diffeomorphism and constant curvature

Sam Gralla

I guess my explanation didn't help. I'll try more a mathematical one. A diffeo maps a points on a manifold to other points. Let's say we have points P1, P2, and P3, and our diffeo is set up to map P1 to P2, P2 to P3, and P3 to P1. By definition, the action of the diffeo on a scalar field (such as the Ricci scalar) is to make the value at P2 be *what it used to be* at P1, and so on.

So if we have the Ricci scalar being 5 at P1, 10 at P2, and 15 at P3, then after the diffeo it is 15 at P1, 5 at P2, and 10 at P3. That's all there is to a diffeomorphism. Being "diffeomorphism invariant" just means this property holds for your scalar field. Diffeomorphism invariance is pretty trivial.

My interpretation of your confusion is that you think a diffeomorphism means just to compare the value at P2 with P1, and that diffeomoprhism invariance would then mean that the values must be equal. But diffeomorphisms tell you to compare the value at P2 with what used to be the value at P1.

It's tricky to explain, but extremely simple.

tom.stoer

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

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

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

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

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.