# Interpretation of the Einstein-Hilbert action

Everyone knows the famous Einstein-Hilbert action $S_{EH} = \int d^4x \sqrt{-g} R$. I'd like to know if, after we first explicit the Ricci scalar in terms of the metric, it could be possible to interpret each one of the terms as a kinetic or potential term, and even more read what could be the free propagator of the graviton, and the graviton-graviton vertexes. If yes, what are the form of those vertexes (how many? are some of them derivatives couplings? how many gravitons per vertex?)

Thanks.

• To clarify: are you asking what the Feynman rules for GR (based on the EH action, and known to be perturbatively unrenormalizable) would look like? – twistor59 Apr 23 '12 at 16:10
• In some way: yes. – toot Apr 23 '12 at 17:30
• The second derivative terms in R are perfect derivatives, the first derivative terms are a form of pseudo-stress-energy and they have kinetic-potential interpretation. The perturbation theory terms are straightforward to work out but practically hopeless--- there are too many vertices that are too complicated. Here supergravity and string theory are essential--- several graviton scattering amplitudes were computed from string theory first. There is a lot of recent literature on this, look up Dixon's recent papers. – Ron Maimon Apr 24 '12 at 8:52

## 1 Answer

The only reference I seem to have to this material is a review article by Duff[*] which states some results of calculations performed using the quantities:

$$\tilde{g}^{\mu\nu}\equiv\sqrt{g}g^{\mu\nu}$$ $$\tilde{g}_{\mu\nu}\equiv\frac{1}{\sqrt{g}}g_{\mu\nu}$$

The graviton field $h^{\mu\nu}$ is defined by a perturbation about flat space $$\tilde{g}^{\mu\nu}=\eta^{\mu\nu}+\kappa h^{\mu\nu}$$ together with the corresponding quantity $$\tilde{g}_{\mu\nu}=\eta_{\mu\nu}-\kappa h_{\mu\nu}+\kappa^2h_{\mu\alpha}h_{\alpha\nu}+...$$ (here $\kappa=\sqrt{16\pi G}$). The free graviton momentum space propagator (after some gauge fixing choices) looks like $$D_{\mu\nu\rho\sigma}(p^2)=\frac{1}{p^2}(\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho}-\eta_{\mu\nu}\eta_{\rho\sigma})$$

There are expressions for 3-point, 4-point etc vertices which look rather complicated.

ETA: I found this online reference. The treatment discussing the propagator is around equation (65) onwards. I suspect that there will be much more detail in the original papers of Feynman and de Witt, but I don't have access to them.

[*] M.J.Duff "Covariant Quantization" in Quantum Gravity-an Oxford Symposium. ed Isham, Penrose, Sciama. OUP 1975