$AdS_{2} \times S^{2}$ geometries and the extreme quantum-corrected black holes

TL;DR

This paper investigates quantum corrections to black hole geometries using the Schwinger-DeWitt method, revealing that quantum effects modify classical solutions and suggesting a minimal approximation from the first two terms.

Contribution

It provides a detailed analysis of second-order quantum corrections to black hole geometries, showing the necessity of a cosmological term and proposing a minimal approximation approach.

Findings

Bertotti-Robinson geometry is not self-consistent with second-order corrections without a cosmological term.

The near-horizon geometry of quantum-corrected extremal Reissner-Nordström black holes is $AdS_2 imes S^2$.

The curvature radii sum is proportional to the trace of the second-order stress-energy tensor.

Abstract

The second-order term of the approximate stress-energy tensor of the quantized massive scalar field in the Bertotti-Robinson and Reissner- Nordstr\"om spacetimes is constructed within the framework of the Schwinger-DeWitt method. It is shown that although the Bertotti- Robinson geometry is a self-consistent solution of the ($\Lambda =0$) semiclassical Einstein field equations with the source term given by the leading term of the renormalized stress-energy tensor, it does not remain so when the next-to-leading term is taken into account and requires the introduction of a cosmological term. The addition of the electric charge to the system does not change this behavior. The near horizon geometry of the extreme quantum-corrected Reissner-Nordstr\"om black hole is analyzed. It has the $AdS_{2} \times S^{2}$ topology and the sum of the curvature radii of the two dimensional submanifolds is proportional to the trace of the second order term. It suggests that the "minimal" approximation should be constructed from the first two terms of the Schwinger-DeWitt expansion.