Does space actually curve?

lugita15

The other gauge forces -- electromagnetism, weak, and strong nuclear -- can be interpreted as curvatures, but not of ordinary space. Instead, the forces correspond to curvatures in the abstract symmetry space associated with each force: U(1) for EM, SU(2) for weak, and SU(3) for the strong force. This is rather far afield from the OP; I mention it here only for completeness.
Where I can find more information on this? It seems really interesting. Can gravity be thought of as the curvature of the Lie group of symmetries under diffeomorphisms?

bapowell

Where I can find more information on this?
Peskin and Schroeder has a section on the geometry of gauge invariance, as does Cheng and Li.

It seems really interesting. Can gravity be thought of as the curvature of the Lie group of symmetries under diffeomorphisms?
Gravity is thought of as the curvature of real space (on which the diffeo group acts), not as a curvature of the group space itself. I am not an expert on gauging gravity, but my understanding is that yes, the group of diffeomorphisms is the symmetry group of gravity. The affine connection in GR that facilitates parallel transport along geodesics in real space is analogous to the vector potential in electromagnetism, which serves the same function in U(1) space. Following the program outlined in the above texts, one finds that it is possible to derive gauge field analogs of the main formal geometric structures found in ordinary differential geometry. This is a beautiful result. Ultimately, the mathematics of gauge theories is rooted in the study of fiber bundles on manifolds. Studying these theories from this vantage point is exciting because you get to see in an clear way how gauge interactions and classical GR arise within the same mathematical framework.

lugita15

Peskin and Schroeder has a section on the geometry of gauge invariance, as does Cheng and Li.

Gravity is thought of as the curvature of real space (on which the diffeo group acts), not as a curvature of the group space itself. I am not an expert on gauging gravity, but my understanding is that yes, the group of diffeomorphisms is the symmetry group of gravity. The affine connection in GR that facilitates parallel transport along geodesics in real space is analogous to the vector potential in electromagnetism, which serves the same function in U(1) space. Following the program outlined in the above texts, one finds that it is possible to derive gauge field analogs of the main formal geometric structures found in ordinary differential geometry. This is a beautiful result. Ultimately, the mathematics of gauge theories is rooted in the study of fiber bundles on manifolds. Studying these theories from this vantage point is exciting because you get to see in an clear way how gauge interactions and classical GR arise within the same mathematical framework.
I was thinking more about quantum gravity, where we have a spin-2 gauge-bosonic field, and you have diffeomorphism invariance as a gauge symmetry. Then would gravity be representable as curvature of the diffeomorphism group?

bapowell

I was thinking more about quantum gravity, where we have a spin-2 gauge-bosonic field, and you have diffeomorphism invariance as a gauge symmetry. Then would gravity be representable as curvature of the diffeomorphism group?
I'm not sure that the distinction between quantum and classical gravity changes things. As I said, I'm not an expert on understanding gravity as a gauge theory (and hopefully someone else can chime in); however, I don't think it's correct to think of gravity as the curvature of the Lie group itself. In gauge theories, the field strength tensor describes the curvature of the connection -- the space of forms that transform under the adjoint representation of the gauge group. So it's the curvature of the space of gauge fields, not of the Lie group manifold itself.

Islam Hassan

Spacetime curvature was a very elegant (and accurate: in predicting other physical observations than itself) proposition until 2004 with the launch of Gravity Probe B:

http://en.wikipedia.org/wiki/Gravity_Probe_B

This experiment finally provided the physical proof that spacetime does indeed curve via measurement of spacetime curvature's geodetic and frame-dragging effects in the vicinity of the earth.

IH

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