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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

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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:

r² = x² + y²

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

x = r cos θ
y = r sin θ
y/x = tan θ
1 = cos² θ + sin² θ

from which it follows that

cos² θ = 1 / (1 + tan² θ)
sin² θ = tan² θ / (1 + tan² θ)

so that

x = r / (1 + tan² θ)^½
y = r tan θ / (1 + tan² θ)^½

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:

x = r / (1 + u²)^½
y = r u / (1 + u²)^½

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

A_x = |A| cos θ
A_y = |A| sin θ

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

x' = x cosα – y sinα
y' = x sinα + y cosα

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

τ² = t² - x²

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:

t = τ cosh θ
x = τ sinh θ
x/t = tanh θ
1 = cosh² θ - sinh² θ

from which it follows that

cosh² θ = 1 / (1 - tanh² θ)
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

A_t = |A| cosh θ
A_x = |A| sinh 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 = m cosh θ and the x-component of momentum p_x = P_x = m sinh θ.

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:

E = m / (1 - v²)^½
p_x = mv / (1 - v²)^½

When v is very small, these become approximately:

E = m + ½mv²
p_x = mv

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:

t' = t coshα - x sinhα
x' = - t sinhα + x coshα

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

r² = Ax² + By² + Cxy

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