Divergences in gravitational-wave emission and absorption from extreme mass ratio binaries

TL;DR

This paper investigates divergences in gravitational-wave emission and absorption calculations for extreme mass ratio binaries, revealing they are artifacts of the point-particle approximation and proposing regularization methods.

Contribution

The study identifies and explains divergences in perturbative calculations of gravitational waves in extreme mass ratio binaries, and introduces a finite-size regularization approach.

Findings

Divergences occur in flux and absorption calculations at the light ring and horizon.

These divergences are artifacts of the point-particle approximation.

Regularization with finite particle size resolves the divergences.

Abstract

A powerful technique to calculate gravitational radiation from binary systems involves a perturbative expansion: if the masses of the two bodies are very different, the "small" body is treated as a point particle of mass $m_p$ moving in the gravitational field generated by the large mass $M$, and one keeps only linear terms in the small mass ratio $m_p/M$. This technique usually yields finite answers, which are often in good agreement with fully nonlinear numerical relativity results, even when extrapolated to nearly comparable mass ratios. Here we study two situations in which the point-particle approximation yields a divergent result: the instantaneous flux emitted by a small body as it orbits the light ring of a black hole, and the total energy absorbed by the horizon when a small body plunges into a black hole. By integrating the Teukolsky (or Zerilli/Regge-Wheeler) equations in the frequency and time domains we show that both of these quantities diverge. We find that these divergences are an artifact of the point-particle idealization, and are able to interpret and regularize this behavior by introducing a finite size for the point particle. These divergences do not play a role in black-hole imaging, e.g. by the Event Horizon Telescope.