Bowen-York Model Solution Redux

TL;DR

This paper revisits and corrects the Bowen-York initial data solution in General Relativity, analyzing the energy distribution and gravitational wave content for systems with linear momentum, with implications for black hole formation.

Contribution

The paper provides a corrected analytical form of the Bowen-York solution and detailed analysis of energy content and gravitational wave emission in dynamical, gravitating systems.

Findings

Non-stationary energy ratio ranges from 0 to 0.592.

Maximum gravitational wave energy is 59.2% of total ADM energy.

Corrected solution impacts energy and wave predictions in initial data models.

Abstract

Initial value problem in General Relativity is often solved numerically; with only a few exceptions one of which is the "model" solution of Bowen and York where an analytical form of the solution is available. The solution describes a dynamical, time-asymmetric, gravitating system with mass and linear momentum. Here we revisit this solution and correct an error which turns out to be important for identifying the energy-content of the solution. Depending on the linear momentum, the ratio of the non-stationary part of the initial energy to the total ADM energy takes values between $[0, 0.592)$. This non-stationary part is expected to be turned into gravitational waves during the evolution of the system to possibly settle down to a black hole with mass and linear momentum. In the ultra-relativistic case (the high momentum limit), the maximum amount of gravitational wave energy is $59.2 \%$ of the total ADM energy. We also give a detailed account of the general solution of the Hamiltonian constraint.