Self-force on extreme mass ratio inspirals via curved spacetime effective field theory

TL;DR

The paper develops a curved spacetime effective field theory (CS-EFT) to study gravitational radiation and backreaction for extreme mass ratio inspirals (EMRIs). By exploiting a scale separation $ $ between the compact object's size and background curvature, it formulates an effective point-particle action and uses the closed-time-path (CTP) in-in formalism to derive real, causal equations of motion, including the first-order gravitational self-force (MST-QW). Ultraviolet divergences are handled via Hadamard's finite part and dimensional regularization, isolating the non-local finite part that drives radiation reaction. The work demonstrates that finite-size (tidal) corrections enter at $O( ^4)$ for non-spinning BHs/NSs, with potential enhancement for white dwarfs due to their large radii, and lays a foundation for higher-order self-force, gravitational radiation, and spinning-object extensions.

Abstract

In this series we construct an effective field theory (EFT) in curved spacetime to study gravitational radiation and backreaction effects. We begin in this paper with a derivation of the self-force on a compact object moving in the background spacetime of a supermassive black hole. The EFT approach utilizes the disparity between two length scales, which in this problem are the size of the compact object and the radius of curvature of the background spacetime, to treat the orbital dynamics of the compact object, described as an effective point particle, separately from its tidal deformations. Ultraviolet divergences are regularized using Hadamard's {\it partie finie} to isolate the non-local finite part from the quasi-local divergent part. The latter is constructed from a momentum space representation for the graviton retarded propagator and is evaluated using dimensional regularization in which only logarithmic divergences are relevant for renormalizing the parameters of the theory. As a first important application of this framework we explicitly derive the first order self-force given by Mino, Sasaki, Tanaka, Quinn and Wald. Going beyond the point particle approximation, to account for the finite size of the object, we demonstrate that for extreme mass ratio inspirals the motion of a compact object is affected by tidally induced moments at $O(ε^4)$, in the form of an Effacement Principle. The relatively large radius-to-mass ratio of a white dwarf star allows for these effects to be enhanced until the white dwarf becomes tidally disrupted, a potentially $O(ε^2)$ process, or plunges into the supermassive black hole. This work provides a new foundation for further exploration of higher order self force corrections, gravitational radiation and spinning compact objects.

Figures (5)

• Figure 1: Interaction vertices. Diagram (a) shows the interaction vertex for $n$ gravitons, denoted by curly lines, coupling to a point particle, denoted by a straight line. Diagram (b) shows the self-interaction vertex of $n$ gravitons. The labels $a_1, a_2, \ldots$ and $b$ are CTP indices, which take values of $1$ and $2$.
• Figure 2: The diagram contributing to the first-order self-force described by the MST-QW equation.
• Figure 3: Graviton scattering off the background of a static and spherically symmetric extended body (e.g., a Schwarzschild black hole, a non-spinning neutron star).
• Figure 4: Lowest order contributions to (a) deviation from (minimal) point particle motion due to the tidal deformations of the compact object and (b) the self-force from the interaction of gravitational radiation with these deformations.
• Figure 5: The effects from the finite size of a white dwarf star can be enhanced as it orbits in closer to the SMBH. The white dwarf seems to undergo some form of tidal disruption by either tidal disintegration (triangle and circle) or Roche lobe overflow (square). In either case, tidal disruption may be numerically equivalent to a second order process.