Gravitational self-force in the ultra-relativistic limit: the “large-N ” expansion

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.