Quantum corrections to the Schwarzschild metric from vacuum polarization

TL;DR

This paper investigates quantum vacuum polarization effects on the Schwarzschild metric, revealing near-horizon modifications and constraints on exotic compact objects through analytical and numerical solutions of semiclassical Einstein equations.

Contribution

It provides a combined analytical and numerical analysis of quantum corrections to black hole metrics, introducing effective equations of state and identifying horizon-scale quantum effects.

Findings

Near-horizon quantum corrections lead to a coordinate singularity at r=2M + O(√ħ).

Quantum effects impose an upper bound on the compactness of stable ECOs.

Vacuum polarization causes wave propagation corrections and introduces a null curvature singularity in the pure vacuum case.

Abstract

We explore static and spherically symmetric solutions of the 4-dimensional semiclassical Einstein's equations using the quantum vacuum polarization of a conformal field as a source. These solutions may be of interest for \black{the study of} exotic, compact objects (ECOs). The full backreaction problem is addressed by solving the semiclassical Tolman-Oppenheimer-Volkoff (TOV) equations making use of effective equations of state inspired \black{by} the trace anomaly and an extra simplifying and reasonable assumption. We combine analytical and numerical techniques to solve the resulting differential equations, both perturbatively and nonperturbatively in $\hbar$. In all cases the solution is similar to the Schwarzschild metric up to the vicinity of the classical horizon $r=2M$. However, at $r=2M + \varepsilon$, with $\varepsilon\sim O(\sqrt{\hbar})$, we find a coordinate singularity. In \black{the} case of matching with a static star, this leads to an upper bound in the compactness, and sets a constraint on the family of stable ECOs. We also study the corrections that the quantum-vacuum polarization induces on the propagation of waves, and discuss the implications. For the pure vacuum case, we can further extend the solution by using appropriate coordinates until we reach another singular point, where this time a null curvature singularity arises and prevents extending beyond. This picture qualitatively agrees with the results obtained in the effective two-dimensional approach, and reinforces the latter as a reasonable method.