Colliding Black Holes: The Close Limit

TL;DR

This work takes initial data due to Misner for close black holes, applies perturbation theory and evolve the data with the Zerilli equation, and computed gravitational radiation agrees with and extends the results of full numerical computations.

Abstract

The problem of the mutual attraction and joining of two black holes is of importance as both a source of gravitational waves and as a testbed of numerical relativity. If the holes start out close enough that they are initially surrounded by a common horizon, the problem can be viewed as a perturbation of a single black hole. We take initial data due to Misner for close black holes, apply perturbation theory and evolve the data with the Zerilli equation. The computed gravitational radiation agrees with and extends the results of full numerical computations.

Figures (3)

• Figure 1: The function $\psi$ of the Cunningham-Price-Moncrief perturbation scheme for Misner's initial data. The values shown are for $p(\mu_{0})=1$, and for units in which $2M=1$. For the $r^*$ coordinate the horizon is at $r^*=-\infty$ and $r^* \sim r$ for large positive values.
• Figure 2: Time evolution of the Misner initial data (with $p(\mu_{0})=1, \ 2M=1$), from the point of view of an observer fixed at $r^*=200$. We see the appearance of quasinormal ringing with the predicted period of 8.4. In the inset we display in a log-log plot the late time behavior of the field, which clearly exhibits a power-law tail form with exponent $-6$ as predicted by theory.
• Figure 3: The solid curve is the prediction for the radiated energy, as a function of $\mu_{0}$, based on linearized perturbation theory. The black dots correspond to the values of numerical relativity results reported by Anninos et al. [5].