Area Invariance of Apparent Horizons under Arbitrary Boosts

TL;DR

This paper explicitly demonstrates that the apparent horizon area of black holes remains invariant under Lorentz boosts in Kerr-Schild coordinates, confirming a fundamental property in general relativity through concrete calculations.

Contribution

It provides the first explicit calculation showing horizon area invariance under boosts for Kerr-Schild black holes, illustrating the physics of horizons and aiding computational modeling.

Findings

Horizon area remains invariant under boosts in Kerr-Schild coordinates.

Explicit calculations confirm theoretical invariance of apparent horizon area.

Results are relevant for initial data in boosted black hole simulations.

Abstract

It is a well known analytic result in general relativity that the 2-dimensional area of the apparent horizon of a black hole remains invariant regardless of the motion of the observer, and in fact is independent of the $ t=constant $ slice, which can be quite arbitrary in general relativity. Nonetheless the explicit computation of horizon area is often substantially more difficult in some frames (complicated by the coordinate form of the metric), than in other frames. Here we give an explicit demonstration for very restricted metric forms of (Schwarzschild and Kerr) vacuum black holes. In the Kerr-Schild coordinate expression for these spacetimes they have an explicit Lorentz-invariant form. We consider {\it boosted} versions with the black hole moving through the coordinate system. Since these are stationary black hole spacetimes, the apparent horizons are two dimensional cross sections of their event horizons, so we compute the areas of apparent horizons in the boosted space with (boosted) $ t = constant $, and obtain the same result as in the unboosted case. Note that while the invariance of area is generic, we deal only with black holes in the Kerr-Schild form, and consider only one particularly simple change of slicing which amounts to a boost. Even with these restrictions we find that the results illuminate the physics of the horizon as a null surface and provide a useful pedagogical tool. As far as we can determine, this is the first explicit calculation of this type demonstrating the area invariance of horizons. Further, these calculations are directly relevant to transformations that arise in computational representation of moving black holes. We present an application of this result to initial data for boosted black holes.