John Norton on Einstein's Move to Four-Dimensional Spacetime
Updated: Jan 19, 2019
This post is to supplement my response in Relativity without Spacetime (Section 184.108.40.206, pages 133–136) to John Norton's account of Einstein’s reasons for embracing four-dimensional spacetime. Norton plausibly argues (Section 6 of “What Was Einstein’s Principle of Equivalence?”) that once rigid frames break down in general relativity, the three-dimensional relative spaces of Einstein’s previous formulation must also break down, since non-rigid frames do not exhibit a well-defined three-dimensional geometry or frame time. Thus was Einstein forced to embrace Minkowski’s four-dimensional geometry. I argue in Relativity without Spacetime (2018) that three-dimensional relative spaces with well-defined geometries are necessary solely to the special gravitational fields associated with the Principle of Equivalence (PE), since the PE regards gravitational fields induced by the accelerated motion of rigid frames in three-dimensional finite Galilean regions. And, as Einstein notes, if we regard as fictitious the special fields associated with the PE, then “the key for understanding the equality of inertial and gravitational mass is missing” (Einstein’s letter to Laue, quoted on page 131 of Relativity without Spacetime). In particular, the gravitational field of the rigidly rotating disk is real, according to Einstein, and inhabits the three-dimensional space of the disk. Moreover, if the gravitational fields underwritten by the PE are real then the four-dimensional interpretation is untenable in general, since it would have to apply to all gravitational fields, including these PE fields. Thus, in general relativity there can be nothing against three-dimensional spaces, including those deprived of a well-defined geometry or frame time. We could well regard such undefined three-dimensional geometries as an element of conventionality in the theory of gravity, deriving from the conventionality of distant simultaneity in Einstein’s 1905 special relativity. Moreover, quite apart from the loss of three-dimensional relative spaces with well-defined geometries, as highlighted by Norton, once we are deprived of rigid reference bodies we are deprived reference bodies as such, since a reference body must have a state of motion as a whole. It is not as if, were there available non-rigid reference bodies (e.g., Einstein's "reference mollusk") with well-defined geometries and frame times, we would not need Gaussian coordinates. A "non-rigid reference body" is not a reference body.
My response above to Norton is correct as far as it goes, but it is not entirely satisfactory because any real three-dimensional space should exhibit a well-defined geometry, not just the special relative spaces associated with the PE. What is needed is a determinate distant simultaneity relation. I argue in Chapter 8 of Relativity without Spacetime that distant simultaneity should not be understood in terms of a frame time, since simultaneity is a topological relation and topological relations are frame-independent. In fact, global and non-frame-relative simultaneity is consistent with general relativity and sufficient to underwrite three-dimensional spaces with well-defined geometries. For the complete argument I refer the reader to Chapter 8 of Relativity without Spacetime.
Parenthetically, this is why I regard as misguided attempts to save global simultaneity based on a preferred cosmological rest frame defined in terms of the homogeneity and isotropy of the universe. Such a proposal could at best could furnish an average cosmological "now." Furthermore, it would be strange if a true present throughout the universe were determined by a factually contingent homogeneity and isotropy. That would suggest that if things had gone differently in the early universe there would be no cosmological present. Rather, the homogeneity and isotropy of the universe gives us an approximate cosmological metrical time, such that we can speak of things like the present age of the universe. If clocks ran at different rates in different parts of the universe, then the universe as a whole would not have an age. But global simultaneity, which is topological rather than metrical, is presupposed in our even raising such a question about metrical time (the present age of the universe). Finally, as regards the concept of a privileged rest frame in general relativity, the local gravitational field itself must be regarded as at rest in the sense that the trajectory of a free body is defined relative to the local gravitational field.