Fierz-Pauli action as an effective action from Einstein-Hilbert acition?

Question

The Fierz-Pauli action $$ S=\frac{1}{16 \pi G} \int d^{4} x\left[-\frac{1}{4} (\partial_{\rho} h_{\mu \nu})( \partial^{\rho} h^{\mu \nu}) + \frac{1}{2} (\partial_{\rho} h_{\mu \nu}) (\partial^{\nu} h^{\rho \mu}) + \frac{1}{4} (\partial_{\mu} h) (\partial^{\mu} h)-\frac{1}{2} (\partial_{\nu} h^{\mu \nu}) (\partial_{\mu} h)\right] $$ describes a spin-2 field. According to the literature (e.g. (Schwartz, 2013)), it should be the leading terms in expansion of the Einstein-Hilbert action. More precisely, if we perturbe the Minkowski spacetime, we'll obtain an effective action of the perturbation $h_{\mu\nu}$, which is exactly the Fierz-Pauli action.

My question is:

• How to expand the determinant of the metric to the second order?
• Are there any resources displaying all the details of the derivation? (I tried hard looking for them but failed. The literature and the Internet just told me it could be done but didn't show how.)

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Ref.

Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2013.

Answer 1

I've been looking for similar info as I'm interested in how to construct Lagrangians for higher spin fields. I came across this link. Hopefully you'll find it helpful too.

http://www.ugr.es/~bjanssen/text/fierz-pauli.pdf

Answer 2

Finally, I found all the details in T. Ortin's Gravity and Strings. Now I'm going to sketch the derivation.

We use $\bar g_{\mu\nu}$ to denote the background metric ($\eta_{\mu\nu}$ in my original question, but here it's generalized to the curved-background case), and $h_{\mu\nu}$ to denote the perturbation, leading to the perturbed metric $g_{\mu\nu}=\bar g_{\mu\nu}+h_{\mu\nu}$.

Firstly, the inverse metric and the determinant to second terms are: \begin{aligned} g^{\mu\nu} &= \bar g^{\mu\nu} - h^{\mu\nu} + \mathcal O(h^3)\\ \sqrt{|g|} &= \sqrt{|\bar g|}\left(1+\frac12 h +\frac18 h^2-\frac14 h_{\mu\nu}h^{\mu\nu}\right) + \mathcal O(h^3) \end{aligned} where $h^{\mu\nu} = \bar g^{\mu\alpha} \bar g^{\nu\beta} h_{\alpha\beta}$ is given by raising the indices of $h_{\mu\nu}$ by the background metric, and $h = \bar g^{\mu\nu} h_{\mu\nu}$ is the trace.

Then observe that, for the Levi-Civita connection, we can write the exact expression $$ \Gamma^{\rho}_{\mu\nu} = \bar \Gamma^{\rho}_{\mu\nu} + g^{\rho\sigma}\gamma_{\mu\nu\sigma},\ \gamma_{\mu\nu\sigma} = \frac12 (\bar\nabla _\mu h_{\nu\sigma}+\bar\nabla _\nu h_{\sigma\mu}-\bar\nabla _\sigma h_{\mu\nu}). $$ Substitue it to the expressions of Riemann curvature tensor and finally Ricci scalar, and expand $g^{\rho\sigma}$ to the second order (or higher order if we want to include interactions).

At last, substitute everything into the Einstein-Hilbert action, we get the Fierz-Pauli action.