In Spacetime and Geometry, Dr. Carroll provides a Lagrangian for Einstein's equations in vacuum assuming that the metric can be written in the form $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$. The Lagrangian is, for reference, $$\mathcal{L}=\frac{1}{2}\left[\left(\partial_\alpha h^{\alpha\beta}\right)\left(\partial_{\beta}h\right)-\left(\partial_\alpha h^{\rho\sigma}\right)\left(\partial_{\rho}h^{\alpha}_{\;\,\sigma}\right)+\frac{1}{2}\eta^{\alpha\beta}\left(\partial_\alpha h^{\rho\sigma}\right)\left(\partial_\beta h_{\rho\sigma}\right)-\frac{1}{2}\eta^{\alpha\beta}\left(\partial_\alpha h\right)\left(\partial_\beta h\right)\right]$$

This, as can be verified, produces the Einstein tensor when varied. Now, later on, Dr. Carroll notes that by treating $h_{\mu\nu}$ as a field propagating over Minkowski spacetime, adding coupling to matter in the Lagrangian, and by requiring it to couple to its own energy-momentum tensor/matter energy-momentum tensor, General Relativity is restored.

The part I am confused about is as follows: how is one to add coupling to matter in the Lagrangian? I assume it's constructed from factors of $h_{\mu\nu}$ and not its derivative, but I'm not sure how to do this. Any assistance would be much appreciated.


The standard Pauli-Fierz Lagrangian density of the spin 2 field $h_{\mu\nu}$ is only the $[...]$ term, without the $1/2$ in front. The expected coupling to matter $\mathcal{L}_{\text{int}} \sim h_{\mu\nu}T^{\mu\nu}$ is "guessed" by Feynman in his lectures notes on gravitation (Lecture 3, page 42, Ed. of 1995).

Kraichnan (Special-Relativistic Derivation of Generally Covariant Gravitation Theory (Physical Review, Volume 98, Issue 4, 1955)), and Gupta (Gravitation and Electromagnetism (Physical Review, Volume 96, Issue 6, 1954)) also put this "by hand" in their articles. I also did not find a direct proof in any of the Deser and Wald articles on gravity.

The only solid proof of this linear coupling that I know of is given by Boulanger et al. in a perturbative cohomological set-up of Lagrangian BRST in Nucl.Phys. B597 (2001) 127-171 for a scalar field (section 9 of the arxiv draft). Of course, a full generality of matter coupling in the absence of own gauge invariance is inferred there, but it is exhibited for example after 20 pages of tedious calculation at the end of section 4 in the JHEP0502:016,2005. I quote as a reference to formula (104): <<Thus, the coupling between a Dirac field and one graviton at the first order in the deformation parameter takes the form $\Theta ^{\mu\nu}h_{\mu\nu}$.We cannot stress enough that is not an assumption, but follows entirely from the deformation approach developed here>>.


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