Planckian $AdS_2 \times S_2$ space is an exact solution of the semiclassical Einstein equations

TL;DR

This paper finds exact solutions to the semiclassical Einstein equations for the $AdS_2 imes S_2$ space with electric charge, revealing quantum deformations and Planck-scale configurations, and discusses implications for black hole structure.

Contribution

It provides explicit solutions for the semiclassical Einstein equations with quantum corrections, extending classical $AdS_2 imes S_2$ space to include quantum and Planck-scale effects.

Findings

Quantum deformation of Bertotti-Robinson space for arbitrary coupling.

Planck-scale charged and uncharged configurations depending on coupling.

Insights into the internal structure of Schwarzschild black holes.

Abstract

The product space configuration $AdS_2\times S_2$ (with $l$ and $r$ being radiuses of the components) carrying the electric charge $Q$ is demonstrated to be an exact solution of the semiclassical Einstein equations in presence of the Maxwell field. If the logarithmic UV divergences are absent in the four-dimensional theory the solution we find is identical to the classical Bertotti-Robinson space ($r=l=Q$) with no quantum corrections added. In general, the analysis involves the quadratic curvature coupling $\lambda$ appearing in the effective action. The solutions we find are of the following types: i) (for arbitrary $\lambda$) charged configuration which is quantum deformation of the Bertotti-Robinson space; ii) ($\lambda >\lambda_{cr}$) Q=0 configuration with $l$ and $r$ being of the Planck order; iii) ($\lambda<\lambda_{cr}$) $Q\neq 0$ configuration ($l$ and $r$ are of the Planck order) not connected analytically to the Bertotti-Robinson space. The interpretation of the solutions obtained and an indication on the internal structure of the Schwarzschild black hole are discussed.