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

We study the gravitational self-force using the effective field theory formalism. We show that in the ultra-relativistic limit γ → ∞, with γ the boost factor, many simplifications arise. Drawing parallels with the large N limit in quantum field theory, we introduce the parameter 1/N ≡ 1/γ and show that the effective action admits a well defined expansion in powers of λ ≡ N at each order in 1/N , where ≡ Em/M with Em = γm the (kinetic) energy of the small mass. Moreover, we show that diagrams with nonlinear bulk interactions first enter at O(1/N) and only diagrams with nonlinearities in the worldline couplings, which are significantly easier to compute, survive in the large N/ultra-relativistic limit. As an example we derive the self-force to O(λ/N) and provide expressions for conservative quantities for circular orbits.