Vacuum polarization in asymptotically anti-de Sitter black hole geometries
Vacuum polarization in asymptotically anti-de Sitter black hole geometries
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DOI：
10.1103/physrevd.78.064011
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发表时间：
2008-03
影响因子：
5
通讯作者：
A. Flachi;Takahiro Tanaka
中科院分区：
文献类型：
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作者：
A. Flachi;Takahiro Tanaka
我们在渐近反德西特黑洞几何上研究标量场 的真空极化。我们遵循的方法使用 WKB 解析展开和点分裂正则化，类似于之前渐近平坦情况下的计算。按照标准程序，我们编写格林函数，通过点分裂对初始发散表达式进行正则化，通过减去几何反项对其进行重新正则化，最后取重合极限。在明确证明了格林函数的散度抵消和规律性之后，我们将结果表示为两部分之和。一种是分析计算，结果用一些广义 zeta 函数表示，这些函数出现在黎曼球上拉普拉斯函数行列式的计算中。我们还描述了一些系统方法来对这些函数进行数值评估。有趣的是，WKB 近似自然地组织为此类 zeta 函数的级数。我们在 WKB 扩展中明确地证明了这一点直至次领先顺序。另一个术语表示 WKB 近似值的“余数”，并且取决于精确（数字）表达式与其 WKB 对应项之间的差异。这必须通过数值近似来处理。一般结果专门针对史瓦西反德西特黑洞几何结构的情况。考虑到渐近反德西特黑洞上量子场的反作用，该方法足以有效地求解半经典爱因斯坦方程。
We study the polarization of the vacuum for a scalar field, , 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 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.