Gravitational scattering amplitudes from curved space

TL;DR

This paper develops a framework for computing gravitational scattering amplitudes in curved backgrounds, specifically Schwarzschild-Tangherlini spacetime, and demonstrates its consistency with flat-space results and known infrared behavior.

Contribution

It introduces a new perturbation theory for gravitational amplitudes in curved backgrounds and applies it to compute post-Minkowskian corrections to the Compton amplitude.

Findings

First and second post-Minkowskian contributions match flat-space results.

Second-order amplitude exhibits expected infrared behavior.

Framework aligns with previous gravitational scattering results.

Abstract

Motivated by the study of extreme mass-ratio binary systems, recent work has explored the use of curved backgrounds in computations of classical gravitational amplitudes [arXiv:2308.15304, arXiv:2308.14832, arXiv:2406.14770]. While these investigations concern the self-force expansion in the ratio of masses of the binaries, the use of curved backgrounds is interesting in its own right. In this thesis, I examine how gravitational computations can be done in a curved background. After having reviewed aspects of general relativity and the $d$-dimensional metric generated by a point mass (known as the Schwarzschild-Tangherlini solution), I quantize general relativity on an arbitrary background and compute Feynman rules for gravity in two cases: when the background is flat, and when it is a Schwarzschild-Tangherlini background. I then outline worldline quantum field theory. Using this newly-developed perturbation theory for the partition function of a worldline coupled to gravity in a curved background, I reformulate the perturbative expansion of the Compton amplitude, which describes the scattering of a graviton off a compact object. Having established this framework, I compute the first and second post-Minkowskian contributions to the Compton amplitude. Both are shown to match the results obtained from a flat-space computation. In addition, the second-order amplitude displays the expected infrared behavior and agrees with earlier results on massless gravitational scattering.