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

Abstract

The second-order term of the approximate stress-energy tensor of the quantized massive scalar field in the Bertotti-Robinson and Reissner- Nordström 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 ($Λ=0$) 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-Nordström 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.