My crude understanding of GR in outline is that spacetime curvature is described by the way the components of the Riemann tensor vary from point to point in spacetime, that such variation is controlled by Einstein's field equations, and that the source of curvature is the energy-momentum tensor together with a possible cosmological constant.(adsbygoogle = window.adsbygoogle || []).push({});

In some respects this outline has vague similarities to a description of how a solid deforms elastically in response to stress. In linear elasticity strains are described by the variation of strain tensor components, controlled by a tensor form of Hooke's law, and sourced by the stress tensor.

Now in the case of a strained solid there are "extra" restrictions on how the components of the strain tensor may vary from point to point. They must do so in a way that does not cause one element of matter to be displaced into a region already occupied by another element --- no matter overlap is allowed -- and also in such a way that holes do not open up in the solid. These are the so-called "conditions of compatibility"

Might there be different but nevertheless similarly "extra" restrictions on the variation of Riemann components in the case of GR?

For instance, for the microscopic laws of physics to be invariant under translations, as we find them, might one require that curved space-(time) sections remain isotropic (on a small enough scale) under translations?

If this were the case one might expect the concomitant spacetime transformations to beconformal, hence arriving at a modified form of SR and GR, eg. de Sitter relativity. Any comments?

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# Restrictions on Riemann components

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