Asymptotic analysis of the "simulated horizon" segment of the Collins spiral

TL;DR

This paper performs an asymptotic analysis of a specific segment of the Collins spiral related to a gravastar model, linking initial conditions to parameters at the simulated horizon with novel power law behaviors.

Contribution

It provides an asymptotic mapping connecting initial values at the spiral's small radius end to parameters at the simulated horizon in a gravastar model.

Findings

Asymptotic relations involve power laws with exponents 1/5 and 1/10.

Results relate initial conditions to the mass and parameters of the black hole mimicker.

Analysis clarifies the structure of the Collins spiral segment associated with the simulated horizon.

Abstract

The Tolman-Oppenheimer-Volkoff (TOV) equations for a massless fluid take the form of a pair of coupled autonomous first order differential equations, which can be employed in a model for a ``dynamical gravastar'' black hole mimicker. The mimicker has no true horizon, but rather a ``simulated horizon'', outside which the geometry resembles a Schwarzschild black hole, but inside which the $g_{00}$ component of the metric is always positive and becomes exponentially small. Collins has reinterpreted the relevant TOV equations in terms of a two-dimensional flow with a spiral form, and Z\"ollner and K\"ampfer have mapped the simulated horizon to a specific segment of the Collins spiral. We give here results of an asymptotic analysis, relating initial values at the small radius end of this spiral segment to the black hole mimicker mass and other parameters that emerge at the large radius kink in the TOV solution, which corresponds to the simulated horizon. A curious feature of this asymptotic mapping, given in Sec. IIB, is the appearance of power law behaviors with exponents of $1/5$ and $1/10$.