate of
change of entropy, S, per unit change in energy, E, must be the same for
both systems.  Otherwise, energy will tend to flow from one subsystem to
another as S3 bounces randomly from one microstate to another, the total
energy E3 being constant, as the combined system moves towards a state of
maximal total entropy.  We define the temperature, T, by 1/T = dS/dE, so
that the equilibrium condition becomes the very simple T_1 = T_2. 

	This statistical mechanical definition of temperature does in fact
correspond to your intuitive notion of temperature for most systems. So
long as dS/dE is always positive, T is always positive.  For common
situations, like a collection of free particles, or particles in a harmonic
oscillator potential, adding energy always increases the number of
available microstates, increasingly faster with increasing total energy. So
temperature increases with increasing energy, from zero, asymptotically
approaching positive infinity as the energy increases. 

Step II: What is "Negative Temperature"?
----------------------------------------

	Not all systems have the property that the entropy increases 
monotonically with energy.  In some cases, as energy is added to the system, 
the number of available microstates, or configurations, actually decreases 
for some range of energies.  For example, imagine an ideal "spin-system", a 
set of N atoms with spin 1/2 on a one-dimensional wire.  The atoms are not 
free to move from their positions on the wire.  The only degree of freedom 
allowed to them is spin-flip:  the spin of a given atom can point up or 
down.  The total energy of the system, in a magnetic field of strength B, 
pointing down, is (N+ - N-)*uB, where u is the magnetic moment of each atom 
and N+ and N- are the number of atoms with spin up and down respectively. 
Notice that with this definition, E is zero when half of the spins are 
up and half are down.  It is negative when the majority are down and 
positive when the majority are up.

	The lowest possible energy state, all the spins pointing down,
gives the system a total energy of -NuB, and temperature of absolute zero. 
There is only one configuration of the system at this energy, i.e., all the
spins must point down.  The entropy is the log of the number of
microstates, so in this case is log(1) = 0.  If we now add a quantum of
energy, size uB, to the system, one spin is allowed to flip up.  There are
N possibilities, so the entropy is log(N).  If we add another quantum of
energy, there are a total of N(N-1)/2 allowable configurations with two
spins up.  The entropy is increasing quickly, and the temperature is rising
as well. 

	However, for this system, the entropy does not go on increasing
forever.  There is a maximum energy, +NuB, with all spins up.  At this
maximal energy, there is again only one microstate, and the entropy is
again zero.  If we remove one quantum of energy from the system, we allow
one spin down.  At this energy there are N available microstates.  The
entropy goes on increasing as the energy is lowered.  In fact the maximal
entropy occurs for total energy zero, i.e., half of the spins up, half
down. 

	So we have created a system where, as we add more and more energy,
temperature starts off positive, approaches positive infinity as maximum
entropy is approached, with half of all spins up.  After that, the
temperature becomes negative infinite, coming down in magnitude toward
zero, but always negative, as the energy increases toward maximum. When the
system has negative temperature, it is *hotter* than when it is has
positive temperature. If you take two copies of the system, one with positive
and one with negative temperature, and put them in thermal contact, heat
will flow from the negative-temperature system into the positive-temperature
system. 

Step III:  What Does This Have to Do With the Real World?
---------------------------------------------------------

	Can this system ever by realized in the real world, or is it just a
fantastic invention of sinister theoretical condensed matter physicists?
Atoms always have other degrees of freedom in addition to spin, usually
making the total energy of the system unbounded upward due to the
translational degrees of freedom that the atom has.  Thus, only certain
degrees of freedom of a particle can have negative temperature.  It makes
sense to define the "spin-temperature" of a collection of atoms, so long as
one condition is met:  the coupling between the atomic spins and the other
degrees of freedom is sufficiently weak, and the coupling between atomic
spins sufficiently strong, that the timescale for energy to flow from the
spins into other degrees of freedom is very large compared to the timescale
for thermalization of the spins among themselves.  Then it makes sense to
talk about the temperature of the spins separately from the temperature of
the atoms as a whole. This condition can easily be met for the case of
nuclear spins in a strong external magnetic field. 

	Nuclear and electron spin systems can be promoted to negative
temperatures by suitable radio frequency techniques.   Various experiments
in the calorimetry of negative temperatures, as well as applications of
negative temperature systems as RF amplifiers, etc., can be found in the
articles listed below, and the references therein. 

References:

        Kittel and Kroemer,_Thermal Physics_, appendix E.
        N.F. Ramsey, "Thermodynamics and statistical mechanics at negative
        absolute temperature,"  Phys. Rev. _103_, 20 (1956).
        M.J. Klein,"Negative Absolute Temperature," Phys. Rev. _104_, 589 (1956).
	
********************************************************************************
Item 19.

Which Way Will my Bathtub Drain?                updated 16-MAR-1993 by SIC
--------------------------------                original by Matthew R. Feinstein

Question: Does my bathtub drain differently depending on whether I live 
in the northern or southern hemisphere?

Answer: No.  There is a real effect, but it is far too small to be relevant
when you pull the plug in your bathtub.

	Because the earth rotates, a fluid that flows along the earth's
surface feels a "Coriolis" acceleration perpendicular to its velocity.
In the northern hemisphere low pressure storm systems spin counterclockwise.
In the southern hemisphere, they spin clockwise because the direction 
of the Coriolis acceleration is reversed.  This effect leads to the 
speculation that the bathtub vortex that you see when you pull the plug 
>from the drain spins one way in the north and the other way in the south.

	But this acceleration is VERY weak for bathtub-scale fluid
motions.  The order of magnitude of the Coriolis acceleration can be
estimated from size of the "Rossby number" (see below).  The effect of the 
Coriolis acceleration on your bathtub vortex is SMALL.  To detect its 
effect on your bathtub, you would have to get out and wait until the motion 
in the water is far less than one rotation per day.  This would require 
removing thermal currents, vibration, and any other sources of noise.  Under 
such conditions, never occurring in the typical home, you WOULD see an 
effect.  To see what trouble it takes to actually see the effect, see the 
reference below.  Experiments have been done in both the northern and 
southern hemispheres to verify that under carefully controlled conditions, 
bathtubs drain in opposite directions due to the Coriolis acceleration from 
the Earth's rotation. 

	Coriolis accelerations are significant when the Rossby number is 
SMALL. 	So, suppose we want a Rossby number of 0.1 and a bathtub-vortex
length scale of 0.1 meter.  Since the earth's rotation rate is about
10^(-4)/second, the fluid velocity should be less than or equal to
2*10^(-6) meters/second.  This is a very small velocity.  How small is it? 
Well, we can take the analysis a step further and calculate another, more
famous dimensionless parameter, the Reynolds number. 

	The Reynolds number is = L*U*density/viscosity

	Assuming that physicists bathe in hot water the viscosity will be
about 0.005 poise and the density will be about 1.0, so the Reynolds Number
is about 4*10^(-2). 

	Now, life at low Reynolds numbers is different from life at high
Reynolds numbers.  In particular, at low Reynolds numbers, fluid physics is
dominated by friction and diffusion, rather than by inertia: the time it
would take for a particle of fluid to move a significant distance due to an
acceleration is greater than the time it takes for the particle to break up
due to diffusion. 

	The same effect has been accused of responsibility for the 
direction water circulates when you flush a toilet.  This is surely 
nonsense.  In this case, the water rotates in the direction which the pipe 
points which carries the water from the tank to the bowl.

Reference: Trefethen, L.M. et al, Nature 207 1084-5 (1965).

********************************************************************************
Item 20.

Why do Mirrors Reverse Left and Right?          updated 04-MAR-1994 by SIC
--------------------------------------          original by Scott I. Chase

	The simple answer is that they don't.  Look in a mirror and wave
your right hand.  On which side of the mirror is the hand that waved?  The
right side, of course. 

	Mirrors DO reverse In/Out.  Imagine holding an arrow in your hand.  
If you point it up, it will point up in the mirror.  If you point it to the 
left, it will point to the left in the mirror.  But if you point it toward 
the mirror, it will point right back at you.  In and Out are reversed. 

	If you take a three-dimensional, rectangular, coordinate system,
(X,Y,Z), and point the Z axis such that the vector equation X x Y = Z is
satisfied, then the coordinate system is said to be right-handed.  Imagine
Z pointing toward the mirror.  X and Y are unchanged (remember the arrows?)
but Z will point back at you.  In the mirror, X x Y = - Z.  The image
contains a left-handed coordinate system. 

	This has an important effect, familiar mostly to chemists and
physicists. It changes the chirality, or handedness, of objects viewed in
the mirror. Your left hand looks like a right hand, while your right hand
looks like a left hand.  Molecules often come in pairs called
stereoisomers, which differ not in the sequence or number of atoms, but
only in that one is the mirror image of the other, so that no rotation or
stretching can turn one into the other.  Your hands make a good laboratory
for this effect.  They are distinct, even though they both have the same
components connected in the same way. They are a stereo pair, identical
except for "handedness". 

	People sometimes think that mirrors *do* reverse left/right, and
that the effect is due to the fact that our eyes are aligned horizontally
on our faces.  This can be easily shown to be untrue by looking in any
mirror with one eye closed! 

Reference:  _The Left Hand of the Electron_, by Isaac Asimov, contains 
a very readable discussion of handedness and mirrors in physics.

********************************************************************************
Item 21.
                                                updated 16-MAR-1992 by SIC
                                                Original by John Blanton
Why Do Stars Twinkle While Planets Do Not? 
-----------------------------------------

	Stars, except for the Sun, although they may be millions of miles 
in diameter, are very far away.  They appear as point sources even when
viewed by telescopes.  The planets in our solar system, much smaller than
stars, are closer and can be resolved as disks with a little bit of
magnification (field binoculars, for example). 

	Since the Earth's atmosphere is turbulent, all images viewed up
through it tend to "swim."  The result of this is that sometimes a single
point in object space gets mapped to two or more points in image space, and
also sometimes a single point in object space does not get mapped into any
point in image space.  When a star's single point in object space fails to
map to at least one point in image space, the star seems to disappear
temporarily. This does not mean the star's light is lost for that moment. 
It just means that it didn't get to your eye, it went somewhere else. 

	Since planets represent several points in object space, it is
highly likely that one or more points in the planet's object space get
mapped to a points in image space, and the planet's image never winks out. 
Each individual ray is twinkling away as badly as any star, but when all of
those individual rays are viewed together, the next effect is averaged out 
to something considerably steadier. 

	The result is that stars tend to twinkle, and planets do not. 
Other extended objects in space, even very far ones like nebulae, do not 
twinkle if they are sufficiently large that they have non-zero apparent
diameter when viewed from the Earth.

********************************************************************************
Item 22.

TIME TRAVEL - FACT OR FICTION?                  updated 07-MAR-1994
------------------------------                  original by Jon J. Thaler

	We define time travel to mean departure from a certain place and
time followed (from the traveller's point of view) by arrival at the same
place at an earlier (from the sedentary observer's point of view) time.
Time travel paradoxes arise from the fact that departure occurs after
arrival according to one observer and before arrival according to another. 
In the terminology of special relativity time travel implies that the
timelike ordering of events is not invariant.  This violates our intuitive
notions of causality.  However, intuition is not an infallible guide, so we
must be careful.  Is time travel really impossible, or is it merely another
phenomenon where "impossible" means "nature is weirder than we think?"  The
answer is more interesting than you might think. 

THE SCIENCE FICTION PARADIGM:

	The B-movie image of the intrepid chrononaut climbing into his time
machine and watching the clock outside spin backwards while those outside
the time machine watch the him revert to callow youth is, according to
current theory, impossible.  In current theory, the arrow of time flows in
only one direction at any particular place.  If this were not true, then
one could not impose a 4-dimensional coordinate system on space-time, and
many nasty consequences would result. Nevertheless, there is a scenario
which is not ruled out by present knowledge.  This usually requires an 
unusual spacetime topology (due to wormholes or strings in general 
relativity) which has not yet seen, but which may be possible.  In 
this scenario the universe is well behaved in every local region; only by 
exploring the global properties does one discover time travel. 

CONSERVATION LAWS:

	It is sometimes argued that time travel violates conservation laws.
For example, sending mass back in time increases the amount of energy that
exists at that time.  Doesn't this violate conservation of energy?  This
argument uses the concept of a global conservation law, whereas
relativistically invariant formulations of the equations of physics only
imply local conservation.  A local conservation law tells us that the
amount of stuff inside a small volume changes only when stuff flows in or
out through the surface.  A global conservation law is derived from this by
integrating over all space and assuming that there is no flow in or out at
infinity.  If this integral cannot be performed, then global conservation
does not follow.  So, sending mass back in time might be all right, but it
implies that something strange is happening.  (Why shouldn't we be able to
do the integral?) 

GENERAL RELATIVITY:

	One case where global conservation breaks down is in general
relativity. It is well known that global conservation of energy does not
make sense in an expanding universe.  For example, the universe cools as it
expands; where does the energy go?  See FAQ article #7 - Energy
Conservation in Cosmology, for details. 

	It is interesting to note that the possibility of time travel in GR
has been known at least since 1949 (by Kurt Godel, discussed in [1], page
168). The GR spacetime found by Godel has what are now called "closed
timelike curves" (CTCs).  A CTC is a worldline that a particle or a person
can follow which ends at the same spacetime point (the same position and
time) as it started.   A solution to GR which contains CTCs cannot have a
spacelike embedding - space must have "holes" (as in donut holes, not holes
punched in a sheet of paper).  A would-be time traveller must go around or
through the holes in a clever way. 

	The Godel solution is a curiosity, not useful for constructing a
time machine.  Two recent proposals, one by Morris, et al. [2] and one by
Gott [3], have the possibility of actually leading to practical devices (if
you believe this, I have a bridge to sell you).   As with Godel, in these
schemes nothing is locally strange; time travel results from the unusual
topology of spacetime.  The first uses a wormhole (the inner part of a
black hole, see fig. 1 of [2]) which is held open and manipulated by
electromagnetic forces.  The second uses the conical geometry generated by
an infinitely long string of mass.  If two strings pass by each other, a
clever person can go into the past by traveling a figure-eight path around
the strings.  In this scenario, if the string has non-zero diameter and
finite mass density, there is a CTC without any unusual topology.

GRANDFATHER PARADOXES:

	With the demonstration that general relativity contains CTCs,
people began studying the problem of self-consistency.  Basically, the
problem is that of the "grandfather paradox":  What happens if our time
traveller kills her grandmother before her mother was born?  In more
readily analyzable terms, one can ask what are the implications of the
quantum mechanical interference of the particle with its future self.
Boulware [5] shows that there is a problem - unitarity is violated. This is
related to the question of when one can do the global conservation integral
discussed above.  It is an example of the "Cauchy problem" [1, chapter 7]. 

OTHER PROBLEMS (and an escape hatch?):

	How does one avoid the paradox that a simple solution to GR has
CTCs which QM does not like?  This is not a matter of applying a theory in
a domain where it is expected to fail.  One relevant issue is the
construction of the time machine.  After all, infinite strings aren't
easily obtained.  In fact, it has been shown [4] that Gott's scenario
implies that the total 4-momentum of spacetime must be spacelike.  This
seems to imply that one cannot build a time machine from any collection of
non-tachyonic objects, whose 4-momentum must be timelike.  There are 
implementation problems with the wormhole method as well.

TACHYONS:

Finally, a diversion on a possibly related topic.

	If tachyons exist as physical objects, causality is no longer
invariant. Different observers will see different causal sequences.  This
effect requires only special relativity (not GR), and follows from the fact
that for any spacelike trajectory, reference frames can be found in which
the particle moves backward or forward in time.  This is illustrated by the
pair of spacetime diagrams below.  One must be careful about what is
actually observed; a particle moving backward in time is observed to be a
forward moving anti-particle, so no observer interprets this as time
travel. 

                t
One reference   |                    Events A and C are at the same
frame:          |                    place.  C occurs first.
                |
                |                    Event B lies outside the causal
                |          B         domain of events A and C.
     -----------A----------- x       (The intervals are spacelike).
                |
                C                    In this frame, tachyon signals
                |                    travel from A-->B and from C-->B.
                |                    That is, A and C are possible causes
                                     of event B.

Another         t
reference       |                    Events A and C are not at the same
frame:          |                    place.  C occurs first.
                |
                |                    Event B lies outside the causal
     -----------A----------- x       domain of events A and C. (The
                |                    intervals are spacelike)
                |
                |   C                In this frame, signals travel from
                |                    B-->A and from B-->C.  B is the cause
                |             B      of both of the other two events.

	The unusual situation here arises because conventional causality
assumes no superluminal motion.  This tachyon example is presented to
demonstrate that our intuitive notion of causality may be flawed, so one
must be careful when appealing to common sense.  See FAQ article # 25 -
Tachyons, for more about these weird hypothetical particles. 

CONCLUSION:

	The possible existence of time machines remains an open question. 
None of the papers criticizing the two proposals are willing to
categorically rule out the possibility.  Nevertheless, the notion of time
machines seems to carry with it a serious set of problems. 

REFERENCES:

1: S.W. Hawking, and G.F.R. Ellis, "The Large Scale Structure of Space-Time,"
   Cambridge University Press, 1973.
2: M.S. Morris, K.S. Thorne, and U. Yurtsever, PRL, v.61, p.1446 (1989).
   --> How wormholes can act as time machines.
3: J.R. Gott, III, PRL, v.66, p.1126 (1991).
   --> How pairs of cosmic strings can act as time machines.
4: S. Deser, R. Jackiw, and G. 't Hooft, PRL, v.66, p.267 (1992).
   --> A critique of Gott.  You can't construct his machine.
5: D.G. Boulware, University of Washington preprint UW/PT-92-04.
   Available on the [email-masked] bulletin board: item number 9207054.
   --> Unitarity problems in QM with closed timelike curves.
6: "Nature", May 7, 1992
   --> Contains a very well written review with some nice figures.

********************************************************************************
Item 23.

Open Questions                                  updated 01-JUN-1993 by SIC
--------------                                  original by John Baez

	While for the most part a FAQ covers the answers to frequently
asked questions whose answers are known, in physics there are also plenty
of simple and interesting questions whose answers are not known. Before you
set about answering these questions on your own, it's worth noting that
while nobody knows what the answers are, there has been at least a little,
and sometimes a great deal, of work already done on these subjects. People
have said a lot of very intelligent things about many of these questions. 
So do plenty of research and ask around before you try to cook up a theory
that'll answer one of these and win you the Nobel prize!  You can expect to
really know physics inside and out before you make any progress on these. 

	The following partial list of "open" questions is divided into two
groups, Cosmology and Astrophysics, and Particle and Quantum Physics. 
However, given the implications of particle physics on cosmology, the
division is somewhat artificial, and, consequently, the categorization is 
somewhat arbitrary.  

	(There are many other interesting and fundamental questions in 
fields such as condensed matter physics, nonlinear dynamics, etc., which 
are not part of the set of related questions in cosmology and quantum 
physics which are discussed below.  Their omission is not a judgement 
about importance, but merely a decision about the scope of this article.)
	
Cosmology and Astrophysics
--------------------------

1.  What happened at or before the Big Bang?  Was there really an initial
singularity?  Of course, this question might not make sense, but it might.
Does the history of the Universe go back in time forever, or only a finite
amount? 

2.  Will the future of the universe go on forever or not?  Will there be a
"big crunch" in the future?  Is the Universe infinite in spatial extent? 

3.  Why is there an arrow of time; that is, why is the future so much
different from the past?

4.  Is spacetime really four-dimensional?   If so, why - or is that just a
silly question?  Or is spacetime not really a manifold at all if examined
on a short enough distance scale? 

5.  Do black holes really exist?  (It sure seems like it.)  Do they really
radiate energy and evaporate the way Hawking predicts?  If so, what happens
when, after a finite amount of time, they radiate completely away?  What's
left?  Do black holes really violate all conservation laws except
conservation of energy, momentum, angular momentum and electric charge? 
What happens to the information contained in an object that falls into a 
black hole?  Is it lost when the black hole evaporates?  Does this require 
a modification of quantum mechanics?

6.  Is the Cosmic Censorship Hypothesis true?  Roughly, for generic
collapsing isolated gravitational systems are the singularities that might
develop guaranteed to be hidden beyond a smooth event horizon?  If Cosmic
Censorship fails, what are these naked singularities like?  That is, what 
weird physical consequences would they have? 

7.  Why are the galaxies distributed in clumps and filaments?  Is most of
the matter in the universe baryonic?  Is this a matter to be resolved by
new physics? 

8.  What is the nature of the missing "Dark Matter"?  Is it baryonic, 
neutrinos, or something more exotic?  

Particle and Quantum Physics
----------------------------

1.  Why are the laws of physics not symmetrical between left and right,
future and past, and between matter and antimatter?  I.e., what is the
mechanism of CP violation, and what is the origin of parity violation in
Weak interactions?  Are there right-handed Weak currents too weak to have
been detected so far?  If so, what broke the symmetry?  Is CP violation 
explicable entirely within the Standard Model, or is some new force or
mechanism required?

2.  Why are the strengths of the fundamental forces (electromagnetism, weak
and strong forces, and gravity) what they are?  For example, why is the
fine structure constant, which measures the strength of electromagnetism,
about 1/137.036?  Where did this dimensionless constant of nature come from? 
Do the forces really become Grand Unified at sufficiently high energy?

3.  Why are there 3 generations of leptons and quarks?  Why are their mass
ratios what they are?  For example, the muon is a particle almost exactly
like the electron except about 207 times heavier.  Why does it exist and
why precisely that much heavier?   Do the quarks or leptons have any 
substructure? 

4.  Is there a consistent and acceptable relativistic quantum field theory
describing interacting (not free) fields in four spacetime dimensions?  For
example, is the Standard Model mathematically consistent?  How about
Quantum Electrodynamics? 

5.  Is QCD a true description of quark dynamics?  Is it possible to
calculate masses of hadrons (such as the proton, neutron, pion, etc.)
correctly from the Standard Model?  Does QCD predict a quark/gluon
deconfinement phase transition at high temperature?  What is the nature of
the transition?  Does this really happen in Nature? 

6.  Why is there more matter than antimatter, at least around here? Is
there really more matter than antimatter throughout the universe? 

7.  What is meant by a "measurement" in quantum mechanics?  Does
"wavefunction collapse" actually happen as a physical process?  If so, how,
and under what conditions?  If not, what happens instead? 

8.  What are the gravitational effects, if any, of the immense (possibly
infinite) vacuum energy density seemingly predicted by quantum field
theory?  Is it really that huge?  If so, why doesn't it act like an
enormous cosmological constant? 

9.  Why doesn't the flux of solar neutrinos agree with predictions?  Is the
disagreement really significant?  If so, is the discrepancy in models of
the sun, theories of nuclear physics, or theories of neutrinos?  Are
neutrinos really massless? 

The Big Question (TM)
---------------------

This last question sits on the fence between the two categories above:

    How do you merge Quantum Mechanics and General Relativity to create a
quantum theory of gravity?  Is Einstein's theory of gravity (classical GR)
also correct in the microscopic limit, or are there modifications
possible/required which coincide in the observed limit(s)?  Is gravity
really curvature, or what else -- and why does it then look like curvature? 
An answer to this question will necessarily rely upon, and at the same time
likely be a large part of, the answers to many of the other questions above.

********************************************************************************
END OF FAQ PART 3/4

Article: 5898 of sci.physics.particle
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Subject: sci.physics Frequently Asked Questions (Part 4 of 4)
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Summary: This posting contains a list of Frequently Asked Questions 
	(and their answers) about physics, and should be read by anyone who 
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--------------------------------------------------------------------------------
               FREQUENTLY ASKED QUESTIONS ON SCI.PHYSICS - Part 4/4
--------------------------------------------------------------------------------
Item 24. Special Relativistic Paradoxes - part (a) 

The Barn and the Pole                   updated 4-AUG-1992 by SIC
---------------------                   original by Robert Firth

	These are the props.  You own a barn, 40m long, with automatic
doors at either end, that can be opened and closed simultaneously by a
switch. You also have a pole, 80m long, which of course won't fit in the
barn. 

	Now someone takes the pole and tries to run (at nearly the speed of
light) through the barn with the pole horizontal.  Special Relativity (SR)
says that a moving object is contracted in the direction of motion: this is
called the Lorentz Contraction.  So, if the pole is set in motion
lengthwise, then it will contract in the reference frame of a stationary
observer. 

	You are that observer, sitting on the barn roof.  You see the pole
coming towards you, and it has contracted to a bit less than 40m. So, as
the pole passes through the barn, there is an instant when it is completely
within the barn.  At that instant, you close both doors.  Of course, you
open them again pretty quickly, but at least momentarily you had the
contracted pole shut up in your barn.  The runner emerges from the far door
unscathed. 

	But consider the problem from the point of view of the runner.  She
will regard the pole as stationary, and the barn as approaching at high
speed. In this reference frame, the pole is still 80m long, and the barn
is less than 20 meters long.  Surely the runner is in trouble if the doors 
close while she is inside.  The pole is sure to get caught. 

	Well does the pole get caught in the door or doesn't it?  You can't
have it both ways.  This is the "Barn-pole paradox."  The answer is buried
in the misuse of the word "simultaneously" back in the first sentence of
the story.  In SR, that events separated in space that appear simultaneous
in one frame of reference need not appear simultaneous in another frame of
reference. The closing doors are two such separate events. 

	SR explains that the two doors are never closed at the same time in
the runner's frame of reference.  So there is always room for the pole.  In
fact, the Lorentz transformation for time is t'=(t-v*x/c^2)/sqrt(1-v^2/c^2).
It's the v*x term in the numerator that causes the mischief here.  In the
runner's frame the further event (larger x) happens earlier.  The far door 
is closed first.  It opens before she gets there, and the near door closes 
behind her. Safe again - either way you look at it, provided you remember 
that simultaneity is not a constant of physics. 

References:  Taylor and Wheeler's _Spacetime Physics_ is the classic. 
Feynman's _Lectures_ are interesting as well.

********************************************************************************
Item 24. Special Relativistic Paradoxes - part (b) 

The Twin Paradox                                updated 04-MAR-1994 by SIC
----------------                                original by Kurt Sonnenmoser

A Short Story about Space Travel:

	Two twins, conveniently named A and B, both know the rules of
Special Relativity.  One of them, B, decides to travel out into space with
a velocity near the speed of light for a time T, after which she returns to
Earth. Meanwhile, her boring sister A sits at home posting to Usenet all
day.  When B finally comes home, what do the two sisters find?  Special
Relativity (SR) tells A that time was slowed down for the relativistic
sister, B, so that upon her return to Earth, she knows that B will be
younger than she is, which she suspects was the the ulterior motive of the
trip from the start. 

	But B sees things differently.  She took the trip just to get away 
>from the conspiracy theorists on Usenet, knowing full well that from her 
point of view, sitting in the spaceship, it would be her sister, A, who 
was travelling ultrarelativistically for the whole time, so that she would 
arrive home to find that A was much younger than she was.  Unfortunate, but 
worth it just to get away for a while. 

	What are we to conclude?  Which twin is really younger?  How can SR
give two answers to the same question?  How do we avoid this apparent
paradox? Maybe twinning is not allowed in SR?  Read on. 

Paradox Resolved:

	Much of the confusion surrounding the so-called Twin Paradox
originates from the attempts to put the two twins into different frames ---
without the useful concept of the proper time of a moving body. 

	SR offers a conceptually very clear treatment of this problem.
First chose _one_ specific inertial frame of reference; let's call it S.
Second define the paths that A and B take, their so-called world lines. As
an example, take (ct,0,0,0) as representing the world line of A, and
(ct,f(t),0,0) as representing the world line of B (assuming that the the
rest frame of the Earth was inertial). The meaning of the above notation is
that at time t, A is at the spatial location (x1,x2,x3)=(0,0,0) and B is at
(x1,x2,x3)=(f(t),0,0) --- always with respect to S. 

	Let us now assume that A and B are at the same place at the time t1
and again at a later time t2, and that they both carry high-quality clocks
which indicate zero at time t1. High quality in this context means that the
precision of the clock is independent of acceleration. [In principle, a
bunch of muons provides such a device (unit of time: half-life of their
decay).] 

	The correct expression for the time T such a clock will indicate at
time t2 is the following [the second form is slightly less general than the
first, but it's the good one for actual calculations]: 

            t2          t2      _______________ 
            /           /      /             2 |
      T  =  | d\tau  =  | dt \/  1 - [v(t)/c]              (1)
            /           /
          t1          t1

where d\tau is the so-called proper-time interval, defined by

              2         2      2      2      2
     (c d\tau)  = (c dt)  - dx1  - dx2  - dx3 .

Furthermore,
                   d                          d
           v(t) = -- (x1(t), x2(t), x3(t)) = -- x(t)
                  dt                         dt

is the velocity vector of the moving object. The physical interpretation
of the proper-time interval, namely that it is the amount the clock time
will advance if the clock moves by dx during dt, arises from considering
the inertial frame in which the clock is at rest at time t --- its
so-called momentary rest frame (see the literature cited below). [Notice
that this argument is only of heuristic value, since one has to assume
that the absolute value of the acceleration has no effect. The ultimate
justification of this interpretation must come from experiment.]

	The integral in (1) can be difficult to evaluate, but certain
important facts are immediately obvious. If the object is at rest with
respect to S, one trivially obtains T = t2-t1. In all other cases, T must
be strictly smaller than t2-t1, since the integrand is always less than or
equal to unity. Conclusion: the traveling twin is younger. Furthermore, if
she moves with constant velocity v most of the time (periods of
acceleration short compared to the duration of the whole trip), T will
approximately be given by      ____________                              
                              /          2 | 
                    (t2-t1) \/  1 - [v/c]    .             (2)

The last expression is exact for a round trip (e.g. a circle) with constant
velocity v. [At the times t1 and t2, twin B flies past twin A and they
compare their clocks.] 

	Now the big deal with SR, in the present context, is that T (or
d\tau, respectively) is a so-called Lorentz scalar. In other words, its
value does not depend on the choice of S. If we Lorentz transform the
coordinates of the world lines of the twins to another inertial frame S',
we will get the same result for T in S' as in S. This is a mathematical
fact. It shows that the situation of the traveling twins cannot possibly
lead to a paradox _within_ the framework of SR. It could at most be in
conflict with experimental results, which is also not the case. 

	Of course the situation of the two twins is not symmetric, although
one might be tempted by expression (2) to think the opposite. Twin A is
at rest in one and the same inertial frame for all times, whereas twin B
is not.  [Formula (1) does not hold in an accelerated frame.]  This breaks 
the apparent symmetry of the two situations, and provides the clearest
nonmathematical hint that one twin will in fact be younger than the other
at the end of the trip.  To figure out *which* twin is the younger one, use
the formulae above in a frame in which they are valid, and you will find
that B is in fact younger, despite her expectations. 

	It is sometimes claimed that one has to resort to General
Relativity in order to "resolve" the Twin "Paradox". This is not true. In
flat, or nearly flat, space-time (no strong gravity), SR is completely
sufficient, and it has also no problem with world lines corresponding to
accelerated motion. 

References: 
        Taylor and Wheeler, _Spacetime Physics_  (An *excellent* discussion)
        Goldstein, _Classical Mechanics_, 2nd edition, Chap.7 (for a good 
        general discussion of Lorentz transformations and other SR basics.) 

********************************************************************************
Item 24. Special Relativistic Paradoxes - part (c) 

The Superluminal Scissors                           updated 31-MAR-1993 
-------------------------                           original by Scott I.Chase

	A Gedankenexperiment:

	Imagine a huge pair of scissors, with blades one light-year long.
The handle is only about two feet long, creating a huge lever arm,
initially open by a few degrees.  Then you suddenly close the scissors. 
This action takes about a tenth of a second.  Doesn't the contact point
where the two blades touch move down the blades *much* faster than the
speed of light? After all, the scissors close in a tenth of a second, but
the blades are a light-year long.  That seems to mean that the contact
point has moved down the blades at the remarkable speed of 10 light-years
per second.  This is more than 10^8 times the speed of light!  But this
seems to violate the most important rule of Special Relativity - no signal
can travel faster than the speed of light.  What's going on here? 

	Explanation:

	We have mistakenly assumed that the scissors do in fact close when
you close the handle.  But, in fact, according to Special Relativity, this
is not at all what happens.  What *does* happen is that the blades of the
scissors flex.  No matter what material you use for the scissors, SR sets a
theoretical upper limit to the rigidity of the material.  In short, when
you close the scissors, they bend. 

	The point at which the blades bend propagates down the blade at
some speed less than the speed of light.  On the near side of this point,
the scissors are closed.  On the far side of this point, the scissors
remain open.  You have, in fact, sent a kind of wave down the scissors,
carrying the information that the scissors have been closed.  But this wave
does not travel faster than the speed of light.  It will take at least one
year for the tips of the blades, at the far end of the scissors, to feel
any force whatsoever, and, ultimately, to come together to completely close
the scissors. 

	As a practical matter, this theoretical upper limit to the rigidity
of the metal in the scissors is *far* higher than the rigidity of any real
material, so it would, in practice, take much much longer to close a real
pair of metal scissors with blades as long as these. 

	One can analyze this problem microscopically as well.  The
electromagnetic force which binds the atoms of the scissors together
propagates at the speeds of light. So if you displace some set of atoms in
the scissor (such as the entire handles), the force will not propagate down
the scissor instantaneously, This means that a scissor this big *must*
cease to act as a rigid body. You can move parts of it without other parts
moving at the same time. It takes some finite time for the changing forces
on the scissor to propagate from atom to atom, letting the far tip of the
blades "know" that the scissors have been closed. 

	Caveat:

	The contact point where the two blades meet is not a physical
object.  So there is no fundamental reason why it could not move faster
than the speed of light, provided that you arrange your experiment correctly.
In fact it can be done with scissors provided that your scissors are short
enough and wide open to start, very different conditions than those spelled
out in the gedankenexperiment above.  In this case it will take you quite
a while to bring the blades together - more than enough time for light to
travel to the tips of the scissors.  When the blades finally come together,
if they have the right shape, the contact point can indeed move faster
than light.  

	Think about the simpler case of two rulers pinned together at an 
edge point at the ends.  Slam the two rulers together and the contact point
will move infinitely fast to the far end of the rulers at the instant
they touch.  So long as the rulers are short enough that contact does not
happen until the signal propagates to the far ends of the rulers, the 
rulers will indeed be straight when they meet.  Only if the rulers are
too long will they be bent like our very long scissors, above, when they
touch.  The contact point can move faster than the speed of light, but
the energy (or signal) of the closing force can not.

	An analogy, equivalent in terms of information content, is, say, a 
line of strobe lights.  You want to light them up one at a time, so that 
the `bright' spot travels faster than light.  To do so, you can send a 
_luminal_ signal down the line, telling each strobe light to wait a 
little while before flashing.  If you decrease the wait time with
each successive strobe light, the apparent bright spot will travel faster
than light, since the strobes on the end didn't wait as long after getting
the go-ahead, as did the ones at the beginning.  But the bright spot
can't pass the original signal, because then the strobe lights wouldn't
know to flash.

********************************************************************************

Item 25.  Can You See the Lorentz-Fitzgerald Contraction?           12-Oct-1995
          Or: Penrose-Terrell Rotation                              by Michael Weiss

People sometimes argue over whether the Lorentz-Fitzgerald contraction is
"real" or not.  That's a topic for another FAQ entry, but here's a short
answer: the contraction can be measured, but the measurement is
frame-dependent.  Whether that makes it "real" or not has more to do with your
choice of words than the physics.

Here we ask a subtly different question.  If you take a snapshot of a rapidly
moving object, will it *look* flattened when you develop the film?  What is the
difference between measuring and photographing?  Isn't seeing believing?  Not
always!  When you take a snapshot, you capture the light-rays that hit the
*film* at one instant (in the reference frame of the film).  These rays may
have left the *object* at different instants; if the object is moving with
respect to the film, then the photograph may give a distorted picture.
(Strictly speaking snapshots aren't instantaneous, but we're idealizing.)

Oddly enough, though Einstein published his famous relativity paper in
1905, and Fitzgerald proposed his contraction several years earlier,
no one seems to have asked this question until the late '50s.  Then
Roger Penrose and James Terrell independently discovered that the
object will *not* appear flattened [1,2].  People sometimes say that
the object appears rotated, so this effect is called the
Penrose-Terrell rotation.

Calling it a rotation can be a bit confusing though.  Rotating an object brings
its backside into view, but it's hard to see how a contraction could do that.
Among other things, this entry will try to explain in just what sense
the Penrose-Terrell effect is a "rotation".

It will clarify matters to imagine *two* snapshots of the same object, taken by
two cameras moving uniformly with respect to each other.  We'll call them *his*
camera and *her* camera.  The cameras pass through each other at the origin at
t=0, when they take their two snapshots.  Say that the object is at rest with
respect to his camera, and moving with respect to hers.  By analysing the
process of taking a snapshot, the meaning of "rotation" will become clearer.

How should we think of a snapshot?  Here's one way: consider a pinhole camera.
(Just one camera, for the moment.)  The pinhole is located at the origin, and
the film occupies a patch on a sphere surrounding the origin.  We'll ignore all
technical difficulties(!), and pretend that the camera takes full spherical
pictures: the film occupies the entire sphere.

We need more than just a pinhole and film, though: we also need a shutter.  At
t=0, the shutter snaps open for an instant to let the light-rays through the
pinhole; these spread out in all directions, and at t=1 (in the rest-frame of
the camera) paint a picture on the spherical film.

Let's call points in the snapshot *pixels*.  Each pixel gets its color due to
an event, namely a light-ray hitting the sphere at t=1.  Now let's consider his
& her cameras, as we said before.  We'll use t for his time, and t' for hers.
At t=t'=0, the two pinholes coincide at the origin, the two shutters snap
simultaneously, and the light rays spread out.  At t=1 for *his* camera, they
paint *his* pixels; at t'=1 for *her* camera, they paint *hers*.  So the
definition of a snapshot is frame-dependent.  But you already knew that.  (Pop
quiz: what shape does *he* think *her* film has?  Not spherical!)  (More
technical difficulties: the rays have to pass right through one film to hit the
other.)

So there's a one-one correspondence between pixels in the two snapshots.  Two
pixels correspond if they are painted by the same light-ray.  You can see now
that her snapshot is just a distortion of his (and vice versa).  You could take
his snapshot, scan it into a computer, run an algorithm to move the pixels
around, and print out hers.

So what does the pixel mapping look like?  Simple: if we put the usual
latitude/longitude grid on the spheres, chosen so that the relative motion is
along the north-south axis, then each pixel slides up towards the north pole
along a line of longitude.  (Or down towards the south pole, depending on
various choices I haven't specified.)  This should ring a bell if you know
about the aberration of light: if our snapshots portray the night-sky, then the
stars are white pixels, and aberration changes their apparent positions.

Now let's consider the object--- let's say a galaxy.  In passing from his
snapshot to hers, the image of the galaxy slides up the sphere, keeping the
same face to us.  In this sense, it has rotated.  Its apparent size will also
change, but not its shape (to a first approximation).

The mathematical details are beautiful, but best left to the textbooks [3,4].
Just to entice you if you have the background: if we regard the two spheres as
Riemann spheres, then the pixel mapping is given by a fractional linear
transformation.  Well-known facts from complex analysis now tell us two things.
First, circles go to circles under the pixel mapping, so a sphere will *always*
photograph as a sphere.  Second, shapes of objects are preserved in the
infinitesimally small limit.  (If you know about the double-covering of SL(2),
that also comes into play.  [3] is a good reference.)

References: [1] and [2] are the original articles.  [3] and [4] are textbook
treatments.  [5] has beautiful computer-generated pictures of the
Penrose-Terrell rotation.  The authors of [5] later made a video [6] of this
and other effects of "SR photography".

[1] Penrose, R.,"The Apparent Shape of a Relativistically Moving Sphere", 
    Proc. Camb. Phil. Soc., vol 55 Jul 1958.
[2] Terrell, J., "Invisibility of the Lorentz Contraction",
    Phys. Rev. vol 116 no. 4 pp. 1041-1045 (1959).    
[3] Penrose, R., and W. Rindler, "Spinors and Space-Time", vol I chapter 1.
[4] Marion, "Classical Dynamics", Section 10.5.
[5] Hsiung, Ping-Kang, Robert H. Thibadeau, and Robert H. P. Dunn,
    "Ray-Tracing Relativity", Pixel, vol 1 no. 1 (Jan/Feb 1990).
[6] Hsiung, Ping-Kang, and Robert H. Thibadeau, "Spacetime
    Visualizations," a video, Imaging Systems Lab, Robotics Institute,
    Carnegie Mellon University.

********************************************************************************

Item 26.  
Tachyons                                        updated: 22-MAR-1993 by SIC
--------                                        original by Scott I. Chase

                There was a young lady named Bright,
                Whose speed was far faster than light.
                She went out one day,
                In a relative way,
                And returned the previous night!

                        -Reginald Buller

	It is a well known fact that nothing can travel faster than the
speed of light. At best, a massless particle travels at the speed of light.
But is this really true?  In 1962, Bilaniuk, Deshpande, and Sudarshan, Am.
J. Phys. _30_, 718 (1962), said "no".  A very readable paper is Bilaniuk
and Sudarshan, Phys. Today _22_,43 (1969).  I give here a brief overview. 

	Draw a graph, with momentum (p) on the x-axis, and energy (E) on
the y-axis.  Then draw the "light cone", two lines with the equations E =
+/- p. This divides our 1+1 dimensional space-time into two regions.  Above
and below are the "timelike" quadrants, and to the left and right are the
"spacelike" quadrants. 

	Now the fundamental fact of relativity is that E^2 - p^2 = m^2. 
(Let's take c=1 for the rest of the discussion.)  For any non-zero value of 
m (mass), this is an hyperbola with branches in the timelike regions.  It 
passes through the point (p,E) = (0,m), where the particle is at rest.  Any 
particle with mass m is constrained to move on the upper branch of this 
hyperbola.  (Otherwise, it is "off-shell", a term you hear in association
with virtual particles - but that's another topic.) For massless particles, 
E^2 = p^2, and the particle moves on the light-cone. 

	These two cases are given the names tardyon (or bradyon in more
modern usage) and luxon, for "slow particle" and "light particle".  Tachyon
is the name given to the supposed "fast particle" which would move with v>c. 

	Now another familiar relativistic equation is E =
m*[1-(v/c)^2]^(-.5).  Tachyons (if they exist) have v > c.  This means that 
E is imaginary!  Well, what if we take the rest mass m, and take it to be
imaginary?  Then E is negative real, and E^2 - p^2 = m^2 < 0.  Or, p^2 -
E^2 = M^2, where M is real.  This is a hyperbola with branches in the
spacelike region of spacetime.  The energy and momentum of a tachyon must
satisfy this relation.

	You can now deduce many interesting properties of tachyons.  For
example, they accelerate (p goes up) if they lose energy (E goes down).
Futhermore, a zero-energy tachyon is "transcendent," or infinitely fast.
This has profound consequences.  For example, let's say that there were
electrically charged tachyons.  Since they would move faster than the speed 
of light in the vacuum, they should produce Cerenkov radiation. This would 
*lower* their energy, causing them to accelerate more!  In other words, 
charged tachyons would probably lead to a runaway reaction releasing an 
arbitrarily large amount of energy.  This suggests that coming up with a 
sensible theory of anything except free (noninteracting) tachyons is likely 
to be difficult.  Heuristically, the problem is that we can get spontaneous 
creation of tachyon-antitachyon pairs, then do a runaway reaction, making 
the vacuum unstable.  To treat this precisely requires quantum field theory, 
which gets complicated.  It is not easy to summarize results here.  However,
one reasonably modern reference is _Tachyons, Monopoles, and Related
Topics_, E. Recami, ed. (North-Holland, Amsterdam, 1978).

	However, tachyons are not entirely invisible.  You can imagine that
you might produce them in some exotic nuclear reaction.  If they are
charged, you could "see" them by detecting the Cerenkov light they produce
as they speed away faster and faster.  Such experiments have been done.  So
far, no tachyons have been found.  Even neutral tachyons can scatter off
normal matter with experimentally observable consequences.  Again, no such
tachyons have been found. 

	How about using tachyons to transmit information faster than the
speed of light, in violation of Special Relativity?  It's worth noting 
that when one considers the relativistic quantum mechanics of tachyons, the 
question of whether they "really" go faster than the speed of light becomes 
much more touchy!  In this framework, tachyons are *waves* that satisfy a
wave equation.  Let's treat free tachyons of spin zero, for simplicity.   
We'll set c = 1 to keep things less messy.  The wavefunction of a single 
such tachyon can be expected to satisfy the usual equation for spin-zero 
particles, the Klein-Gordon equation:

                (BOX + m^2)phi = 0

where BOX is the D'Alembertian, which in 3+1 dimensions is just

                BOX = (d/dt)^2 - (d/dx)^2 - (d/dy)^2 - (d/dz)^2.

The difference with tachyons is that m^2 is *negative*, and m is
imaginary.

To simplify the math a bit, let's work in 1+1 dimensions, with
coordinates x and t, so that

                BOX = (d/dt)^2 - (d/dx)^2

Everything we'll say generalizes to the real-world 3+1-dimensional case.
Now - regardless of m, any solution is a linear combination, or
superposition, of solutions of the form

                phi(t,x) = exp(-iEt + ipx)

where E^2 - p^2 = m^2.  When m^2 is negative there are two essentially
different cases.  Either |p| >= |E|, in which case E is real and
we get solutions that look like waves whose crests move along at the
rate |p|/|E| >= 1, i.e., no slower than the speed of light.  Or |p| <
|E|, in which case E is imaginary and we get solutions that look waves
that amplify exponentially as time passes!

We can decide as we please whether or not we want to consider the second
sort of solutions.   They seem weird, but then the whole business is
weird, after all.

1)	If we *do* permit the second sort of solution, we can solve the
Klein-Gordon equation with any reasonable initial data - that is, any
reasonable values of phi and its first time derivative at t = 0.  (For
the precise definition of "reasonable," consult your local
mathematician.)  This is typical of wave equations.  And, also typical
of wave equations, we can prove the following thing: If the solution phi
and its time derivative are zero outside the interval [-L,L] when t = 0,
they will be zero outside the interval [-L-|t|, L+|t|] at any time t.
In other words, localized disturbances do not spread with speed faster
than the speed of light!  This seems to go against our notion that
tachyons move faster than the speed of light, but it's a mathematical
fact, known as "unit propagation velocity".

2)	If we *don't* permit the second sort of solution, we can't solve the
Klein-Gordon equation for all reasonable initial data, but only for initial
data whose Fourier transforms vanish in the interval [-|m|,|m|].  By the
Paley-Wiener theorem this has an odd consequence: it becomes
impossible to solve the equation for initial data that vanish outside
some interval [-L,L]!  In other words, we can no longer "localize" our
tachyon in any bounded region in the first place, so it becomes
impossible to decide whether or not there is "unit propagation
velocity" in the precise sense of part 1).    Of course, the crests of
the waves exp(-iEt + ipx) move faster than the speed of light, but these
waves were never localized in the first place!

	The bottom line is that you can't use tachyons to send information 
faster than the speed of light from one place to another.  Doing so would 
require creating a message encoded some way in a localized tachyon field,
and sending it off at superluminal speed toward the intended receiver. But 
as we have seen you can't have it both ways - localized tachyon disturbances 
are subluminal and superluminal disturbances are nonlocal.

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Item 27.

The Particle Zoo                                updated 4-JUL-1995 by MCW
----------------                                original by Matt Austern

	If you look in the Particle Data Book, you will find more than 150
particles listed there.  It isn't quite as bad as that, though... 

	The (observed) particles are divided into two major classes:
the material particles, and the gauge bosons.  We'll discuss the gauge
bosons further down.  The material particles in turn fall into three
categories: leptons, mesons, and baryons. Leptons are particles that
are like the electron: they have spin 1/2, and they do not undergo the
strong interaction.  There are three charged leptons, the electron,
muon, and tau, and three corresponding neutral leptons, or neutrinos.
(The muon and the tau are both short-lived.)

	Mesons and baryons both undergo strong interactions.  The
difference is that mesons have integral spin (0, 1,...), while baryons have
half-integral spin (1/2, 3/2,...).  The most familiar baryons are the
proton and the neutron; all others are short-lived.  The most familiar
meson is the pion; its lifetime is 26 nanoseconds, and all other mesons
decay even faster. 

	Most of those 150+ particles are mesons and baryons, or,
collectively, hadrons.  The situation was enormously simplified in the
1960s by the "quark model," which says that hadrons are made out of
spin-1/2 particles called quarks.  A meson, in this model, is made out
of a quark and an anti-quark, and a baryon is made out of three
quarks.  We don't see free quarks, but only hadrons; nevertheless, the
evidence for quarks is compelling. Quark masses are not very well
defined, since they are not free particles, but we can give estimates.
The masses below are in GeV; the first is current mass and the second
constituent mass (which includes some of the effects of the binding
energy):

      Generation:       1             2            3
      U-like:     u=.006/.311   c=1.50/1.65   t=91-200/91-200
      D-like:     d=.010/.315   s=.200/.500   b=5.10/5.10

	In the quark model, there are only 12 elementary particles,
which appear in three "generations."  The first generation consists of
the up quark, the down quark, the electron, and the electron
neutrino. (Each of these also has an associated antiparticle.)  These
particles make up all of the ordinary matter we see around us.  There
are two other generations, which are essentially the same, but with
heavier particles.  The second consists of the charm quark, the
strange quark, the muon, and the muon neutrino; and the third consists
of the top quark, the bottom quark, the tau, and the tau neutrino.
These three generations are sometimes called the "electron family",
the "muon family", and the "tau family."

	Finally, according to quantum field theory, particles interact by
exchanging "gauge bosons," which are also particles.  The most familiar on
is the photon, which is responsible for electromagnetic interactions. 
There are also eight gluons, which are responsible for strong interactions,
and the W+, W-, and Z, which are responsible for weak interactions. 

The picture, then, is this:

                FUNDAMENTAL PARTICLES OF MATTER
  Charge        -------------------------
    -1          |  e    |  mu   |  tau  |
     0          | nu(e) |nu(mu) |nu(tau)|
                -------------------------       + antiparticles
   -1/3         | down  |strange|bottom |
    2/3         |  up   | charm |  top  |
                -------------------------

                        GAUGE BOSONS
  Charge                                                Force
     0                  photon                          electromagnetism
     0                  gluons (8 of them)              strong force
    +-1                 W+ and W-                       weak force
     0                  Z                               weak force

	The Standard Model of particle physics also predicts the
existence of a "Higgs boson," which has to do with breaking a symmetry
involving these forces, and which is responsible for the masses of all the
other particles.  It has not yet been found.  More complicated theories
predict additional particles, including, for example, gauginos and sleptons
and squarks (from supersymmetry), W' and Z' (additional weak bosons), X and
Y bosons (from GUT theories), Majorons, familons, axions, paraleptons,
ortholeptons, technipions (from technicolor models), B' (hadrons with
fourth generation quarks), magnetic monopoles, e* (excited leptons), etc. 
None of these "exotica" have yet been seen.  The search is on! 

REFERENCES:

	The best reference for information on which particles exist,
their masses, etc., is the Particle Data Book.  It is published every
two years; the most recent edition is Physical Review D vol.50 No.3
part 1 August 1994.  The Web version can be accessed through
http://pdg.lbl.gov/.

	There are several good books that discuss particle physics on a
level accessible to anyone who knows a bit of quantum mechanics.  One is
_Introduction to High Energy Physics_, by Perkins.  Another, which takes a
more historical approach and includes many original papers, is
_Experimental Foundations of Particle Physics_, by Cahn and Goldhaber. 

	For a book that is accessible to non-physicists, you could try _The
Particle Explosion_ by Close, Sutton, and Marten.  This book has fantastic
photography. 

	For a Web introduction by the folks at Fermilab, take a look
at http://fnnews.fnal.gov/hep_overview.html .
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Item 28.                                         original by Scott I. Chase

Does Antimatter Fall Up or Down?
--------------------------------

This question has never been subject to a successful direct experiment. 
In other words, nobody has ever directly measured the gravititational 
acceleration of antimatter.  So the bottom line is that we don't know yet.  
However, there is a lot more to say than just that, with regard to both
theory and experiment.  Here is a summary of the current state of affairs.

(1) Is is even theoretically possible for antimatter to fall up?

Answer: According to GR, antimatter falls down.

If you believe that General Relativity is the exact true theory of 
gravity, then there is only one possible conclusion - by the equivalence
principle, antiparticles must fall down with the same acceleration as 
normal matter.

On the other hand: there are other models of gravity which are not ruled out 
by direct experiment which are distinct from GR in that antiparticles can 
fall down at different rates than normal matter, or even fall up, due to 
additional forces which couple to the mass of the particle in ways which are 
different than GR.  Some people don't like to call these new couplings 
'gravity.'  They call them, generically, the 'fifth force,' defining gravity 
to be only the GR part of the force.  But this is mostly a semantic 
distinction.  The bottom line is that antiparticles won't fall like normal 
particles if one of these models is correct.  

There are also a variety of arguments, based upon different aspects of 
physics, against the possibility of antigravity.  These include constraints
imposed by conservation of energy (the "Morrison argument"), the detectable 
effects of virtual antiparticles (the "Schiff argument"), and the absense
of gravitational effect in kaon regeneration experiments.  Each of these
does in fact rule out *some* models of antigravity.  But none of them 
absolutely excludes all possible models of antigravity.  See the reference
below for all the details on these issues.

(2) Haven't people done experiments to study this question?

There are no valid *direct* experimental tests of whether antiparticles
fall up or down.  There was one well-known experiment by Fairbank at 
Stanford in which he tried to measure the fall of positrons.  He found that
they fell normally, but later analyses of his experiment revealed that
he had not accounted for all the sources of stray electromagnetic fields.
Because gravity is so much weaker than EM, this is a difficult experimental
problem.  A modern assessment of the Fairbank experiment is that it was
inconclusive.  

In order to reduce the effect of gravity, it would be nice to repeat the
Fairbank experiment using objects with the same magnitude of electric 
charge as positrons, but with much more mass, to increase the relative
effect of gravity on the motion of the particle.  Antiprotons are 1836
times more massive than positrons, so give you three orders of magnitude
more sensitivity.  Unfortunately, making many slow antiprotons which you
can watch fall is very difficult.  An experiment is under development
at CERN right now to do just that, and within the next couple of years
the results should be known.

Most people expect that antiprotons *will* fall.  But it is important
to keep an open mind - we have never directly observed the effect of 
gravity on antiparticles.  This experiment, if successful, will definitely
be "one for the textbooks."

Reference: Nieto and Goldman, "The Arguments Against 'Antigravity' and 
the Gravitational Acceleration of Antimatter,"  Physics Reports, v.205,
No. 5, p.221.

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Item 29.

What is the Mass of a Photon?                   updated 24-JUL-1992 by SIC
                                                original by Matt Austern

Or, "Does the mass of an object depend on its velocity?"

	This question usually comes up in the context of wondering whether
photons are really "massless," since, after all, they have nonzero energy. 
The problem is simply that people are using two different definitions of
mass.  The overwhelming consensus among physicists today is to say that 
photons are massless.  However, it is possible to assign a "relativistic 
mass" to a photon which depends upon its wavelength.  This is based upon 
an old usage of the word "mass" which, though not strictly wrong, is not 
used much today.

	The old definition of mass, called "relativistic mass," assigns
a mass to a particle proportional to its total energy E, and involved
the speed of light, c, in the proportionality constant:

                m = E / c^2.                                        (1) 

This definition gives every object a velocity-dependent mass.

	The modern definition assigns every object just one mass, an 
invariant quantity that does not depend on velocity.  This is given by

                m = E_0 / c^2,                                      (2)

where E_0 is the total energy of that object at rest.  

	The first definition is often used in popularizations, and in some
elementary textbooks.  It was once used by practicing physicists, but for
the last few decades, the vast majority of physicists have instead used the
second definition.  Sometimes people will use the phrase "rest mass," or
"invariant mass," but this is just for emphasis: mass is mass.  The
"relativistic mass" is never used at all.  (If you see "relativistic mass"
in your first-year physics textbook, complain! There is no reason for books
to teach obsolete terminology.) 

	Note, by the way, that using the standard definition of mass, the
one given by Eq. (2), the equation "E = m c^2" is *not* correct.  Using the
standard definition, the relation between the mass and energy of an object
can be written as 

                E   = m c^2 / sqrt(1 -v^2/c^2),                     (3) 
or as

                E^2 = m^2 c^4  +  p^2 c^2,                          (4)

where v is the object's velocity, and p is its momentum.

	In one sense, any definition is just a matter of convention.  In
practice, though, physicists now use this definition because it is much
more convenient.  The "relativistic mass" of an object is really just the
same as its energy, and there isn't any reason to have another word for
energy: "energy" is a perfectly good word.  The mass of an object, though,
is a fundamental and invariant property, and one for which we do need a
word. 

	The "relativistic mass" is also sometimes confusing because it
mistakenly leads people to think that they can just use it in the Newtonian
relations 
                F = m a                                             (5) 
and
                F = G m1 m2 / r^2.                                  (6)

In fact, though, there is no definition of mass for which these
equations are true relativistically: they must be generalized.  The
generalizations are more straightforward using the standard definition
of mass than using "relativistic mass."

	Oh, and back to photons: people sometimes wonder whether it makes
sense to talk about the "rest mass" of a particle that can never be at
rest.  The answer, again, is that "rest mass" is really a misnomer, and it
is not necessary for a particle to be at rest for the concept of mass to
make sense.  Technically, it is the invariant length of the particle's
four-momentum.  (You can see this from Eq. (4).)  For all photons this is
zero. On the other hand, the "relativistic mass" of photons is frequency
dependent. UV photons are more energetic than visible photons, and so are
more "massive" in this sense, a statement which obscures more than it
elucidates. 

	Reference: Lev Okun wrote a nice article on this subject in the 
June 1989 issue of Physics Today, which includes a historical discussion 
of the concept of mass in relativistic physics. 

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Item 30.                                           original by David Brahm

Baryogenesis - Why Are There More Protons Than Antiprotons?
-----------------------------------------------------------

(I) How do we really *know* that the universe is not matter-antimatter 
symmetric?  

(a) The Moon:  Neil Armstrong did not annihilate, therefore the moon
is made of matter.
(b) The Sun:  Solar cosmic rays are matter, not antimatter.
(c) The other Planets:  We have sent probes to almost all.  Their survival 
demonstrates that the solar system is made of matter.
(d) The Milky Way:  Cosmic rays sample material from the entire galaxy.
In cosmic rays, protons outnumber antiprotons 10^4 to 1.
(e) The Universe at large: This is tougher.  If there were antimatter
galaxies then we should see gamma emissions from annihilation.  Its absence
is strong evidence that at least the nearby clusters of galaxies (e.g., Virgo)
are matter-dominated.  At larger scales there is little proof.
    However,  there is a problem, called the "annihilation catastrophe"
which probably eliminates the possibility of a matter-antimatter symmetric
universe.  Essentially, causality prevents the separation of large chucks
of antimatter from matter fast enough to prevent their mutual annihilation
in in the early universe.  So the Universe is most likely matter dominated.

(II) How did it get that way?  

	Annihilation has made the asymmetry much greater today than in the
early universe.  At the high temperature of the first microsecond, there 
were large numbers of thermal quark-antiquark pairs.  K&T estimate 30 
million antiquarks for every 30 million and 1 quarks during this epoch.  
That's a tiny asymmetry.  Over time most of the antimatter has annihilated 
with matter, leaving the very small initial excess of matter to dominate 
the Universe.

	Here are a few possibilities for why we are matter dominated today:

a) The Universe just started that way.
   Not only is this a rather sterile hypothesis, but it doesn't work under
   the popular "inflation" theories, which dilute any initial abundances.
b) Baryogenesis occurred around the Grand Unified (GUT) scale (very early).
   Long thought to be the only viable candidate, GUT's generically have
   baryon-violating reactions, such as proton decay (not yet observed).
c) Baryogenesis occurred at the Electroweak Phase Transition (EWPT).
   This is the era when the Higgs first acquired a vacuum expectation value
   (vev), so other particles acquired masses.  Pure Standard Model physics.

	Sakharov enumerated 3 necessary conditions for baryogenesis:

	(1)  Baryon number violation.  If baryon number is conserved in all
reactions, then the present baryon asymmetry can only reflect asymmetric
initial conditions, and we are back to case (a), above.
	(2) C and CP violation.  Even in the presence of B-violating
reactions, without a preference for matter over antimatter the B-violation
will take place at the same rate in both directions, leaving no excess.
	(3) Thermodynamic Nonequilibrium.  Because CPT guarantees equal
masses for baryons and antibaryons, chemical equilibrium would drive the 
necessary reactions to correct for any developing asymmetry.  

	It turns out the Standard Model satisfies all 3 conditions:

	(1) Though the Standard Model conserves B classically (no terms in
the Lagrangian violate B), quantum effects allow the universe to tunnel
between vacua with different values of B.  This tunneling is _very_
suppressed at energies/temperatures below 10 TeV (the "sphaleron mass"),
_may_ occur at e.g. SSC energies (controversial), and _certainly_ occurs at
higher temperatures.

	(2) C-violation is commonplace.  CP-violation (that's "charge
conjugation" and "parity") has been experimentally observed in kaon
decays, though strictly speaking the Standard Model probably has
insufficient CP-violation to give the observed baryon asymmetry.

	(3) Thermal nonequilibrium is achieved during first-order phase
transitions in the cooling early universe, such as the EWPT (at T = 100 GeV
or so).  As bubbles of the "true vacuum" (with a nonzero Higgs vev)
percolate and grow, baryogenesis can occur at or near the bubble walls.

	A major theoretical problem, in fact, is that there may be _too_
_much_ B-violation in the Standard Model, so that after the EWPT is
complete (and condition 3 above is no longer satisfied) any previously
generated baryon asymmetry would be washed out.

References: Kolb and Turner, _The Early Universe_;
  Dine, Huet, Singleton & Susskind, Phys.Lett.B257:351 (1991);
  Dine, Leigh, Huet, Linde & Linde, Phys.Rev.D46:550 (1992).

********************************************************************************
Item 31.                                         

The EPR Paradox and Bell's Inequality Principle    updated 31-AUG-1993 by SIC
-----------------------------------------------    original by John Blanton

	In 1935 Albert Einstein and two colleagues, Boris Podolsky and
Nathan Rosen (EPR) developed a thought experiment to demonstrate what they
felt was a lack of completeness in quantum mechanics.  This so-called "EPR
paradox" has led to much subsequent, and still on-going, research. This 
article is an introduction to EPR, Bell's inequality, and the real 
experiments which have attempted to address the interesting issues raised 
by this discussion.

	One of the principal features of quantum mechanics is that not all
the classical physical observables of a system can be simultaneously known, 
either in practice or in principle.  Instead, there may be several sets of 
observables which give qualitatively different, but nonetheless complete
(maximal possible) descriptions of a quantum mechanical system.  These sets 
are sets of "good quantum numbers," and are also known as "maximal sets of 
commuting observables."  Observables from different sets are "noncommuting 
observables."  

	A well known example of noncommuting observables is position and
momentum.  You can put a subatomic particle into a state of well-defined
momentum, but then you cannot know where it is - it is, in fact, everywhere
at once.  It's not just a matter of your inability to measure, but rather,
an intrinsic property of the particle.  Conversely, you can put a particle
in a definite position, but then its momentum is completely ill-defined.
You can also create states of intermediate knowledge of both observables:
If you confine the particle to some arbitrarily large region of space,
you can define the momentum more and more precisely.  But you can never 
know both, exactly, at the same time.

	Position and momentum are continuous observables.  But the same
situation can arise for discrete observables such as spin.  The quantum
mechanical spin of a particle along each of the three space axes is a set
of mutually noncommuting observables.  You can only know the spin along one
axis at a time.  A proton with spin "up" along the x-axis has undefined
spin along the y and z axes.  You cannot simultaneously measure the x and y
spin projections of a proton. EPR sought to demonstrate that this
phenomenon could be exploited to construct an experiment which would
demonstrate a paradox which they believed was inherent in the
quantum-mechanical description of the world. 

	They imagined two physical systems that are allowed to interact
initially so that they subsequently will be defined by a single Schrodinger
wave equation (SWE).   [For simplicity, imagine a simple physical 
realization of this idea - a neutral pion at rest in your lab, which decays
into a pair of back-to-back photons.  The pair of photons is described
by a single two-particle wave function.]  Once separated, the two systems
[read: photons] are still described by the same SWE, and a measurement of 
one observable of the first system will determine the measurement of the 
corresponding observable of the second system.  [Example:  The neutral pion
is a scalar particle - it has zero angular momentum.  So the two photons
must speed off in opposite directions with opposite spin. If photon 1 
is found to have spin up along the x-axis, then photon 2 *must* have spin
down along the x-axis, since the total angular momentum of the final-state,
two-photon, system must be the same as the angular momentum of the intial
state, a single neutral pion.  You know the spin of photon 2 even without
measuring it.] Likewise, the measurement of another observable of the first
system will determine the measurement of the corresponding observable of the
second system, even though the systems are no longer physically linked in
the traditional sense of local coupling. 

	However, QM prohibits the simultaneous knowledge of more than one
mutually noncommuting observable of either system.  The paradox of EPR is
the following contradiction:  For our coupled systems, we can measure
observable A of system I [for example, photon 1 has spin up along the
x-axis; photon 2 must therefore have x-spin down.] and observable B of 
system II [for example, photon 2 has spin down along the y-axis; therefore
the y-spin of photon 1 must be up.] thereby revealing both observables for 
both systems, contrary to QM.  

	QM dictates that this should be impossible, creating the
paradoxical implication that measuring one system should "poison" any
measurement of the other system, no matter what the distance between
them. [In one commonly studied interpretation, the mechanism by which
this proceeds is 'instantaneous collapse of the wavefunction'.  But
the rules of QM do not require this interpretation, and several
other perfectly valid interpretations exist.]  The second system
would instantaneously be put into a state of well-defined observable A,
and, consequently, ill-defined observable B, spoiling the measurement.
Yet, one could imagine the two measurements were so far apart in
space that special relativity would prohibit any influence of one
measurement over the other.  [After the neutral-pion decay, we can wait until
the two photons are a light-year apart, and then "simultaneously" measure
the x-spin of photon 1 and the y-spin of photon 2.  QM suggests that if,
for example, the measurement of the photon 1 x-spin happens first, this
measurement must instantaneously force photon 2 into a state of ill-defined
y-spin, even though it is light-years away from photon 1. 

	How do we reconcile the fact that photon 2 "knows" that the x-spin
of photon 1 has been measured, even though they are separated by 
light-years of space and far too little time has passed for information
to have travelled to it according to the rules of Special Relativity?  
There are basically two choices.  You can accept the postulates of QM
as a fact of life, in spite of its seemingly uncomfortable coexistence 
with special relativity, or you can postulate that QM is not complete, 
that there *was* more information available for the description of the 
two-particle system at the time it was created, carried away by both 
photons, and that you just didn't know it because QM does not properly 
account for it.  

	So, EPR postulated that the existence of hidden variables, some 
so-far unknown properties, of the systems should account for the discrepancy. 
Their claim was that QM theory is incomplete; it does not completely
describe the physical reality.  System II knows all about System I 
long before the scientist measures any of the observables, thereby
supposedly consigning the other noncommuting observables to obscurity.
No instantaneous action-at-a-distance is necessary in this picture, 
which postulates that each System has more parameters than are 
accounted by QM. Niels Bohr, one of the founders of QM, held the opposite 
view and defended a strict interpretation, the Copenhagen Interpretation, 
of QM. 

	In 1964 John S. Bell proposed a mechanism to test for the existence
of these hidden parameters, and he developed his inequality principle as
the basis for such a test. 

	Use the example of two photons configured in the singlet state,
consider this:  After separation, each photon will have spin values for
each of the three axes of space, and each spin can have one of two values;
call them up and down.  Call the axes A, B and C and call the spin in the A
axis A+ if it is up in that axis, otherwise call it A-.  Use similar
definitions for the other two axes. 

	Now perform the experiment.  Measure the spin in one axis of one
particle and the spin in another axis of the other photon.  If EPR were
correct, each photon will simultaneously have properties for spin in each
of axes A, B and C. 

	Look at the statistics.  Perform the measurements with a number of
sets of photons.  Use the symbol N(A+, B-) to designate the words "the
number of photons with A+ and B-."  Similarly for N(A+, B+), N(B-, C+),
etc.  Also use the designation N(A+, B-, C+) to mean "the number of photons
with A+, B- and C+," and so on.  It's easy to demonstrate that for a set of
photons 

(1)    N(A+, B-) = N(A+, B-, C+) + N(A+, B-, C-)

because all of the (A+, B-, C+) and all of the (A+, B-, C-) photons are
included in the designation (A+, B-), and nothing else is included in N(A+,
B-).  You can make this claim if these measurements are connected to some
real properties of the photons. 

	Let n[A+, B+] be the designation for "the number of measurements of
pairs of photons in which the first photon measured A+, and the second
photon measured B+."  Use a similar designation for the other possible
results.  This is necessary because this is all it is possible to measure. 
You can't measure both A and B of the same photon.  Bell demonstrated that
in an actual experiment, if (1) is true (indicating real properties), then
the following must be true: 

(2)    n[A+, B+] <= n[A+, C+] + n[B+, C-].

	Additional inequality relations can be written by just making the
appropriate permutations of the letters A, B and C and the two signs.  This
is Bell's inequality principle, and it is proved to be true if there are
real (perhaps hidden) parameters to account for the measurements. 

    	At the time Bell's result first became known, the experimental
record was reviewed to see if any known results provided evidence against
locality. None did. Thus an effort began to develop tests of Bell's
inequality. A series of experiments was conducted by Aspect ending with one
in which polarizer angles were changed while the photons were `in flight'.
This was widely regarded at the time as being a reasonably conclusive
experiment confirming the predictions of QM. 

	Three years later Franson published a paper showing that the timing
constraints in this experiment were not adequate to confirm that locality
was violated. Aspect measured the time delays between detections of photon
pairs. The critical time delay is that between when a polarizer angle is
changed and when this affects the statistics of detecting photon pairs.
Aspect estimated this time based on the speed of a photon and the distance
between the polarizers and the detectors. Quantum mechanics does not allow
making assumptions about *where* a particle is between detections. We
cannot know *when* a particle traverses a polarizer unless we detect the
particle *at* the polarizer. 

	Experimental tests of Bell's inequality are ongoing but none has
yet fully addressed the issue raised by Franson. In addition there is an
issue of detector efficiency. By postulating new laws of physics one can
get the expected correlations without any nonlocal effects unless the
detectors are close to 90% efficient. The importance of these issues is a
matter of judgement. 

	The subject is alive theoretically as well.  In the 1970's 
Eberhard derived Bell's result without reference to local hidden variable 
theories; it applies to all local theories.  Eberhard also showed that the 
nonlocal effects that QM predicts cannot be used for superluminal 
communication.  The subject is not yet closed, and may yet provide more 
interesting insights into the subtleties of quantum mechanics.

REFERENCES:

1.  A. Einstein, B. Podolsky, N. Rosen:  "Can quantum-mechanical 
description of physical reality be considered complete?"
Physical Review 41, 777 (15 May 1935).  (The original EPR paper)

2.  D. Bohm:  Quantum Theory, Dover, New York (1957).  (Bohm
discusses some of his ideas concerning hidden variables.)

3.  N. Herbert:  Quantum Reality, Doubleday.  (A very good
popular treatment of EPR and related issues)

4.  M. Gardner: Science - Good, Bad and Bogus, Prometheus Books. 
(Martin Gardner gives a skeptics view of the fringe science
associated with EPR.)

5.  J. Gribbin:  In Search of Schrodinger's Cat, Bantam Books. 
(A popular treatment of EPR and the paradox of "Schrodinger's
cat" that results from the Copenhagen interpretation)

6.  N. Bohr:  "Can quantum-mechanical description of physical
reality be considered  complete?" Physical Review 48, 696 (15 Oct
1935).  (Niels Bohr's response to EPR)

7.  J. Bell:  "On the Einstein Podolsky Rosen paradox" Physics 1
#3, 195 (1964).

8.  J. Bell:  "On the problem of hidden variables in quantum
mechanics" Reviews of  Modern Physics 38 #3, 447 (July 1966). 

9.  D. Bohm, J. Bub:  "A proposed solution of the measurement
problem in quantum  mechanics by a hidden variable theory"
Reviews of Modern Physics 38  #3, 453 (July 1966).

10.  B. DeWitt:  "Quantum mechanics and reality" Physics Today p.
30 (Sept 1970).

11.  J. Clauser, A. Shimony:  "Bell's theorem: experimental
tests and implications" Rep.  Prog. Phys. 41, 1881 (1978).

12.  A. Aspect, Dalibard, Roger:  "Experimental test of Bell's
inequalities using time- varying analyzers" Physical Review
Letters 49 #25, 1804 (20 Dec 1982).

13.  A. Aspect, P. Grangier, G. Roger:  "Experimental realization
of Einstein-Podolsky-Rosen-Bohm gedankenexperiment; a new
violation of Bell's inequalities" Physical  Review Letters 49
#2, 91 (12 July 1982).

14.  A. Robinson: "Loophole closed in quantum mechanics test"
Science 219, 40 (7 Jan 1983).

15.  B. d'Espagnat:  "The quantum theory and reality" Scientific
American 241 #5 (November 1979).

16. "Bell's Theorem and Delayed Determinism", Franson, Physical Review D,
pgs. 2529-2532, Vol. 31, No. 10, May 1985.

17. "Bell's Theorem without Hidden Variables", P. H. Eberhard, Il Nuovo 
Cimento, 38 B 1, pgs. 75-80, (1977).

18. "Bell's Theorem and the Different Concepts of Locality", P. H. 
Eberhard, Il Nuovo Cimento 46 B, pgs. 392-419, (1978).

********************************************************************************
Item 32.

Some Frequently Asked Questions About Virtual Particles 
-------------------------------------------------------
                                                       original By Matt McIrvin

Contents:

     1. What are virtual particles?
     2. How can they be responsible for attractive forces?
     3. Do they violate energy conservation?
     4. Do they go faster than light?  Do virtual particles contradict
        relativity or causality?
     5. I hear physicists saying that the "quantum of the gravitational
        force" is something called a graviton.  Doesn't general
        relativity say that gravity isn't a force at all?

1. What are virtual particles?

	One of the first steps in the development of quantum mechanics was
Max Planck's idea that a harmonic oscillator (classically, anything that
wiggles like a mass bobbing on the end of an ideal spring) cannot have just
any energy.  Its possible energies come in a discrete set of equally spaced
levels. 

	An electromagnetic field wiggles in the same way when it possesses
waves.  Applying quantum mechanics to this oscillator reveals that it must
also have discrete, evenly spaced energy levels.  These energy levels are
what we usually identify as different numbers of photons. The higher the
energy level of a vibrational mode, the more photons there are.  In this
way, an electromagnetic wave acts as if it were made of particles.  The
electromagnetic field is a quantum field. 

	Electromagnetic fields can do things other than vibration.  For
instance, the electric field produces an attractive or repulsive force
between charged objects, which varies as the inverse square of distance. 
The force can change the momenta of the objects. 

	Can this be understood in terms of photons as well?  It turns out
that, in a sense, it can.  We can say that the particles exchange "virtual
photons" which carry the transferred momentum.  Here is a picture (a
"Feynman diagram") of the exchange of one virtual photon. 

                    \                /
                     \   <- p       / 
                      >~~~         /          ^ time
                     /    ~~~~    /           |
                    /         ~~~<            |
                   /              \            ---> space
                  /                \

	The lines on the left and right represent two charged particles,
and the wavy line (jagged because of the limitations of ASCII) is a virtual
photon, which transfers momentum from one to the other.  The particle that
emits the virtual photon loses momentum p in the recoil, and the other
particle gets the momentum. 

	This is a seemingly tidy explanation.  Forces don't happen because
of any sort of action at a distance, they happen because of virtual
particles that spew out of things and hit other things, knocking them
around.  However, this is misleading.  Virtual particles are really not
just like classical bullets. 

2. How can they be responsible for attractive forces?

	The most obvious problem with a simple, classical picture of
virtual particles is that this sort of behavior can't possibly result in
attractive forces.  If I throw a ball at you, the recoil pushes me back;
when you catch the ball, you are pushed away from me.  How can this attract
us to each other?  The answer lies in Heisenberg's uncertainty principle. 

	Suppose that we are trying to calculate the probability (or,
actually, the probability amplitude) that some amount of momentum, p, gets
transferred between a couple of particles that are fairly well- localized. 
The uncertainty principle says that definite momentum is associated with a
huge uncertainty in position.  A virtual particle with momentum p
corresponds to a plane wave filling all of space, with no definite position
at all.  It doesn't matter which way the momentum points; that just
determines how the wavefronts are oriented.  Since the wave is everywhere,
the photon can be created by one particle and absorbed by the other, no
matter where they are.  If the momentum transferred by the wave points in
the direction from the receiving particle to the emitting one, the effect
is that of an attractive force. 

	The moral is that the lines in a Feynman diagram are not to be
interpreted literally as the paths of classical particles.  Usually, in
fact, this interpretation applies to an even lesser extent than in my
example, since in most Feynman diagrams the incoming and outgoing particles
are not very well localized; they're supposed to be plane waves too. 

3. Do they violate energy conservation?

	We are really using the quantum-mechanical approximation method
known as perturbation theory.  In perturbation theory, systems can go
through intermediate "virtual states" that normally have energies different
>from that of the initial and final states.  This is because of another
uncertainty principle, which relates time and energy. 

	In the pictured example, we consider an intermediate state with a
virtual photon in it.  It isn't classically possible for a charged particle
to just emit a photon and remain unchanged (except for recoil) itself.  The
state with the photon in it has too much energy, assuming conservation of
momentum.  However, since the intermediate state lasts only a short time,
the state's energy becomes uncertain, and it can actually have the same
energy as the initial and final states.  This allows the system to pass
through this state with some probability without violating energy
conservation. 

	Some descriptions of this phenomenon instead say that the energy of
the *system* becomes uncertain for a short period of time, that energy is
somehow "borrowed" for a brief interval.  This is just another way of
talking about the same mathematics.  However, it obscures the fact that all
this talk of virtual states is just an approximation to quantum mechanics,
in which energy is conserved at all times.  The way I've described it also
corresponds to the usual way of talking about Feynman diagrams, in which
energy is conserved, but virtual particles can carry amounts of energy not
normally allowed by the laws of motion. 

	(General relativity creates a different set of problems for energy
conservation; that's described elsewhere in the sci.physics FAQ.)

4. Do they go faster than light?  Do virtual particles contradict
   relativity or causality?

	In section 2, the virtual photon's plane wave is seemingly created
everywhere in space at once, and destroyed all at once.  Therefore, the
interaction can happen no matter how far the interacting particles are from
each other.  Quantum field theory is supposed to properly apply special
relativity to quantum mechanics.  Yet here we have something that, at least
at first glance, isn't supposed to be possible in special relativity: the
virtual photon can go from one interacting particle to the other faster
than light!  It turns out, if we sum up all possible momenta, that the
amplitude for transmission drops as the virtual particle's final position
gets further and further outside the light cone, but that's small
consolation.  This "superluminal" propagation had better not transmit any
information if we are to retain the principle of causality. 

	I'll give a plausibility argument that it doesn't in the context of
a thought experiment.  Let's try to send information faster than light with
a virtual particle. 

	Suppose that you and I make repeated measurements of a quantum
field at distant locations.  The electromagnetic field is sort of a
complicated thing, so I'll use the example of a field with just one
component, and call it F.  To make things even simpler, we'll assume that
there are no "charged" sources of the F field or real F particles
initially.  This means that our F measurements should fluctuate quantum-
mechanically around an average value of zero.  You measure F (really, an
average value of F over some small region) at one place, and I measure it a
little while later at a place far away.  We do this over and over, and wait
a long time between the repetitions, just to be safe. 

                                .
                                .
                                .
                                   ------X
                             ------
                      X------

                                                     ^ time
                                   ------X me        |
                             ------                  |
                  you X------                         ---> space

After a large number of repeated field measurements we compare notes.
We discover that our results are not independent; the F values are
correlated with each other-- even though each individual set of
measurements just fluctuates around zero, the fluctuations are not
completely independent.  This is because of the propagation of virtual
quanta of the F field, represented by the diagonal lines.  It happens
even if the virtual particle has to go faster than light.

	However, this correlation transmits no information.  Neither of us
has any control over the results we get, and each set of results looks
completely random until we compare notes (this is just like the resolution
of the famous EPR "paradox"). 

	You can do things to fields other than measure them.  Might you
still be able to send a signal?  Suppose that you attempt, by some series
of actions, to send information to me by means of the virtual particle. If
we look at this from the perspective of someone moving to the right at a
high enough speed, special relativity says that in that reference frame,
the effect is going the other way: 

           .
            .
             .

          X------
                 ------
                       ------X

            you X------                        ^ time
                       ------                  |
                             ------X me        |
                                                ---> space

Now it seems as if I'm affecting what happens to you rather than the
other way around.  (If the quanta of the F field are not the same as
their antiparticles, then the transmission of a virtual F particle
>from you to me now looks like the transmission of its antiparticle
>from me to you.)  If all this is to fit properly into special
relativity, then it shouldn't matter which of these processes "really"
happened; the two descriptions should be equally valid.

	We know that all of this was derived from quantum mechanics, using
perturbation theory.  In quantum mechanics, the future quantum state of a
system can be derived by applying the rules for time evolution to its
present quantum state.  No measurement I make when I "receive" the particle
can tell me whether you've "sent" it or not, because in one frame that
hasn't happened yet!  Since my present state must be derivable from past
events, if I have your message, I must have gotten it by other means.  The
virtual particle didn't "transmit" any information that I didn't have
already; it is useless as a means of faster-than-light communication. 

	The order of events does *not* vary in different frames if the
transmission is at the speed of light or slower.  Then, the use of virtual
particles as a communication channel is completely consistent with quantum
mechanics and relativity.  That's fortunate: since all particle
interactions occur over a finite time interval, in a sense *all* particles
are virtual to some extent. 

5. I hear physicists saying that the "quantum of the gravitational
   force" is something called a graviton.  Doesn't general relativity
   say that gravity isn't a force at all?

	You don't have to accept that gravity is a "force" in order to
believe that gravitons might exist.  According to QM, anything that behaves
like a harmonic oscillator has discrete energy levels, as I said in part 1.
 General relativity allows gravitational waves, ripples in the geometry of
spacetime which travel at the speed of light.  Under a certain definition
of gravitational energy (a tricky subject), the wave can be said to carry
energy.  If QM is ever successfully applied to GR, it seems sensible to
expect that these oscillations will also possess discrete "gravitational
energies," corresponding to different numbers of gravitons. 
	
	Quantum gravity is not yet a complete, established theory, so
gravitons are still speculative.  It is also unlikely that individual
gravitons will be detected anytime in the near future. 

	Furthermore, it is not at all clear that it will be useful to think
of gravitational "forces," such as the one that sticks you to the earth's
surface, as mediated by virtual gravitons.  The notion of virtual particles
mediating static forces comes from perturbation theory, and if there is one
thing we know about quantum gravity, it's that the usual way of doing
perturbation theory doesn't work. 

	Quantum field theory is plagued with infinities, which show up in
diagrams in which virtual particles go in closed loops.  Normally these
infinities can be gotten rid of by "renormalization," in which infinite
"counterterms" cancel the infinite parts of the diagrams, leaving finite
results for experimentally observable quantities. Renormalization works for
QED and the other field theories used to describe particle interactions,
but it fails when applied to gravity. Graviton loops generate an infinite
family of counterterms.  The theory ends up with an infinite number of free
parameters, and it's no theory at all.  Other approaches to quantum gravity
are needed, and they might not describe static fields with virtual
gravitons. 

********************************************************************************
END OF FAQ