〈 Berger | Dillon 〉

Curvature tensor nonzero

Curvature tensor of the surface or of spacetime though? Idk how one would measure the curvature tensor of the surface directly but I assume you could construct it from studying the curvature tensor of spacetime, as that’s much easier to observe.

It’s the exact same as GR but a different manifold

I know, but how do would you measure it locally? Even for spacetime, it seems that measuing curvature requires you to make use of nonzero intervals and therefore can only be expressed locally mathematically but not measured to be so.

It depends how you’re asking.. like practically or in principle? You could measure the radius by just measuring shadows

Practically, same question applies to spacetime: how could I measure the curvature of a physical manifold like spacetime or the surface of the earth locally? And by locally I mean at a point, rather than by extending geodesics through the manifold.

I assume something to do woth holonomy, but still, how would one measure this at a point?

You can’t measure it at a point, that’s what it means to be a locally flat space (as in GR)