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

Chad R. Galley, Rafael A. Porto

TL;DR

This paper develops an effective field theory framework to study gravitational self-force in the ultra-relativistic limit, introducing a large-N expansion with 1/N = 1/γ^2 and λ = Nε (ε = E_m/M, E_m = γ m). It demonstrates that bulk nonlinearities are suppressed at leading order, and derives the self-force to O(λ^4/N) using a master-source approach that yields a regular, finite h^R on the worldline, with divergences vanishing under dimensional regularization. The formalism provides both a general self-force expression and concrete conservative quantities for circular orbits near the light ring, illustrating the method's promise for high-order ultra-relativistic calculations. The results offer a systematic route to refine EMRI modelling in regimes of large γ and lay groundwork for cross-checks with numerical relativity and waveform modeling.

Abstract

We study the gravitational self-force using the effective field theory formalism. We show that in the ultra-relativistic limit γ\to \infty, with γthe boost factor, many simplifications arise. Drawing parallels with the large N limit in quantum field theory, we introduce the parameter 1/N = 1/γ^2 and show that the effective action admits a well defined expansion in powers of λ= Nε, at each order in 1/N, where ε= E_m/M and E_m=γm is the (kinetic) energy of the small mass. Moreover, we show that diagrams with nonlinear bulk interactions first enter at O(λ^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(λ^4/N) and provide expressions for some conservative quantities for circular orbits.

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

TL;DR

This paper develops an effective field theory framework to study gravitational self-force in the ultra-relativistic limit, introducing a large-N expansion with 1/N = 1/γ^2 and λ = Nε (ε = E_m/M, E_m = γ m). It demonstrates that bulk nonlinearities are suppressed at leading order, and derives the self-force to O(λ^4/N) using a master-source approach that yields a regular, finite h^R on the worldline, with divergences vanishing under dimensional regularization. The formalism provides both a general self-force expression and concrete conservative quantities for circular orbits near the light ring, illustrating the method's promise for high-order ultra-relativistic calculations. The results offer a systematic route to refine EMRI modelling in regimes of large γ and lay groundwork for cross-checks with numerical relativity and waveform modeling.

Abstract

We study the gravitational self-force using the effective field theory formalism. We show that in the ultra-relativistic limit γ\to \infty, with γthe boost factor, many simplifications arise. Drawing parallels with the large N limit in quantum field theory, we introduce the parameter 1/N = 1/γ^2 and show that the effective action admits a well defined expansion in powers of λ= Nε, at each order in 1/N, where ε= E_m/M and E_m=γm is the (kinetic) energy of the small mass. Moreover, we show that diagrams with nonlinear bulk interactions first enter at O(λ^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(λ^4/N) and provide expressions for some conservative quantities for circular orbits.

Paper Structure

This paper contains 17 sections, 137 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Power counting rules
  3. Gravitational self-force in the large $N$ limit
  4. Gravitational perturbations and self-force to ${\cal O}(\lambda^4/N)$
  5. Conservative self-force effects for circular orbits near light ring
  6. Concluding remarks
  7. The causal variational principle of stationary action
  8. Green functions in curved spacetime
  9. Feynman rules
  10. Feynman diagram calculations of gravitational perturbations
  11. Leading order
  12. Next-to-leading order
  13. Next-to-next-to-leading order
  14. Next-to-next-to-next-to-leading order
  15. Gravitational perturbations to NNNLO
  16. ...and 2 more sections

Figures (1)

  • Figure 1: The normal neighborhood of $x^\mu = z^\mu(\tau')$. All points within the boundary can be connected to $x$ by a unique geodesic. Within the normal neighborhood one can construct a momentum space representation for the retarded Green function.