Planckian $Ad{S}_2\times S_2$ space is an exact solution of the semiclassical Einstein equations

Abstract

The product space configuration $Ad{S}_2\times S_2$ (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) ($\lambda <{\lambda }_{cr}$) $Q\ne 0$ 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.

14 pages, latex, 1 figure; v2: a note on S2*S2 type solutions added

References in corpus (5)

• Universality of Quantum Entropy for Extreme Black Holes
• Hawking radiation by effective two-dimensional theories
• Geometry of nonextreme black holes near the extreme state
• A discrete Hubbard-Stratonovich decomposition for general, fermionic two-body interactions
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• Do Black Holes Exist?
• Nonextreme black holes near the extreme state and acceleration horizons: thermodynamics and quantum-corrected geometry
• Charged Randall-Sundrum black holes and N=4 super Yang-Mills in AdS(2)xS(2)
• Semiclassical ultraextremal horizons
• $Ad{S}_2\times S^2$ geometries and the extreme quantum-corrected black holes