2008/03/21 by Antonino Flachi, Takahiro Tanaka · 1 citation
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #Noncommutative and Quantum Gravity Theories #gr-qc #hep-th #math-ph #math.MP
paper · pdf · doi:10.1103/physrevd.78.064011
published as Phys.Rev.D78:064011,2008 · 12 pages, 1 figure
arxiv created 2008/03/21 · openalex publication_date 2008/09/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the polarization of the vacuum for a scalar field, ⟨\ensuremathφ2⟩, on a asymptotically anti-de Sitter black hole geometry. The method we follow uses the WKB analytic expansion and point-splitting regularization, similar to previous calculations in the asymptotically flat case. Following standard procedures, we write the Green function, regularize the initial divergent expression by point-splitting, renormalize it by subtracting geometrical counterterms, and take the coincidence limit in the end. After explicitly demonstrating the cancellation of the divergences and the regularity of the Green function, we express the result as a sum of two parts. One is calculated analytically and the result is expressed in terms of some generalized zeta functions, which appear in the computation of functional determinants of Laplacians on Riemann spheres. We also describe some systematic methods to evaluate these functions numerically. Interestingly, the WKB approximation naturally organizes ⟨\ensuremathφ2⟩ as a series in such zeta functions. We demonstrate this explicitly up to next-to-leading order in the WKB expansion. The other term represents the ``remainder'' of the WKB approximation and depends on the difference between an exact (numerical) expression and its WKB counterpart. This has to be dealt with by means of numerical approximation. The general results are specialized to the case of Schwarzschild-anti-de Sitter black hole geometries. The method is efficient enough to solve the semiclassical Einstein's equations taking into account the backreaction from quantum fields on asymptotically anti-de Sitter black holes.