Semiclassical corrections to a regularized Schwarzschild metric

TL;DR

This paper proposes a semiclassical extension of the Schwarzschild metric incorporating quantum effects, resulting in a regularized black hole model with novel properties like anisotropic pressures and finite entropy at extremality.

Contribution

It introduces a modified Schwarzschild metric valid in microphysics, featuring anisotropic fluids and quantum corrections, and analyzes its properties including horizons and thermodynamics.

Findings

The metric is regular at the origin due to quantum corrections.

Extremal black holes have finite horizon entropy despite zero temperature.

The model predicts violations of energy conditions at small scales.

Abstract

A version of the Schwarzschild metric to be valid in microphysics is proposed. The source fluid is anisotropic with $p_{r} = -\rho$ and fluctuating tangential pressures. At large distances with respect to the Compton wavelength associated to the source particle, they do not depend on the mass $m$ of the source and everywhere depend on $\hbar$ and the velocity of light $c$ but not on the Newton constant $G$. The particle may be a black hole for $m \geq m_{P}$ only and when $m = m_{P}$ it becomes an extremal black hole. The Komar energy $W$ of the gravitational fluid is $mc^{2}$ for $\hbar = 0$ and at large distances and vanishes at $r_{0} = 2\hbar/emc$. The WEC is violated when $r < r_{0}/2$ due to the negative tangential pressures. The horizon entropy for the extremal black hole is finite though $W$ and the temperature $T$ are vanishing there.