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Dmytro Taranovsky

Date Started: March 16, 2001

Last Modified: May 21, 2001

This paper continues the
discussion on the structure of general relativity and presents the most
relevant applications such as gravitational waves. The paper does not cover
all applications, and does not cover electromagnetism. Its style is informal.
Reading it requires understanding of __General Relativity__ by Dmytro
Taranovsky or an equivalent.

**Section 1:** Additional Basic Concepts

The choice of 8π in

**G=**8π**T**

defines what is
meant by mass of amount 1. It is selected so that the gravitational constant
is equal 1. For fast unit conversion, you may want to note that 1=G/c^{2}≈7.425*10^{-28} m/kg.

(Meter is abbreviated 'm', kilogram--'kg', second--'s').

Some
physicists believe that space has an inherent stress energy tensor: Λ**g**,
where Λ is **cosmological constant**. Λ is very close to 0.
Cosmological constant is important primarily for the second derivative of the
size of the universe.

A
**generalized Lorentz transformation** is transformation under which if g=η
before the transformation, then g=η after the transformation. It can be
viewed as a rotation in space-time.

**Contravariant
derivatives** are written like covariant except that they appear as
superscripts rather than subscripts; they are obtained through raising the
index in the contravariant derivative:

a^{,u}≡g^{uv}*a_{,v}.

Metric is often represented as space-time interval in terms of the coordinate changes. For example, in

ds^{2}=‑dt^{2}+dx^{2}+dy^{2}+dz^{2},

**ds ^{2}
**represents the metric for the locally normal frame of reference. Note:
When coordinates are distinguished by name rather than index, superscripts are
in most cases interpreted as powers.

Two manifolds are called physically equivalent if a one-to-one transformation exists between the manifolds that preserves continuity and metric. Geometry of space-time is whether space-time is physically equivalent to a particular manifold.

Symbol
'<<' means in many times smaller. In daily life, most velocities are
<<1≈3*10^{8}m/s. That is
why speed is often expressed in terms of m/s as opposed to a real number.

4-dimensional
**Laplacian** is defined by the equation

⃞f≡f_{,u}^{,u}.

(Symbol ∇^{2} is usually
used instead of ⃞; however, in a locally inertial frame of reference, ∇^{2} is sometimes
used as ∂^{2}/∂(x^{1})^{2}+∂^{2}/∂(x^{2})^{2}+∂^{2}/∂(x^{3})^{2}).

For a non-interacting particle, rest mass is conserved.

A trajectory is called:

**time-like**
if ||u||<0,

**light-like**
if u is undefined and

**space-like**
if ||u||>0.

No time-like or
light-like trajectories are closed--no travel back in time. Lack of closure
allows defining **time**: a continuous function T (event→real number) that if A
causes B, then T(A)<T(B). *(If travel back in time would be possible, then
the following object could be set up: The object receives signal 0 or 1 and
sends itself (back in time) the opposite signal: If the object receives 0, it
sends 1, and thus receives 1--a contradiction; if it receives 1, it sends 0, and
receives 0--a contradiction. Since such object is contradictory, it does not
exist, and travel back in time is impossible.) *All objects of nonzero rest
mass follow time-like trajectories. All objects of zero rest mass (light has
zero rest mass) follow light-like trajectories.

**Section 2:** Linearization and Newtonian Approximation

A coordinate
system is called **nearly Lorentz** if

**g**=η+h
where h_{ab}<<η_{ab}.

h_{ab}
transforms like a tensor of rank (0, 2) in a generalized Lorentz
transformation. Solar system can be described accurately by a nearly Lorentz
coordinate system. For a nearly Lorentz coordinate system,

R_{abuv}
≈½*(h_{av,bu}
+ h_{bu,av} - h_{au,bv} - h_{bv,au}).

h^{u}_{b}
≡ η_{ua}* h_{ab}; h^{uv} ≡ η^{vb}
* h^{u}_{b};

_{}. (Schutz
204)

Coordinates can
always be selected to adhere to the **Lorenz gauge condition**:

_{}

(do not forget the summation). The linear approximation to Einstein field equation is

**⃞ _{}=-16πT**
(Schutz 205)

in such coordinates. The equation, being linear in terms of the metric, is more computationally friendly then the exact equation. Its use requires almost Cartesian coordinates system and thus absence of strong gravity.

When
velocities are <<1 (such as one thousand kilometers per second), _{}^{00}
is the dominant term of _{}and mass density (which is
approximately rest mass density),

ρ=T^{00},

is the dominant part of the stress-energy tensor.

⃞=(-∂^{2}/∂t^{2}+∇^{2}).

∂/∂t=∂/∂x^{i}*∂x^{i}/∂t=
v∇.

Since v<<1, -∂^{2}/∂t^{2}=v^{2}∇^{2} can be
ignored. Thus,

∇^{2}_{}^{00}=-16πρ.
(Schutz 206)

**Gravitational
potential** (a real number),

φ≡-_{}^{00}/4.

Thus, the non-relativistic equation for gravity is

∇^{2}φ=4πρ.

Through substitution, the metric can be shown to be

ds^{2}=-(1+2φ)dt^{2}+(1-2φ)(dx^{2}+dy^{2}+dz^{2}).
(Schutz 206)

For a spherically symmetric object with radius R<r,

φ=-M/r.

Thus, by linearity, for a continuous mass distribution,

_{}.

Non-interacting particles move so that

dp^{i}/dt=
‑p^{4}∂φ/∂x^{i}

(the equation does not assume that 3-velocity is small) where p is 4-momentum.

**Section 3:** Metric for static spherically symmetric cases

In
such cases, the field equation can often be solved exactly. The cases occur in
non-rotating stars, heavy compact objects (such as white dwarfs and neutron
stars), and black holes. The coordinates chosen are usually t (time), r (radius,
whose value is derived from the formula for area of the sphere), and Ω
(that covers the two angle coordinates on a sphere: dΩ^{2}=dθ^{2}+sin^{2}θ
dφ^{2}). Then, the metric is

ds^{2}=-f(r)dt^{2}+g(r)dr^{2}+
r^{2}dΩ^{2}. (Schutz 254)

If the entire mass (M) is at the origin,

f(R)=(1-2M/R) and

g(R)=1/(1-2M/R). (Schutz 258)

If R<2M
(f(R)<0 and g(r)<0), then all objects (including light) must move
inward. Such objects are soon destroyed; all that remains for external
observation is total mass, total electric charge, and total angular momentum.
Such object is called a **black hole**. A black hole (whose mass is
millions of times larger than that of the sun) is located in the center of most
galaxies including Milky Way (the galaxy in which Earth resides).

**Section 4:** Gravitational Waves

The field equation allows wavelike solutions
in empty space. Since **gravitational waves** have not been detected (as of
2001), the waves passing through the Earth are very weak. Weakness of
gravitational waves allows linear approximation. Gravitational waves are
created through acceleration of heavy objects, such as when two stars rotate
around each other at a small distance.

In free space,
the linearized equation (⃞_{}=-16πT) has a general
solution that is (an infinite but converging) sum of plane waves. For best
examination of gravitational waves, a nearly Lorentz frame of reference
satisfying Lorenz gauge condition and h^{a}_{a}=0 is used. A
plane wave has

_{},

where g^{ab}k_{a}k_{b}=0.

Note: g^{ab}k_{a}k_{b}=||k||=(k_{1})^{2}+(k_{2})^{2}+(k_{3})^{2}-(k_{4})^{2}.

Its **phase** is φ, **angular
frequency**--k_{4}, **frequency**--k_{4}/(2π), and its **wavelength**:

λ=sqrt((k_{1})^{2}+(k_{2})^{2}+(k_{3})^{2}),

which is equal its frequency.

For _{}, let x, y, and z be coordinates
of space such that the wave travels in z direction. Then,

A_{xx}=-A_{yy}, A_{xy}=A_{yx},
(Schutz 217)

and other components are zero. (Note: Here coordinate names rather than indexes identify the components.)

Using
the coordinate system selected, gravitational waves do not change locations of
non-interacting objects at rest. Instead, they distort distance between
objects through change in the metric. (Distance squared for locations that are
not far apart and that are separated by (∆x, ∆y, ∆z) is (∆x)^{2}+(∆y)^{2}+(∆z)^{2}+h_{xx}∆x∆x+
h_{yy}∆y∆y+ h_{zz}∆x∆x+ 2h_{xy}∆x∆y+2h_{xz}∆x∆z+2h_{yz}∆y∆z).
Note that gravitational waves do not affect interval in time in the chosen
coordinate system.

** **By
selectively choosing x and y coordinates, A_{xy} can be made to equal
0. The relative value of A_{xx} and A_{xy} is called **polarization**
of the wave. Polarization states whether the oscillation is along the
coordinate axes or the diagonals.

Gravitational waves can be detected by measuring oscillations in proper distance (based on the metric) of objects. They can also be measured by holding proper distance fixed (by, say, using a rod) and measuring the forces--to maintain fixed proper distance, objects have to accelerate.

Since gravitational waves cause force, they carry energy. Energy flux per unit of area of a plane gravitational plane wave is (in the direction of propagation)

F=Ω^{2}<h_{uv}h^{uv}>/(32π),
(Schutz 238)

Where Ω is frequency, and '<...>' indicates average over time.

Radiation of energy through gravitational waves causes binary stars to increase period of revolution through decreased distance. Such increase in period was detected.

**Section 5:** Additional Applications

Item 1: Gravitational Red shift.

In approximation in which gravitational potential is defined, light frequency changes approximately according to

υ′=(φ-φ′+1)υ,

that is as light escapes from gravity, its frequency decreases. Its energy is proportional to frequency. The frequency is also dependent on the coordinate system:

υ′=sqrt((1+v)/(1-v))υ,

where υ is frequency in the initial coordinate system, υ′ is frequency in the new coordinate system, and v is the component of 3-velocity of the new coordinate system towards the old. Historically, one of the confirmations of general relativity was that light of heavy stars had lower frequency than predicted from the dynamics of the stars.

Item 2: Cosmology:

**Cosmology** is the
theory about the development of the universe as a whole, and it is based on
solving Einstein Field Equation for the entire universe with certain simplistic
assumptions. Combined with observational data, the solutions tell that as time
increase, the size of the universe increases; the universe has a finite size in
space; and at certain time (12-15 billion years ago) the space was only one
point. Like other models, cosmology is relevant not because it suggests the
past, but because it allows for better analysis of the data about the present
and because it suggests the future.

According to observations, on the scale comparable to the size of the universe, matter is homogenous and isotropic (no preferred direction or position). The metric is approximately,

ds^{2}=-dt^{2}+R^{2}(t)*(dr^{2}/(1-k*r^{2})+r^{2}dΩ^{2}),
(Schutz 324)

where R(t) is size of universe at time t and where k depends only on time.

Unfortunately, much of the cosmology is uncertain, so the information in this item is subject to change.

Item 3: Miscellaneous Applications

**Frame
dragging**: When a body rotates, incoming objects are pressured to rotate
with the body, even if the incoming objects are in vacuum.

**General
Positioning System** (GPS) consists of satellites that send electromagnetic
signals. The signals arrive with delay because speed of light is 1, not
infinity. The delay allows the receiver to calculate its location up to
10meters. General theory of relativity is needed in these calculations.

Galaxies
sometimes act as **gravitational lenses** for faraway objects. (Because of
non-Euclidean geometry due to galaxies, angular size and intensity of the
object as viewed from Earth can be increased because of the galaxy.)

As predicted by classical gravity, a planet rotates around a star in an ellipse. As general relativity shows, however, the axis of the ellipse slowly rotates.

**Section 6:** Preface to Quantum Mechanics

Much
of the universe is described in terms of tensor fields. Can these fields be
split up into independent elements indefinitively? As was discovered in the
20^{th} century, no: All fields are quantized. The quantization
is essential for the structure of the universe. It is described by quantum
field theory and the standard model (to be covered in a later paper).

**Section 7:** Exercises

Please note: The exercises have not been tested for difficulty to the readers, and are not sufficient for some readers.

1. Fill in the missing steps in derivation of

ds^{2}=-(1+2φ)dt^{2}+(1-2φ)(dx^{2}+dy^{2}+dz^{2}).

_{}.

2. The coordinate system used for black holes does not work when R=2M. Find
and justify a coordinate system that covers the surface R=2M as well. (Note:
If you cannot solve this difficult problem, you can find such a coordinate
system in Schutz (292).)

3. For the black holes, prove that no coordinate system can include a point at
R=0. Thus, R=0 is not part of the manifold--it is a singularity.

4. a) Show that a plane wave is a
solution of the linear field equation for free space (⃞_{}=0).

b) Show that polarization so that A_{xy}=0 can be obtained through
rotation along z-axis.

**Section 8:** Readings about General Relativity

*Introduction to Tensor Calculus
and Continuum Mechanics* by Heinbockel provides an excellent introduction to
the mathematical background: conventions, definition of tensors, tensor
operations, metric, covariant differentiation, Riemann Christoffel tensor, and
other topics.

*A First Course in General
Relativity* provides a good introduction to general relativity. Almost half
of the book develops needed mathematical background.
Applications--gravitational waves, stars, black holes, and cosmology--are
included. The book is easy to read (but still requires careful studying and solving
problems).

The author used the following sources in learning general relativity:

Baez, John. *General Relativity
Tutorial*. 1996-1997. Online. Internet. Accessed: February 7,

2001. Available: http://mat.ucr.edu/home/baez/gr/gr.html.

Dunsby, Peter. *Tensors and
Relativity*. 1996-2000. Online. Internet. Accessed: February 22,

2001. Available: http://vishnu.mth.uct.ac.za/omei/gr.

Heinbockel, John H. *Introduction
to Tensor Calculus and Continuum Mechanics*. 1997.

Online. Internet. Accessed: February 12, 2001. Available:

http://www.math.odu.edu/~jhh/counter2.html.

Misner, Charles W., Thorne, Kip S.,
Wheeler, John Archibald. *Gravitation*. W. H. Freeman and

Company: San Francisco, 1973.

Schutz, Bernand F. *A First
Course in General Relativity*. Cambridge University Press:

Cambridge MA, 1986.

Waner, Stefan. *Introduction to
Differential Geometry and General Relativity*. 1998. Online.

Internet. Accessed: February 27, 2001. Available:

http://147.4.150.5/~matscw/diff_geom/tc.html.

Acknowledgement: Jay Fogleman spent several hours with me to help me understand the theory and to make my paper easy to read.

Notation specifications: Comma (not semicolon) is used for contravariant derivatives. Metric is selected so that ||U||=-1.