Marginally trapped tubes and dynamical horizons

TL;DR

This paper investigates the generic behavior of marginally trapped tubes (MTTs) and dynamical horizons in strong gravity by analyzing simple, spherically symmetric models of dust and scalar-field collapse onto pre-existing black holes. It derives a compact condition, $C = \frac{T_{ab} \ell^a \ell^b}{1/(2A) - T_{ab} \ell^a n^b}$, that determines the MTT signature (spacelike, null, or timelike) and thereby whether the horizon expands or undergoes a jump to a new horizon. Through analytic Tolman-Bondi solutions and extensive scalar-field simulations, the work demonstrates that spacelike, expanding MTTs are common, while high-density episodes can produce horizon jumps via dynamical horizons–timelike membranes pairs; timelike membranes can also emerge in smooth OS-like dust collapse under suitable conditions. The results deepen intuition about quasi-local horizon dynamics, relate horizon behavior to local matter content, and offer a useful testing ground for horizon flux laws and related theorems in dynamical spacetimes. The insights have potential implications for understanding black hole growth, gravitational fluxes, and the causal structure of dynamical horizons in astrophysical settings.

Abstract

We investigate the generic behaviour of marginally trapped tubes (roughly time-evolved apparent horizons) using simple, spherically symmetric examples of dust and scalar field collapse/accretion onto pre-existing black holes. We find that given appropriate physical conditions the evolution of the marginally trapped tube may be either null, timelike, or spacelike and further that the marginally trapped two-sphere cross-sections may either expand or contract in area. Spacelike expansions occur when the matter falling into a black hole satisfies $ρ- P \leq 1/A$, where $A$ is the area of the horizon while $ρ$ and $P$ are respectively the density and pressure of the matter. Timelike evolutions occur when $(ρ- P)$ is greater than this cut-off and so would be expected to be more common for large black holes. Physically they correspond to horizon "jumps" as extreme conditions force the formation of new horizons outside of the old.

Figures (11)

• Figure 1: A schematic of an MTT as informed by our later examples. In this diagram $45^\circ$ lines are null, time increases in the vertical direction, while horizon area increases as one moves to the right.
• Figure 2: Collapse of a dust ball
• Figure 3: Penrose-Carter diagram for dust-ball collapse. To obtain the full spacetime diagram, including the corresponding white hole and $\mathscr{I}^-$, reflect this diagram through the $\tau = \eta = 0$ instant of time symmetry. Note that $r=0$ starts as a regular timelike geodesic in the centre of the ball and then changes into a spacelike singularity at $\tau = \tau_c (0)$ as the central density goes to infinity.
• Figure 4: Approach to Oppenheimer-Snyder collapse
• Figure 5: Small dust shell