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Diffeomorphism and constant curvature

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tom.stoer
#19
Mar18-11, 02:15 AM
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Think about a surface like an ellipsoid with non-constant curvature. You can introduce coordinates with origin at the north pole; then you can shift the origin slightly which means you introduce a second coordinate system. Such coordinate systems can be related to each other using diffeomorphisms, but there is no reason why this should force the ellispoid to have constant curvature i.e. to "become" a sphere.
atyy
#20
Mar18-11, 03:23 AM
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P: 8,400
Perhaps jfy4 is using the (correct) terminology that a diffeomorphism moves points on the manifold only. The physics is not invariant under such an operation. Physically, this would mean that one can stand at a point on the blank manifold. However, this isn't possible, because a person is a collection of physical fields that include the metric, and once he is on the manifold, it is no longer blank.

To be very correct, the physics is invariant under a diffeomorphism on the manifold and the corresponding pullback of the metric. Physicists, perhaps sloppily, call this "diffeomorphism invariance" with the pullback implied.
atyy
#21
Mar18-11, 04:01 PM
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P: 8,400
Quote Quote by atyy View Post
Perhaps jfy4 is using the (correct) terminology that a diffeomorphism moves points on the manifold only. The physics is not invariant under such an operation. Physically, this would mean that one can stand at a point on the blank manifold. However, this isn't possible, because a person is a collection of physical fields that include the metric, and once he is on the manifold, it is no longer blank.

To be very correct, the physics is invariant under a diffeomorphism on the manifold and the corresponding pullback of the metric. Physicists, perhaps sloppily, call this "diffeomorphism invariance" with the pullback implied.
Ooops, please ignore "Physically, this would mean that one can stand at a point on the blank manifold. However, this isn't possible, because a person is a collection of physical fields that include the metric, and once he is on the manifold, it is no longer blank."
jfy4
#22
Mar21-11, 01:28 AM
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P: 647
I have here, an example from Rovelli's Quantum gravity on Diffeomorphism invariance. We can have a gravitational field ( lets say flat-space) and we act on it with a diffeomorphism. The result is a new gravitational field, that is, it is the flat-space gravitational field expressed in terms of new coordinates. But notice that it is still the flat-space metric.

Then by the definition of diffeomorphism, the gravitational field has been transplanted across a manifold to a new location, however, the gravitational field is still the flat-space metric. Then the curvature would be the curvature of the flat-space metric (none).

Then the manifold would be a space of constant curvature.

Does this example not work for all metrics? it seems true...
Sam Gralla
#23
Mar21-11, 10:09 AM
P: 95
No, that doesn't work for all metrics. It works for flat metrics, where the curvature is zero before (and after) the diffeomorphism. With a curved metric, the curvature will be non-zero before and after the diffeomoprhism.


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