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I am familiar with the tools that appear in (linear) perturbation theory for general relativity, that is namely that one writes:

$$g_{\mu \nu} = g^{(0)}_{\mu \nu} + \epsilon g^{(1)}_{\mu \nu} + \mathcal{O}(\epsilon^2) \tag{*}$$

Where $g^{(0)}_{\mu \nu}$ is typically assumed, and then one assumes a perturbation of some kind and solves some equations. Let us now perturb the background space-time manifold, $(M,\mathbf{g}^{(0)})$, by a (massless) scalar field, we have that the equations of motion for the scalar field $\Phi$ are given by the Klein-Gordon equation:

$$(\square + \xi R) \Phi := (\nabla_{\mu} \nabla^{\mu} + \xi R) \Phi = 0 \tag{**}$$

Where the covariant derivative is taken with respect to the background metric tensor $g^{(0)}_{\mu \nu}$. Suppose we can solve these.

How does one translate the solution for $\Phi$ into terms of the metric components $g^{(1)}_{\mu \nu}$? In the sense that we have considered a physical perturbation on the space-time and now the metric tensor field must be modified via $(*)$. What if we generalise a bit and consider a spin $\sigma$ perturbation; electromagnetic, Dirac or gravitational (via the Teukolsky equation)? Following the excellent review article of Kokkotas and Schmidt (http://arxiv.org/abs/gr-qc/9909058) he says that (on page 10): "The variation of the Einstein equations:

$$\delta G_{\mu \nu} = 4 \pi \delta T_{\mu \nu}$$

by assuming a decomposition into tensor spherical harmonics for each (in my notation) $g^{(1)}_{\mu \nu}$ of the form $\chi(t,r,\theta,\phi) = \sum_{\ell m} \chi_{\ell m}(r,t) Y_{\ell}^{m}(\theta,\phi)$ the perturbation is reduced to a single equation." I believe this immediatly in light of separation of variables and equation $(**)$ above. But they do not describe how the metric components in $g^{(1)}_{\mu \nu}$ are recovered. Is there a general procedure by which I can match a spin $\sigma$ perturbation solution (given that I can solve the EoM for the spin $\sigma$ field) with the metric?

Question: How does one match $\Phi$ in $(**)$ with $g^{(1)}_{\mu \nu}$? How do the quasi-normal modes in terms of $\omega$ explicitly couple to the metric?

Edit: I have re-written the question slightly in the hope that it is now more cogent.

Thanks!

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  • $\begingroup$ What do you mean by "translate"? The scalar field is something different than the metric, there is no direct relation between the two. In principle, you have to solve for both the perturbations of the metric and the scalar field independently. $\endgroup$ – Frederic Brünner May 15 '14 at 8:40
  • $\begingroup$ The scalar field that is introduced as a perturbation generates stress-energy for the space-time. The perturbed Einstein equations then say that the metric will have its behaviour modified. What I want to know is how exactly this happens. Sure, the scalar field abides by some laws of motion (in particular, the Klein-Gordon equation), but the space-time manifold must necessarily be modified too. Otherwise, you have the idea that the presence of a non-zero scalar field does not disturb the geodesics associated with $g_{\mu \nu}$, which is nonsense. $\endgroup$ – Arthur Suvorov May 15 '14 at 8:44
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    $\begingroup$ In light of your comments Frederic I have now re-written the question slightly. Thank you for your input $\endgroup$ – Arthur Suvorov May 15 '14 at 9:12
  • $\begingroup$ You are welcome! Another comment: you should make explicit what you mean by $\omega$. Furthermore, your question appears to be applicable not only to quasinormal modes, but linearized gravity in general. Whether you have quasinormal modes or not depends on the precise form of the spacetime and corresponding boundary conditions you impose. $\endgroup$ – Frederic Brünner May 15 '14 at 11:29
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After some searching I found a paper by Bini et al that explores these ideas. It can be found here (http://arxiv.org/abs/gr-qc/0609041) on the arXiv.

While they only explore spin-1 perturbations, I think there is enough in there to proceed.

For all of those who are interested, the main equations in Bini et al are eq. (303) and its derivation on page 34, along with equation (9) for motivation and the mechanics of the set up.

Any more comments would still be warmly welcomed!

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