Spacetime near isolated and dynamical trapping horizons

TL;DR

This paper develops a formalism to analyze the near-horizon spacetime geometry of isolated and dynamical trapping horizons, providing explicit calculations and demonstrating applications to known solutions and slowly evolving horizons.

Contribution

It introduces a second-order expansion formalism for near-horizon spacetime of trapping horizons applicable to arbitrary dimensions and field equations, with practical demonstrations.

Findings

Explicit second-order near-horizon metric expansion.

Validation against Kerr-Newman and extremal black holes.

Identification of event horizon candidates near slowly evolving horizons.

Abstract

We study the near-horizon spacetime for isolated and dynamical trapping horizons (equivalently marginally outer trapped tubes). The metric is expanded relative to an ingoing Gaussian null coordinate and the terms of that expansion are explicitly calculated to second order. For the spacelike case, knowledge of the intrinsic and extrinsic geometry of the (dynamical) horizon is sufficient to determine the near-horizon spacetime, while for the null case (an isolated horizon) more information is needed. In both cases spacetime is allowed to be of arbitrary dimension and the formalism accomodates both general relativity as well as more general field equations. The formalism is demonstrated for two applications. First, spacetime is considered near an isolated horizon and the construction is both checked against the Kerr-Newman solution and compared to the well-known near-horizon limit for stationary extremal black hole spacetimes. Second, spacetime is examined in the vicinity of a slowly evolving horizon and it is demonstrated that there is always an event horizon candidate in this region. The geometry and other properties of this null surface match those of the slowly evolving horizon to leading order and in this approximation the candidate evolves in a locally determined way. This generalizes known results for Vaidya as well as certain spacetimes known from studies of the fluid-gravity correspondence.