### Abstract

We study the polarization of the vacuum for a scalar field, 2, 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 2 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.

Original language | English |
---|---|

Article number | 064011 |

Journal | Physical Review D - Particles, Fields, Gravitation and Cosmology |

Volume | 78 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2008 Sep 2 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Nuclear and High Energy Physics
- Physics and Astronomy (miscellaneous)

### Cite this

**Vacuum polarization in asymptotically anti-de Sitter black hole geometries.** / Flachi, Antonino; Tanaka, Takahiro.

Research output: Contribution to journal › Article

*Physical Review D - Particles, Fields, Gravitation and Cosmology*, vol. 78, no. 6, 064011. https://doi.org/10.1103/PhysRevD.78.064011

}

TY - JOUR

T1 - Vacuum polarization in asymptotically anti-de Sitter black hole geometries

AU - Flachi, Antonino

AU - Tanaka, Takahiro

PY - 2008/9/2

Y1 - 2008/9/2

N2 - We study the polarization of the vacuum for a scalar field, 2, 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 2 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.

AB - We study the polarization of the vacuum for a scalar field, 2, 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 2 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.

UR - http://www.scopus.com/inward/record.url?scp=51649115990&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51649115990&partnerID=8YFLogxK

U2 - 10.1103/PhysRevD.78.064011

DO - 10.1103/PhysRevD.78.064011

M3 - Article

AN - SCOPUS:51649115990

VL - 78

JO - Physical review D: Particles and fields

JF - Physical review D: Particles and fields

SN - 1550-7998

IS - 6

M1 - 064011

ER -