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I am interested in linearizing actions that are constructed out of geometrical objects. By this I mean perturbing the metric (or vielbein) and keeping in the action terms which are quadratic in the perturbation.

For the purpose of this question, let's consider the well known Einstein-Hilbert action, $$S_{\text{EH}}=\kappa\int\text{d}^4x\sqrt{-g}R~.$$ and perturb the metric around an arbitrary background, $$ \tilde{g}_{mn}=g_{mn}+h_{mn}~,$$ where $g_{mn}$ is the background metric and $h_{mn}$ is the perturbation, $|h_{mn}|\ll 1$.

As I said, we need to keep terms quadratic in the perturbation. It seems to me that this would require us to expand the scalar curvature $R$, and hence the Riemann tensor $R_{kpmn}$ to quadratic order. Expanding to linear order is not that bad, but expanding to quadratic order (particularly around an arbitrary background and not flat) is quite an arduous task. So I would like to know if there is an easier way.

We know that the equation of motion that ought to result from the linearized action is $$R_{ab}^{\text{lin.}}=0~.$$ So by expanding $R_{ab}$ to linear order (much easier than expanding to second order), we can then deduce that the variation of the action (w.r.t the perturbation) is of the form $$\delta S_{\text{EH}}^{\text{lin.}}=\kappa\int\text{d}^4x\sqrt{-g}\delta h^{ab}R_{ab}^{\text{lin.}}~.$$ This already gives us some information on what the action should look like when expanded to second order. But I am not sure where to take it from here, or if there is an even easier way to proceed.

Do you know of a shortcut to obtaining the quadratic expansion of the action? Is your method applicable to a broader range of action functionals (not just EH)? For illustrative purposes, it would be fine if an answer expands around a flat background instead.

Edit: See comments for a little more detail on what I am looking for.

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  • $\begingroup$ Are you looking to find the term proportional to (del h)^2, or variations of higher order in R? I think you mean the former but just want to be sure. What do you expect to get from the second variation? A measure of the stability of the system? $\endgroup$ – ggcg Nov 14 '18 at 3:05
  • $\begingroup$ Sorry, I will try to be more precise. If the EH action for the metric $\tilde{g}_{mn}$ is $$\tilde{S}_{\text{EH}}=\kappa\int\text{d}^4x\sqrt{-\tilde{g}}\tilde{R}~,$$ then I would like express this action in terms of the background metric $g_{mn}$ plus terms up to quadratic order in the perturbation $h_{mn}$. Such terms would come from expanding $\tilde{R}$ to quadratic order in $h_{mn}$, which will look something like $\tilde{R}=R+\mathcal{O}(h^2)$. $\endgroup$ – NormalsNotFar Nov 14 '18 at 3:16
  • $\begingroup$ As for my motivation, I am interested in applying this to higher spin theories in 3D. $\endgroup$ – NormalsNotFar Nov 14 '18 at 3:22
  • $\begingroup$ I see. So, it seems like a higher order gravitational wave. $\endgroup$ – ggcg Nov 14 '18 at 4:20
  • $\begingroup$ I am still considering gravitational waves; the equations of motion that will result are only linear in the perturbation (i.e. it is the same equation which describes gravitational waves). The problem is that if you want to derive them from an action principle (and not just linearize the non-linear EoM), then your action has to be quadratic in the perturbation. $\endgroup$ – NormalsNotFar Nov 14 '18 at 4:53
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I’m not aware of shortcuts, but equations 2.15-2.18 of this paper

https://arxiv.org/abs/hep-th/9411092v1

give the expansion of the Einstein-Hilbert action around a flat metric in de Donder gauge. Through quartic order, because they want to compute graviton-graviton scattering! At quadratic order there are only two terms:

$$-\frac{1}{4}\partial_\mu h \partial^\mu h+\frac{1}{2}\partial_\mu h^{\sigma\nu}\partial^\mu h_{\sigma\nu}$$

I suppose that computer algebra programs are a must for the cubic and quartic terms, but it looks like the quadratic terms might be doable by hand.

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