Mass formulas for individual black holes in merging binaries

TL;DR

This paper derives formulas for individual black hole masses in merging binaries using Newtonian physics combined with general relativistic gravitational wave power, providing a pedagogical approach to interpret gravitational wave data.

Contribution

It introduces a method to estimate individual black hole masses in mergers by combining Newtonian physics with a perturbative GR power formula, addressing a subtlety in previous approaches.

Findings

Formulas relate initial and maximum frequencies to black hole masses.

A new average power formula from perturbative GR is proposed.

Method simplifies understanding of gravitational wave observations.

Abstract

We give formulas for individual black hole masses in a merger, by using Newtonian physics, in terms of the three measured quantities in the detector: the initial wave frequency $f_1$, the maximum detected frequency (chirp frequency) $f_2$, and the time elapse $\tau$ between these two frequencies. Newtonian gravity provides an excellent pedagogical tool to understand the basic features of gravitational wave observations, but it must be augmented with the assumption of gravitational radiation from General Relativity for accelerating masses as there is no gravitational wave in Newtonian gravity. The simplest approach would be to consider a binary system of two non-spinning masses (two black holes) circling their common center of mass. All the computations can be done within Newtonian physics, but the General Relativistic formula for the power carried by gravitational waves is required in this scheme. It turns out there is a subtle point: for the consistency of this simple, yet pedagogical computation, taking the lowest order power formula from General Relativity leads to complex individual masses. Here we remedy this problem and {suggest} a way to write down an average power formula coming from perturbative General Relativity.