Restrictions on Riemann components

[oldman]

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.

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 be conformal, hence arriving at a modified form of SR and GR, eg. de Sitter relativity. Any comments?

 

[pervect]

It sounds like you may be looking for the bianchi identities? See for example http://www.mth.uct.ac.za/omei/gr/chap6/node12.html

 

[oldman]

It sounds like you may be looking for the bianchi identities? See for example http://www.mth.uct.ac.za/omei/gr/chap6/node12.html

Thanks for the suggestion that the Bianchi identities might be the sort of condition I’m looking for. And thanks for the UCT reference.

But I don’t think the identities have anything to do with a requirement that transformations in GR should be angle and shape preserving on a local scale, that is to say should be conformal transformations. As far as I’m aware the Bianchi identities guided Einstein in formulating his tensor, and are involved with conservation of momentum and energy – importantly enough --- but they don’t look like the conditions I had in mind.

These (if such exist) should bear on the question of why there is a need to extend Einsteinian relativity to de Sitter relativity, as proposed recently by Aldrovandi and Pereira in http://arxiv.org/abs/0711.2274 (If their proposal has the considerable merit I think it has).