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Local energy of a gravitational wave is not gauge invariant?

  1. Aug 11, 2012 #1
    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?
  2. jcsd
  3. Aug 12, 2012 #2


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    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, hμν → hμν + ξμ,ν + ξν,μ.
  4. Aug 12, 2012 #3


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    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.
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