Initial data for a head on collision of two Kerr-like black holes with close limit

TL;DR

This paper constructs a family of initial data for two Kerr-like black holes in vacuum Einstein equations, capturing various limits such as single Kerr black holes and merged configurations, with smooth dependence on parameters.

Contribution

It introduces a new family of initial data for two Kerr-like black holes with arbitrary positions and spins, extending previous models and capturing key limiting cases.

Findings

Reduces to Kerr initial data when one mass is zero or at infinite separation.

Results in a single Kerr black hole when the two are at zero separation.

Family of data depends smoothly on parameters.

Abstract

We prove the existence of a family of initial data for the Einstein vacuum equation which can be interpreted as the data for two Kerr-like black holes in arbitrary location and with spin in arbitrary direction. This family of initial data has the following properties: (i) When the mass parameter of one of them is zero or when the distance between them goes to infinity, it reduces exactly to the Kerr initial data. (ii) When the distance between them is zero, we obtain exactly a Kerr initial data with mass and angular momentum equal to the sum of the mass and angular momentum parameters of each of them. The initial data depends smoothly on the distance, the mass and the angular momentum parameters.