# Calculating Forces via Feynman diagrams?

How would one go about calculating forces that test objects feel using Feynman diagram methods?

For example, say we have a massive object in GR so that the metric takes on the standard Schwarzschild form, $g^S_{\mu\nu}$. Now add some test matter in which is described by some lagrangian $\mathcal L'$ and has corresponding stress tensor $T'^{\mu\nu}$. In the action, the first order perturbation will then be $S\supset \int d^4x\, \sqrt{-g^S}\, h_{\mu\nu}T'^{\mu\nu}$ where $h_{\mu\nu}$ is the perturbation of the metric away from the Schwarzschild metric. I know how to calculate the force on the test particle in GR, but how would one do so by using field theory techniques like Feynman diagrams?

Relatedly, in some modified gravity theories you get some action like $\mathcal L\sim \frac{1}{2}(\partial\phi)^2+\mathcal{L}_{\rm int}(\phi)+\phi T/M_{\rm pl}$ where $\phi$ is related to the modification of gravity. If $\phi$ takes on a non-zero profile, say $\phi\sim1/r$, then this results in a fifth force on test particles, according to the literature. How do I see this by using Feynman diagrams?

I know that usually to figure out the potential generated by an interaction you can used the Born approximation, but I dont' quite see how that will work out in this case.

• You can try something along these lines : arxiv.org/abs/gr-qc/9310024 To get the leading order, solve the classical solution of the action, expend the action to second order in $h$, and do the gaussian integral over $h$. – Adam Dec 20 '13 at 15:34