Black holes in a dense infinite medium: a toy-model regularizing the Schwarzschild metric

TL;DR

The paper addresses the finite-time mass divergence of a black hole growing in a dense, infinite medium under the Schwarzschild approximation. It proposes a Newtonian toy-model that defines the horizon from the observer's viewpoint, yielding a density-dominated regime where the horizon scales as $R_H^{\text{dens}} = \left( \frac{3 M c^2}{4 \pi \rho} \right)^{1/3}$ and removing the finite-time singularity; the mass then grows as a power law $M(t) \propto t^3$ at late times. The authors also analyze a binary system, showing that gravitational-wave-driven inspiral remains non-pathological and can be described analytically within the model. They further derive a metric and thermodynamics in the dense regime, obtaining a Hawking-like temperature $T \propto M^{-1/3}$ and a faster evaporation law, and they discuss potential implications for quantum gravity at horizons and for contracting-universe (bouncing) scenarios. While the work is a simplified guide, it provides a coherent framework to pursue a more rigorous GR treatment and highlights novel connections between horizon physics, dense environments, and cosmological bounces.

Abstract

We revisit the dynamics of a black hole accreting energy from a surrounding homogeneous and infinite space. We argue for a simple heuristic modification of the Schwarzschild approximation when the density of the medium is not negligible anymore. The resulting behavior is drastically modified: the mass divergence at finite time is cured and the thermodynamical properties are deeply changed. Some potential consequences for quantum gravity and bouncing models are also pointed out. Those conclusions being mostly obtained from a Newtonian approach, they only aim at guiding toward a more rigorous treatment. Still, interestingly, the behavior is far more convincing that the one usually obtained.

Figures (4)

• Figure 1: Sketch of the Gauss approach to the question: the gravitational field at the surface of the hole vanishes.
• Figure 2: Sketch of the alternative approach to the same question: he gravitational field at the surface of the hole does not vanish.
• Figure 3: Horizon radius as a function of the mass, in double logarithmic scale and in Planck units -- setting $\rho=1$. The exact solution is in solid black, the low mass approximation is in gray dashed, and the large mass approximation is in black dot-dashed.
• Figure 4: Mass of the black hole as a function of time, in double logarithmic scale and in Planck units -- setting $\rho=1$. The exact solution is in solid black, the low mass approximation is in gray dashed, and the large mass approximation is in black dot-dashed. The divergence of the Schwarzschild solution -- visible on the dashed grey curve -- is not anymore present.