Some uniqueness results for dynamical horizons

TL;DR

The paper establishes a geometric maximum principle that yields a unique marginally trapped surface foliation for any dynamical horizon (DH) and derives NEC-based constraints that limit the occurrence and placement of MTSs and DHs near a regular DH. It connects these foundational results to numerical relativity by clarifying how a DH is generated from a spacelike foliation and why DHs do not undergo bifurcation within a fixed slicing. It further analyzes the interplay between DHs and spacetime symmetries via Killing vectors, showing that regular DHs cannot be transversal to a Killing field on leaves and that tangential Killing fields must align with the DH foliation, thereby restricting possible DH configurations. Overall, the work demonstrates that DHs are robust, non-foliating structures with limited freedom, offering practical guidance for simulations and insights into the quasi-local description of evolving black holes.

Abstract

We first show that the intrinsic, geometrical structure of a dynamical horizon is unique. A number of physically interesting constraints are then established on the location of trapped and marginally trapped surfaces in the vicinity of any dynamical horizon. These restrictions are used to prove several uniqueness theorems for dynamical horizons. Ramifications of some of these results to numerical simulations of black hole spacetimes are discussed. Finally several expectations on the interplay between isometries and dynamical horizons are shown to be borne out.

Figures (4)

• Figure 1: Examples of situations ruled out by Theorem \ref{['basic3']} as well as of those that are not. Here $H$ is the dynamical horizon, $H^-(H)$, its past Cauchy horizon, and $D^-(H)$, its past domain of dependence. The compact surface $\Sigma_1$ lies in $D^-(H)\setminus H$ while $\Sigma_3$ lies entirely to the future of $H$. Compact surfaces $\Sigma_2$ and $\Sigma_4$ intersect $H$ and $R^*$ is the largest value of $R$ on their intersections with $H$. $\Sigma_2$ meets the causal past of $H_{R > R^*}$. $\Sigma_4$ meets $H^-(H)$. The theorem implies that $\Sigma_1$ and $\Sigma_2$ can not be WTSs but does not rule out the possibility that $\Sigma_3$ and $\Sigma_4$ are WTSs.
• Figure 2: Null hypersurfaces ${\bf N_1}$ and ${\bf N_2}$. ${\bf N_1}$ is generated by future directed null geodesics issuing orthogonally to $\Sigma_0$ from points near $p$ in $\Sigma_0$. ${\bf N_2}$ is generated by null geodesics along $\ell$, passing through points of $S_{R_0}$ near $q$. The segment near $q'$ of the null geodesic $\eta$ lies on both null surfaces.
• Figure 3: $H$ is a dynamical horizon. Theorem \ref{['dual']} implies that a 2-surface like $\Sigma$ can not be marginally trapped.
• Figure 4: Illustration of a dynamical horizon $H$ which is generated by a foliation $M_t$. The figure shows one leaf of the foliation which intersects $H$ in a MTS $S_R$ of area radius $R$. $\ell$ is the outward pointing null normal to the MTSs and $n$ the inward pointing one.

Theorems & Definitions (17)

• Proposition 3.1
• Corollary 3.2
• Theorem 4.1
• Theorem 4.2
• Theorem 4.3
• Theorem 4.4
• Corollary 4.5
• Corollary 4.6
• Theorem 5.1
• Corollary 5.2