Head-on collision of compact objects in general relativity: Comparison of post-Newtonian and perturbation approaches

TL;DR

This paper compares post-Newtonian and perturbation methods for calculating gravitational-wave energy flux during head-on collisions of compact objects, revealing insights into their convergence and approximation accuracy.

Contribution

It provides a detailed comparison of two approaches for modeling gravitational radiation in head-on collisions, highlighting the convergence behavior and effective approximations.

Findings

Post-Newtonian method converges slowly to perturbation results.

Quadrupole approximation with exact equations closely matches perturbation results.

Perturbation theory offers a surprisingly accurate estimate of radiated energy.

Abstract

The gravitational-wave energy flux produced during the head-on infall and collision of two compact objects is calculated using two approaches: (i) a post-Newtonian method, carried to second post-Newtonian order beyond the quadrupole formula, valid for systems of arbitrary masses; and (ii) a black-hole perturbation method, valid for a test-body falling radially toward a black hole. In the test-body case, the methods are compared. The post-Newtonian method is shown to converge to the ``exact'' perturbation result more slowly than expected {\it a priori\/}. A surprisingly good approximation to the energy radiated during the infall phase, as calculated by perturbation theory, is found to be given by a Newtonian, or quadrupole, approximation combined with the exact test-body equations of motion in the Schwarzschild spacetime.