Fourier Methods in General Relativity

Question

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.)

This post has been migrated from (A51.SE)

Comments

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.

This post has been migrated from (A51.SE)

Answer 1

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.

This post has been migrated from (A51.SE)

Comments

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.

This post has been migrated from (A51.SE)

---

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.

This post has been migrated from (A51.SE)

---

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...

This post has been migrated from (A51.SE)