Nonexistence of conformally flat slices of the Kerr spacetime

TL;DR

This paper proves that within certain symmetry and smoothness constraints, there are no conformally flat slices of the Kerr spacetime, impacting initial data methods for rotating black holes.

Contribution

It demonstrates the nonexistence of conformally flat, axisymmetric slices of Kerr spacetime under specified conditions, clarifying limitations of common initial data approaches.

Findings

No conformally flat slices exist for Kerr spacetime under the given restrictions.

Limits the applicability of Bowen-York initial data for rotating black holes.

Provides a rigorous mathematical proof of nonexistence.

Abstract

Initial data for black hole collisions are commonly generated using the Bowen-York approach based on conformally flat 3-geometries. The standard (constant Boyer-Lindquist time) spatial slices of the Kerr spacetime are not conformally flat, so that use of the Bowen-York approach is limited in dealing with rotating holes. We investigate here whether there exist foliations of the Kerr spacetime that are conformally flat. We limit our considerations to foliations that are axisymmetric and that smoothly reduce in the Schwarzschild limit to slices of constant Schwarzschild time. With these restrictions, we show that no conformally flat slices can exist.