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

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 $1/10$.