Is an isometry always a statement about the principle of covariance?

1. Dec 2, 2008

quasar_4

Can someone help me out with this idea?

Let's say we have a diffeomorphism. We know that under certain circumstances (invariance of the metric) this diffeomorphism is an isometry. Here is the part I'm not sure about. Is an isometry always just a statement about the principle of covariance? I.e., under an isometry, do the laws of physics always look identical in any coordinate system? Or is the isometry a necessary, but not sufficient condition, and there are more restrictions on the diffeomorphism?

2. Dec 2, 2008

Fredrik

Staff Emeritus
General covariance is just the idea that the laws of physics are tensor equations. Isometries don't have anything to do with it.

Special covariance in SR is the fact that if you consider the components of a tensor equation in an inertial frame and replace the symbols $g_{\mu\nu}$ that represent the components of the metric with the corresponding numbers, the result will look the same no matter what inertial frame you're using.

The connection to isometries is that the isometry group of Minkowski space is the Poincaré group. So there's exactly one isometry for each inertial frame and vice versa.

Last edited: Dec 2, 2008
3. Dec 2, 2008

quasar_4

Oh, that's right... I forgot about that.

I must be thinking of something else that was special about isometries... but... if they are only interesting in that they give us the Poincare group, then why bother to go looking for them? Is there any relation to symmetry and conservation laws, or are those completely different?

4. Dec 2, 2008

atyy

General covariance is very general (to the point of being trivial), and applies to Newtonian mechanics, Maxwell's equations, and General Relativity. It just means you can make the dynamical equations look the same under all coordinate transformations.

A solution of Einstein's equations consists of a manifold, a metric and a stress-energy tensor. You can create another solution by using a diffeomorphism to move the points, metric and stress-energy tensor to another manifold, ie. with regard to the metric, you use the diffeomorphism to create an isometry. Even in Newtonian physics the metric is a geometric coordinate-independent object that defines a physical situation, so it's no surprise that isometric situations should be physically the same.

Some people make a big deal about having to fix a gauge for unique evolution ("hole argument"), but that is apparently true for any differential equation that arises from geometry (Jaramillo et al, From Geometry to Numerics http://arxiv.org/abs/0712.2332)

The difference between Newtonian and Einstein gravity is not general covariance, but "no prior geometry", ie. metric and matter must be determined together (MTW; Giulini, Some remarks on the notions of general covariance and background independence http://arxiv.org/abs/gr-qc/0603087)