General relativistic time

Space-time versus spacetime

Over the millennia, humanity’s view of time has changed in a series of steps, each stimulated by a paradigm shift, or way of looking at the universe. Aristotelian space-time A4 was followed by Galilean-Newtonian space-time N4 and then by Minkowski spacetime M4. We discussed these in earlier chapters. In this chapter we discuss the step after M4, which we will refer to as general relativistic (GR) spacetime, denoted by G4.

An important difference between G4 and the other models is that they are specific in their mathematical structures, whereas G4 is really a class of spacetimes, each with its characteristic metric or distance structure. There is, unfortunately, no unique GR spacetime that we can say for sure models the universe. There are two reasons for this. First, each GR spacetime is dynamically coupled to the local distribution of energy and mass via Einstein’s field equations [Einstein, 1915a], so each different distribution of energy and mass leads to a different G4 spacetime. Second, Einstein’s field equations, being local, do not say anything about the topology, or large-scale structure, of spacetime: we don’t know whether it is spherical, flat, or hyperbolic at very large distances, or even like a vast four-dimensional doughnut with many holes. Recent astrophysical data suggests that at the largest observable spatial distances it is flat [WMAP, 2013], but we can have no idea what it is like beyond those distances.

Another, more disturbing, point is that whilst it could be said that the spacetime we are in now is well-defined locally, that is no more than a classical approximation to what is surely a much different reality [Donoghue, 1994], in the same way that local temperature is a simple indicator of thermal equilibrium in our unimaginably complex local environment. It surely would be hubris of the worse kind to image that we know what ‘reality’ is really like. In Chapter 27 we discuss Snyder’s quantized spacetime theory, a theory that suggests that a geometrical model is not the last word on the nature of time and space.

To understand G4, it helps to understand M4. There is good evidence [Petkov, 2012] that Hermann Minkowski [1864-1909] was thinking about a unified, absolute four-dimensional spacetime structure well before Einstein came around to

Images of Time. First Edition. George Jaroszkiewicz.

© George Jaroszkiewicz 2016. Published in 2016 by Oxford University Press.

a geometric perspective. For some reason, Arnold Sommerfeld in 1907 ‘. . . was unable to resist rewriting Minkowski’s judgement of Einstein’s formulation of the principle of relativity.’51 Sommerfeld ‘. . . also suppressed Minkowski’s conclusion, where Einstein was portrayed as the clarifier, but by no means as the principal expositor, of the principle of relativity.’ [Petkov, 2012]. The evidence is that Minkowski alone understood the significance of the spacetime (no hyphen) perspective when he spoke at a scientific meeting in 1908 and made his ideas on spacetime public. At that talk, he famously declared:

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

[Minkowski, 1908]

He went on to propose that, not only should time and space be thought of as a single four-dimensional manifold, but that it should have a distance structure now known as the Minkowski metric. If At, Ax, Ay, and Az are the coordinate intervals between any two events in Minkowski spacetime, relative to coordinates in a given standard inertial frame, then Minkowski’s distance As between those events is given by a ‘line element’ As written in the form

where c is the speed of light. After 1908, the four-dimensional spacetime continuum of SR with the above line element was referred to by relativists as Minkowski spacetime (with no hyphen) and denoted M4.

A refinement of this concept is to consider infinitesimal coordinate displacements dx?“ = (dx0, dx1, dx2, dx3), where x0 = ct, x1 = x, x2 = y, and x3 = z.52 The choice of upper index is conventional. Then the M4 line element is given by (17.10).

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