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

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.