# Fourier Methods in General Relativity

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.)

• 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. – MBN Feb 17 '12 at 11:45
• 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? – R. Rankin Jan 4 at 12:54

• 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... – Kyle Feb 16 '12 at 20:07