On Coordinate Singularities Induced by Trapping Horizons

TL;DR

This work analyzes coordinate singularities induced by trapping horizons in dynamically evolving, spherically symmetric spacetimes and shows that such singularities arise from the orthogonal $\{t,r\}$ foliation rather than from intrinsic horizon physics. A covariant framework based on the Kodama vector is developed to disentangle coordinate effects from physical content, and the Ellis drainhole is used as an explicit, solvable example to demonstrate that a commonly cited near-horizon EMT limit is not universal. The results emphasize that horizon regularity alone does not enforce a single EMT structure, and they advocate using Kodama-based covariant methods to study horizon evolution and NEC implications in dynamical black-hole spacetimes. These insights refine the interpretation of horizon dynamics and have potential bearing on black-hole evolution in realistic, time-dependent spacetimes.

Abstract

The trapping (or apparent) horizon serves as a key tool for tracing the complete evolution of black holes. We investigate a class of coordinate singularities induced by such trapping (or apparent) horizons in a spherically symmetric, dynamic spacetime, which are distinct from the well-known coordinate singularities associated with the Killing horizon. In particular, we clarify the geometric structure of this coordinate singularity by means of the Kodama vector field, thereby avoiding unphysical artifacts. We further employ the evolving Ellis drainhole as an analytical model to illustrate key details of this phenomenon.

Figures (3)

• Figure 1: The gray curves represent surfaces of constant $r$, and the muted teal dashed curve represents a time slice, which is orthogonal to every surface of constant $r$. These curves become tangent to each other at the MTS represented by the black dot. The green arrow denotes a future-directed null vector with vanishing expansion. For instance, if $\theta_{k}=0$, the green arrow represents $k^a$; if $\theta_{l}=0$,it represents $l^a$. In contrast, the brown arrow denotes another future-directed null vector.
• Figure 2: We set $\alpha=1.2$. Gray contours represent the surfaces of constant $r$, and muted teal contours represent those time slices $t$ that are orthogonal to the surfaces of constant $r$. The red line denotes the trapping horizon $\theta_k=0$, while the blue line denotes the trapping horizon $\theta_l=0$. On these horizons, the $t$-slices are tangent to the surfaces of constant $r$, as their normal vectors become null. This implies that $dt$ and $dr$ become collinear, giving rise to the coordinate singularity in the $\{t, r\}$ coordinate system.
• Figure 3: (a) Black dots denote two trapped surfaces, and the vertical dashed line represents a regular center. The horizontal black curve corresponds to the hypersurface containing the initial data, while the orange vertical curve depicts a distant observer that never falls into the trapped region. Green arrows indicate two radial outgoing null geodesic congruences, and the blue curve denotes the trapping horizon. The visibility of the trapping horizon necessitates that the null geodesics attain a local minimum in areal radius before reaching the observer. When extending the null geodesics into the past, they may either be captured by the initial hypersurface or converge to the regular center. (b ) The variation of $r$ and $\theta_{k}$ with the affine parameter $\lambda$ are illustrated. We assume $\lambda$ increases along the future direction of the outgoing null geodesic congruence. Since $\theta_{k}$ has two roots, one corresponds to the local minimum of $r$ (on the outer part of the horizon), and the other to the local maximum (on the inner part of the horizon).