I am looking for some references which discuss Fourier transform methods in GR. Specifically supposing you have a metric $g_{\mu \nu}(x)$ and its Fourier transform $\tilde{g}_{\mu \nu}(k)$, what does this tell you about the Fourier transform of the inverse metric $\tilde{g}^{\mu \nu}(k)$ or the Riemann tensor $\tilde{R}^{\mu}{}_{\nu \rho \sigma}(k)$. There are some obvious identities you can derive and I am looking for a references which discusses these and says if they are useful or not.

An example of what I mean is the following identity:

$g^{\mu \alpha}(x)g_{\alpha \nu}(x) = \delta^{\mu}_{\nu} \implies (\tilde{g}^{\mu \alpha} \ast \tilde{g}_{\alpha \nu})(k) = \delta^{\mu}_{\nu} \delta^{4}(k)$. (This can be made sensible on a compact manifold or for metrics which are asymptotically flat etc.)

  • $\begingroup$ You'd need the soacetime to have extra structure. For example if it is a homogenous space, then you can do harmonic anlysis on it. For example the flat Minkowski spacetime, it is even a group. But I don't think you can do much on a general Lorentzian manifold. $\endgroup$ – MBN Feb 17 '12 at 11:45
  • $\begingroup$ I think you just need to specify a global topology and go from there. Assuming the space is simply connected you really only have a few choices (in four dimensions). Spatially closed, open, or flat, and temporally closed, open (or flat???). For the globally flat case-standard type of fourier transform. The closed case is a more generalized Fourier transform (you can Find with the Peter-Weyl theorem). The open case I don't believe is compact so I don't know what you can do? $\endgroup$ – R. Rankin Jan 4 at 12:54

I would have added this as a comment, but I don't have enough reputation.

The Fourier transform is not a terribly useful thing to do on a generic GR background. In flat space the Fourier transform is useful because we have translation symmetry and momentum is conserved. But in a generic solution of Einstein's equations, the are no such symmetries (or Killing vectors).

As a consequence of this, most of the time when doing QFT on a curved space one works in position space, not in momentum space.

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    $\begingroup$ Yes a Fourier decomposition will be dependent on the coordinate system you choose, and yes without any symmetries then there aren't obvious coordinate choices. Nevertheless, it isn't clear that this means there is absolutely no value in studying such things. You may not be able to assign any "physical" meaning to frequencies etc., and this can be problematic for QFT. In this, I'm more interested in studying classical GR than QFT on a fixed background. But thanks for your comment/answer. $\endgroup$ – Kyle Feb 16 '12 at 6:53
  • $\begingroup$ What do you have in mind for the Fourier transform of the flat metric? How would you ascribe a metric interpretation to the resulting object? It is rare to have a coordinate-dependent method of general import. If you do something on a single patch, you at least need boundary conditions to ensure that your construction does not interact with other patches -- but then it's unlikely that your Fourier transforms would respect the same boundary conditions. And if you restrict to rapidly decreasing functions, then your metric becomes degenerate at the boundary. $\endgroup$ – Eric Zaslow Feb 16 '12 at 18:47
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    $\begingroup$ Well for instance you could write the metric $g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$ with $h_{\mu \nu}$ not necessarily a perturbation, falling off rapidly enough at infinity. Then in a sense $\tilde{\eta}_{\mu \nu} = diag\{-\delta(k),\delta(k),\delta(k),\delta(k)\}$. And then study $\tilde{h}_{\mu \nu}$. Or you could say look at metrics on manifold with toroidal topology for instance... $\endgroup$ – Kyle Feb 16 '12 at 20:07

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