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SPACE AND TIME: Epilogue

EPILOGUE:

דורי נסע ונגלה מני כאהל רעי קפּדתּי כארג חיי מדלה יבצעני מיום עד-לילה תּשׁלימני

*My generation is at an end,
and it is rolled away from me, as a shepherd's tent.
My life is cut off, as by a weaver:
Whilst I was yet but beginning, he cut me off:
From morning even to night thou wilt make an end of me. *

Isaiah 38:12

Hermann Minkowski's personal world-line will be cut short in January of 1909; the thread of his ideas will continue unbroken. His address to the GDNÄ will produce a sensation; many German physicists will later speak of it as the beginning of relativity the research area. The geometrical elegance and metaphysical implications which Minkowski has found in this previously obscure branch of electrodynamics will also attract the attention of mathematicians, philosophers, and even the general public; 1909 will mark the start of the relativity craze in popular culture.

The heyday of four-dimensional physics will arrive in 1916, with Einstein's
theory of gravity; he will take the flat, non-Euclidean space-time of
Minkowski and add to it Riemann's notion of curvature. In 1918, Hermann
Weyl will publish a set of immensely influential lectures under the deliberately
Minkowskian title *Space, Time, Matter*; Arthur Stanley Eddington
will soon write both advanced textbooks
and popular expositions of the new world-view
which (like Einstein's own) will remain among the best available even a century later.
In the 1920s seekers after
the unified field will take it for granted that all of physics can be obtained
from some appropriate generalisation of Minkowski's space-time, with
even more dimensions, perhaps, or with some exotic, antisymmetric way
of multiplying vectors.

There will be ups and downs in the road ahead. P. A. M. Dirac will unify
special relativity with quantum mechanics, but in a way owing more to
quantum theory's anti-pictorial bias than to Minkowski's vision. Physicists
of the "shut up
and calculate" post-war era will pay lip service to relativity, but
marginalise it as a research area and play down its philosophical
implications. Nevertheless, a great (but eccentric)
successor to Minkowski will arise, John Archibald Wheeler, prophet of
"geometrodynamics". The huge black paperback *Gravitation*
by Misner, Thorne, and Wheeler (known to relativists of my
generation as "the telephone book") will fail in its goal of making
general relativity a universal component of post-graduate education in
physics, but will at least manage to restore the subject's
intellectual cachet within the community. At the close of the
1900s Kaluza-inspired geometric unifications will be all the rage,
although they will invariably be presented as classical approximations to
some underlying -- and unvisualisable -- quantum-mechanical model.
In the 2000s ... too soon to say.

Graphic by Drschawrz

What, from the perspective of nearly a century, is the fundamental content of Minkowski's vision (for "flat", gravity-free space-time)? We might summarise it as follows:

Space-time is a continuum with four degrees of freedom. Three of these
correspond to space; the fourth to time. The three spatial dimensions
obey the usual laws of Euclidean geometry and plane trigonometry.
There are no preferred directions in space, so we can point the
*x*-axis anywhere we choose, with the following proviso: the
unit vector in the *x*-direction, when multiplied by itself (as an
inner product) must yield + 1. (If it yields a negative number,
we are pointing into the timelike sector of space-time, which the
*x*-axis should not do.)

Obviously to have an "unit" vector one must first
have some units: meters, inches, etc.
This will also apply to the time axis: seconds, minutes, etc. However,
if space and time are unified, the units should be the same on the spatial
axes as on the time axis. A (possibly) convenient choice is to measure everything
in light-years. For spatial distances this is no different than choosing any other
unit, such as nanometers. For time, we note that the speed of light is by
definition one light-year per year, so that *c* = 1, and *t*
years = *ct* light-years =* t* light-years, so we can measure time in years
and automatically get the correct result. (Of course, light-years may be
very big or very small units for a particular system, but the same is true
of meters and seconds.) Using this system of units, we can ignore
factors of *c*, which is a great simplification.

There is no natural centre to space,
so we can pick any point along the *x*-axis as our origin.
At the origin, we can choose any unit-vector which squares to
− 1 and call it the unit vector in the *t*-direction.
This leaves two-degrees of freedom, both space-like: the *yz*-plane.
Any two mutually-perpendicular vectors in this plane can be taken as
the unit *y* and *z* vectors.

Let us look for a moment at the *xy* plane (or any other plane
containing only space-like vectors). The geometry of this plane is
*Euclidean*, that is, the distance *r* between the origin
and the point (*x, y*) is given by the Law of Pythagoras:

Graphic by Mets501 and HereToHelp

Also, the laws of plane trigonometry apply. If θ is the
angle between the *x* axis and the vector pointing to
(*x, y*), then

1 = cos² θ + sin² θ

from which it follows that

sin² θ = tan² θ / (1 + tan² θ)

so that

If for some reason we chose to write *u* = tan θ, the formulæ
above give us *x* and *y* as functions of *r* and *u*:

Any vector **A** can be written in terms of its magnitude |A|
and direction θ

A

which could also be written in terms of tan θ, as before.

Graphic by
Dr. David P. Stern

Finally, we could choose some other set of axes *x′* and
*y′* in the same plane as alternative coordinates. If the *y′*
axis is tipped by an angle α, the *x′* axis will be
tipped down by the same amount. It is an elementary exercise in analytic geometry
to show that

Obviously, we could decide arbitrarily to write this in terms of
*v =* tan α using the same identities relating the
sine and cosine to the tangent as before; the result would be
a moderately complicated expression which the reader is encouraged to work out
independently.

Graphic by Philc 0780

So much for space. Now consider the *xt*-plane. According to Minkowski,
in this plane the geometry is *(hyperbolic) non-Euclidean*, meaning
that instead of the usual Pythagorean formula we must use

where τ is the proper time. By analogy with plane trigonometry
we introduce *hyperbolic trigonometry*:

Video by "Math is Power 4U" [7 minutes]

We write:

1 = cosh² θ - sinh² θ

from which it follows that

sinh² θ = tanh² θ / (1 - tanh² θ)

This is very like the ordinary Euclidean case, only with certain minus-signs in it.

Any vector **A** in the *xt*-plane
can be written in terms of its magnitude |A|
and (hyperbolic) direction θ

Notice that these components are in general *bigger* than
the magnitude, the opposite of what we are familiar with on the
Euclidean plane.

Let **P** be the relativistic momentum of a particle.
According to relativity, **P** has
magnitude *m* (in our units where *c* = 1).
Thus the *energy E = **P _{t}* =

Suppose the particle is moving in the *x*-direction with constant
speed *v*
light-years per year. If it started at the origin and has reached *x*
at time *t*, we have *v = x/t* = tanh θ. (Reassuringly,
the hyperbolic tangent is always less than 1, becoming 1 at θ = ∞,
so the speed of the particle is always less than light-speed.) We can insert
*v* into the formulæ for *x*-momentum and energy to obtain:

When *v* is very small, these become approximately:

The second formula is familiar from Newton's mechanics, as is the second
(kinetic energy) term of the first formula. The
other part of the first formula,
with *c* put back in it, is the famous *E = mc²*, which
is only true for an object at rest relative to the observer.

Wikimedia Commons

Finally, we could imagine a second *observer* moving with velocity
tanh α in the *x*-direction. Then we have new coordinates
*x′* and *t′* obeying the ("Lorentz transformation")
rule:

As before, the reader is urged to express this in terms of the
second observer's velocity. On a Minkowski diagram, both axes
tilt *toward* the line *x = t*.

To add curvature to a Euclidean plane, one merely weights the
various terms of the Pythagorean formula (and includes cross-terms).
Instead of
*r² = x² + y² *
one writes

for some functions *A, B, C*. In general relativity
one does the same for all the terms in the Minkowski
four-dimensional distance formula and identifies the
weights *A, B, ... J* with the gravitational potential.
(Of course this means there are ten different numbers needed to
describe gravity, instead of one as in Newton's theory.
Newton's potential turns out to be a combination of four of
Einstein's potentials; the other six are usually too weak to
have observable effects.)

Here we end our synopsis of relativity as seen from the geometric
point of view; those who might wish to explore it further
at an introductory level are encouraged
to read the Twentieth-Century classic
*Spacetime Physics* by Edward F. Taylor and
John Archibald Wheeler [San Francisco: Freeman, 1963].

J. A. Wheeler, 1981

TIME AND SPACE, by Hermann Minkowski