Spacetime near isolated and dynamical trapping horizons

TL;DR

The authors develop a universal near-horizon metric expansion in Gaussian null coordinates for horizons of arbitrary signature and dimension, deriving how horizon intrinsic and extrinsic data determine spacetime in a neighbourhood. For spacelike (dynamical) horizons this data suffices, while null (isolated) horizons require additional boundary information; the framework is then tested with two applications. First, extremal isolated horizons reproduce the standard near-horizon form of extremal black holes, demonstrating the perturbative method's consistency with known results. Second, slowly evolving horizons admit an event-horizon candidate that hugs the horizon, with explicit examples in Vaidya spacetimes and fluid-gravity–type black branes, illustrating the practical utility of the construction for understanding black-hole mechanics and causal structure in dynamical settings.

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.

Figures (3)

• Figure 1: Time evolution of a spacelike slice. The time-evolution vector $T^a$ deforms an initial surface $\Sigma_o$ into $\Sigma_{\Delta t}$. It can be decomposed its parts perpendicular and parallel to $\Sigma$, hence defining a lapse function $N$ and shift-vector field $N^a = e^a_i N^i$.
• Figure 2: This figure is similar in appearance to FIG. \ref{['SpacelikeDef']} and represents a similar situation. The bottom sheet $H$ is foliated by $(n\!-\!1)$-dimensional surfaces $S_v$ (drawn as solid lines). On $H$, $\mathscr{V}^a$ is the evolution vector field evolving surfaces into each other from the left-hand to the right-hand sides and identifying points on those surfaces. The null normal $n^a$ is used to deform $H$ into $H_{\Delta \rho}$. The evolution vector field $\mathscr{V}^a$ on $H_{\Delta \rho}$ will usually no longer be normal to the $S_v$ after the translation.
• Figure 3: A schematic that plots both the (spherically symmetric) FOTH and event horizon for a typical Vaidya spacetime in which a shell of dust (the shaded gray region) falls into a pre-existing black hole. In this figure, horizontal position records the areal radius of the associated spherical shell while the direction of increasing time is roughly vertical outside the event horizon but tipping horizontal-and-to-the-left inside. On both sides, inward-moving null geodesics are horizontal while "outward-pointing" null geodesics are represented by gray dashed lines.