[Physics] What transformation is the metric of general relativity invariant under


My limited understanding of metrics comes from Cartan. From there, I understand that a metric is something invariant under certain transformations, e.g. Lorentz in special relativity. But with the metric varying from point to point (event-to-event), what is the meaning of the metric?

Best Answer

I don't like to think of general relativity as allowing more coordinate systems or transformations than special relativity, or even Newtonian physics. You can do Newtonian physics in any strange, silly coordinate system you can cook up. You can do special relativity in any strange coordinate system you want, too. However, you will discover that you need to introduce machinery that is usually introduced in the context of general relativity to do so. Those extra terms in the vector Laplacian in curvilinear coordinates are precisely the Christoffel symbols -- non-zero, even though space(time) is flat.

Coordinate independence is a basic, fundamental requirement of a sensible physical theory. The generalization of general relativity from special relativity lies not in that it allows more coordinate systems but in that it allows more geometries and for geometry to be a dynamical quantity. Galilean space and Minkowski spacetime come with privileged coordinate systems, inertial frames, a spacetime in general does not.

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