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- TL;DR Summary
- Axiom or theorem, part two.

I brought up this subject here about a decade ago so this time I'll try to be more specific to avoid redundancy.

In chapter five of Bernard F. Schutz's A First Course In General Relativity, he arrives at the conclusion that in flat space the covariant derivative of the metric tensor is zero. This can be done very simply, it seems to me. Then in chapter six he argues that since Riemannian manifolds are locally flat, this relation holds there as well, and says no more about it — evidentally satisfied with this reasoning.

I have read elsewhere, however, (including here) that [itex] \nabla g = 0 [/itex] should be taken as an axiom instead since without it funny things happen, like the lengths of vectors and the angles between them changing when they're parallel transported, which quite reasonably seems like something to avoid.

So what I'm wondering is . . what are the premises of flat space that make [itex] \nabla g = 0 [/itex] true, such that there one does not have to bother positing it as a separate axiom? Obviously in Cartesian coordinates the coefficients of [itex] ds^2 = dx^2 + dy^2 + . . . [/itex] do not change as we move around a flat manifold, and equally obviously the basis vectors parallel transport, so the Christoffel symbols are all zero. But maybe there's something deeper to it? I'm just troubled by the idea that one can derive [itex] \nabla g = 0 [/itex] in flat space but outside of flat space — or at least outside of Riemannian manifolds — one should posit it as an axiom instead.

In chapter five of Bernard F. Schutz's A First Course In General Relativity, he arrives at the conclusion that in flat space the covariant derivative of the metric tensor is zero. This can be done very simply, it seems to me. Then in chapter six he argues that since Riemannian manifolds are locally flat, this relation holds there as well, and says no more about it — evidentally satisfied with this reasoning.

I have read elsewhere, however, (including here) that [itex] \nabla g = 0 [/itex] should be taken as an axiom instead since without it funny things happen, like the lengths of vectors and the angles between them changing when they're parallel transported, which quite reasonably seems like something to avoid.

So what I'm wondering is . . what are the premises of flat space that make [itex] \nabla g = 0 [/itex] true, such that there one does not have to bother positing it as a separate axiom? Obviously in Cartesian coordinates the coefficients of [itex] ds^2 = dx^2 + dy^2 + . . . [/itex] do not change as we move around a flat manifold, and equally obviously the basis vectors parallel transport, so the Christoffel symbols are all zero. But maybe there's something deeper to it? I'm just troubled by the idea that one can derive [itex] \nabla g = 0 [/itex] in flat space but outside of flat space — or at least outside of Riemannian manifolds — one should posit it as an axiom instead.