Seminar on Spacetime, 2003 Fall
by Prof. Soshichi Uchii
Part 3 The Deep Structure of General Relativity
In Part 3, Barbour begins to survey two theories of relativity, and begins to point out some questions raised by Mach, Poncare and others, but neglected by Einstein and recent philosophers who followed Einstein's line. The relativity of simultaneity is well discussed, but what is duration, measured by a clock? And what is clock in the first place? Or for that matter, what is rod, for measuring length? They are simply assumed, independently from relativity theories, and Einstein himself was well aware of this deficiency. The talk of spacetime as fundamental entity may easily hide these questions.
Einstein, in his "Autobiographical Notes" (1949; paperback edition, Open Court,1979 is still available) says as follows:
It is striking that the theory (except for the four dimensional space) introduces two kinds of physical things, i.e., (1) measuring rods and clocks, (2) all other things, e.g., the electromagnetic field, material point, etc. This, in a certain sense, is inconsistent; strictly speaking, measuring rods and clocks should emerge as solutions of the basic equations (objects consisting of moving atomic configurations), not, as it were, as theoretically self-sufficient entities. The procedure justifies itself, however, because it was clear from the very beginning that the postulates of the theory are not strong enough to deduce from them equations for physical events sufficiently complete and sufficiently free from arbitrariness in order to base upon such a foundation a theory of measuring rods and clocks. If one did not wish to forego a physical interpretation of the cooordinates in general (something that, in itself, would be possible), it was better to permit such insonsistency---with the obligation, however, of eliminating it at a later stage of the theory. But one must not legitimize the sin just described so as to imagine that distances are physical entities of a special type, intrinsically different from other physical variables ("reducing physics to geometry," etc.).
Barbour is talking about Einstein's stance in this passage. Whereas Barbour's stance in this regard is quite different:
All that exists are things that change. What we call time is---in classical physics at least--- simply a complex of rules that govern the change. (Barbour, 137)
For Minkowski space, see: Minkowski Space and Minkowski Note
Some people may feel difficulties for understanding how we can construct Platonia for relativity theories (chapter 9). For instance, how does the Triangle Land look like for special relativity, usually expressed in terms of Minkowski Space? Here is an illustration (by myself). Suppose Alice and Bob are moving, relative to each other, at constant speed. But each has a right to assume that she/he is at rest, and they can construct the same sort of spacetime diagram, as follows:
Now, Barbour's point is that even in special relativity, 3-dimensional configurations are indispensable; for, unless we can identify hyperplanes of simultaneity (i.e., a 3-dimensional space, but in our illustration, reduced to a 2-dimensional plane), Einstein's theory would not work. Thus, although triangles (formed of 3 particles) are different for Alice and Bob, they can be easily represented in the same Triangle Land. However, unlike the case of classical mechanics, we have to take into consideration that Alice's and Bob's triangles are related (one by one) by means of Lorentz transformation. And you may recall that Lorentz transformation can be derived from the two assumptions of special relativity. Thus Einstein's special relativity can be regarded as giving sufficient means for relating, in conformity with laws of nature, Alice's and Bob's triangles. In a word, Barbour has no difficulty for extending his idea from classical to relativistic physics.
But what does Barbour mean by the "deep structure" of general relativity? There still seems to be a great gap between special and general relativity. After a few digressions, Barbour comes to this question in chapter 11. In a word, general relativity can be interpreted as a dynamical theory of 3-dimensional geometry (Riemannian geometry): a 3 space develops according to a law, and general relativity gives that law. This idea was developed by Dirac, Wheeler, and Wheeler's collaborators, and another name is geometrodynamics. For Wheeler and his company's version, see Misner, Thorne, and Wheeler's famous Gravitation (1973). But Barbour wants to emphasize the timeless character of this theory in his own manner. Just as Newtonian mechanics can be reconstructed in Platonia (as a dynamics of relative configurations), so can be general relativity as a dynamics of 3-dimensional geometry (space, i.e. a relative configuration, in essense); thus no need for time.
See Evolution of 3-Space, The Machian Problem; for Einstein's struggle for general relativity, see The Genesis of General Relativity.
