Gravitational Wave Local Energy: Not Gauge Invariant?

[alemsalem]

I'm reading wald page 85, and he defines a stress-energy tensor for the linearized gravitational field. he mentions that it not gauge invariant as a problem. but isn't that a general property of any tensor (except scalars). so any stress-energy tensor will not be gauge invariant (change of coordinates).
is it because we're using η and not the full metric?

 

[Bill_K]

When you're dealing with the linearized approximation, you have two metrics: the flat pace metric and the curved space metric. In this context, a gauge transformation refers to a transformation that alters the relationship between the two. In infinitesimal form, h_μν → h_μν + ξ_μ,ν + ξ_ν,μ.

 

[bcrowell]

If you think it might be helpful to get a different presentation of the same thing, Carroll has a discussion on p. 162: http://arxiv.org/abs/gr-qc/?9712019

Although it's true that any tensor changes with a change of coordinates (in the sense that its representation in those coordinates is different), tensors do have a couple of important properties: (1) a change of coordinates never changes a zero tensor to a nonzero tensor, and (2) you can form scalars by contracting indices on tensors.

I think what Wald and Carroll have in mind as being objectionable about the stress-energy "tensor" t amounts to saying that it fails to have these properties.

You typically only get meaningful results from t by averaging over a region of space that is large compared to a wavelength, and the failure of #1 clearly shows what could go wrong if you didn't. As a non-fancy way of discussing this, just imagine trying to calculate the local energy density of a gravitational wave from the square of the ordinary gravitational field vector g from freshman physics. We know this can't be right, by the equivalence principle, since we can always make g=0 at any given spacetime location by going into a free-falling frame.

As an example of why #2 is important, see Carroll's discussion on p. 169. To get the total energy radiated to infinity, he has to form something that isn't a scalar according to the grammar of index gymnastics, and then a separate calculation is required in order to show that this thing really is gauge-invariant.