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I know that the Klein-Gordon equation in general relativity takes the form (a massless field)

$\nabla_\mu \nabla^\mu \phi=\sum_{a,b} \frac{1}{\sqrt{-g}}\partial_a(\sqrt{-g}g^{ab}\partial_b\phi) =0$

and that the spectrum of energies is $\omega_p=\sqrt{p^2+m^2}$, is it the same spectrum for curved space-time?

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    $\begingroup$ Why not plug in a plane wave and see what you get. $\endgroup$ Commented Mar 7, 2023 at 16:43
  • $\begingroup$ I read the section of Klein-Gordon in Peskin's book but I don't understand where to "insert" the metric $\endgroup$
    – TTT
    Commented Mar 7, 2023 at 16:47
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    $\begingroup$ You are missing the mass term in your KG equation, how could you infer it in omega? $\endgroup$
    – DanielC
    Commented Mar 7, 2023 at 20:05
  • $\begingroup$ I don't understand where to "insert" the metric. Where the $g$ and the $g^{ab}$ are. $\endgroup$
    – Ghoster
    Commented Mar 8, 2023 at 0:30
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    $\begingroup$ Is it the same spectrum for curved space-time? Do you have some reason to think that it should be? $\endgroup$
    – Ghoster
    Commented Mar 8, 2023 at 0:34

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No, the spectrum is different and in general depends on the global properties of the pseudo-Riemannian manifold. Questions you should ask yourself to better understand this:

  1. What exactly is the spectrum, and how to derive it from the e.o.m. in the Minkowski space-time case? Hint: the missing ingredient is the Fourier transform.
  2. Which part of this derivation doesn't extend to curved spacetime? What does it give instead?

Don't expect the result to be "nice". It should and will have a very complicated nonlocal dependence on the metric.

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