Gravitational self-force in the ultra-relativistic limit: The 'large-N' expansion

TL;DR

This paper develops a new expansion method for calculating the gravitational self-force in ultra-relativistic regimes, simplifying computations by focusing on dominant diagrams and introducing a large-N analogy to quantum field theory.

Contribution

It introduces a novel large-N expansion framework for the gravitational self-force in the ultra-relativistic limit, simplifying the calculation of nonlinear effects.

Findings

Derived the self-force to order /N

Identified dominant diagrams with worldline nonlinearities

Provided expressions for conservative quantities in circular orbits

Abstract

We study the gravitational self-force using the effective field theory formalism. We show that in the ultra-relativistic limit \gamma \to \infty, with \gamma the boost factor, many simplifications arise. Drawing parallels with the large N limit in quantum field theory, we introduce the parameter 1/N = 1/\gamma^2 and show that the effective action admits a well defined expansion in powers of \lambda = N\epsilon, at each order in 1/N, where \epsilon = E_m/M and E_m=\gamma m is the (kinetic) energy of the small mass. Moreover, we show that diagrams with nonlinear bulk interactions first enter at O(\lambda^2/N^2) and only diagrams with nonlinearities in the worldline couplings, which are significantly easier to compute, survive in the large N/ultra-relativistic limit. Finally, we derive the self-force to O(\lambda^4/N) and provide expressions for some conservative quantities for circular orbits.