Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Fourier Methods in General Relativity

+ 4 like - 0 dislike
930 views

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)
asked Feb 16, 2012 in Theoretical Physics by Kyle (335 points) [ no revision ]
retagged Mar 18, 2014 by dimension10
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)

1 Answer

+ 4 like - 0 dislike

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)
answered Feb 16, 2012 by Sidious Lord (160 points) [ no revision ]
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)

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\varnothing$ysicsOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...