Black hole one-loop determinants in the large dimension limit
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Springer
Received: May 4, Revised: April 16, Accepted: May 17, Published: June 16,
2019 2020 2020 2020
Cynthia Keeler and Alankrita Priya Physics Department, Arizona State University, Tempe, AZ, 85287, U.S.A.
E-mail: [email protected], [email protected] Abstract: We calculate the contributions to the one-loop determinant for transverse traceless gravitons in an n + 3-dimensional Schwarzschild black hole background in the large dimension limit, due to the SO(n + 2)-type tensor and vector fluctuations, using the quasinormal mode method. Accordingly we find the quasinormal modes for these fluctuations as a function of a fiducial mass parameter ∆. We show that the behavior of the one-loop determinant at large ∆ accords with a heat kernel curvature expansion in one lower dimension, lending further evidence towards a membrane picture for black holes in the large dimension limit. Keywords: Black Holes, Classical Theories of Gravity, Field Theories in Higher Dimensions ArXiv ePrint: 1904.09299
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)099
JHEP06(2020)099
Black hole one-loop determinants in the large dimension limit
Contents 1 Introduction
1
2 Review 2.1 The large dimension limit 2.2 Quasinormal mode method
3 3 6 7 7 11 11 13 15 16
4 Writing the one-loop determinant 4.1 Expressing ZV in terms of Hurwitz ζ 4.2 Matching with the heat kernel expression 4.2.1 Large ∆ limit using QNM method 4.2.2 Heat kernel calculation
17 18 19 19 20
5 Conclusion
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A Master equation A.1 SO(n + 2) tensor modes A.2 SO(n + 2) vector modes
24 24 25
B Larger k modes
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C Scalar modes
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1
Introduction
Quantum effects in nontrivial gravitational backgrounds are of great interest to the theoretical physics community. Even the leading one-loop effects can contain important physical results, such as quantum corrections to the entropy of black holes, which a series of papers [1–10] found via calculations of one-loop determinants. Since the computation of one-loop determinants, and thus one-loop partition functions, is technically difficult in generic curved spacetimes, several methods have been developed to handle the computations. There are three primary strategies: heat kernel methods, group
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3 Calculating the quasinormal modes 3.1 Setting up the equations 3.2 SO(n + 2) vector modes 3.2.1 Decoupled vector modes 3.2.2 Non-decoupled vector modes 3.3 SO(n + 2) tensor modes 3.4 Quasinormal mode results
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theoretic approaches, and the quasinormal mode method. Heat kernel methods may be exact, as in the eigenfunction expansion (e.g. [11–14]) or the method of images (e.g. [15]); alternatively, the heat kernel curvature approximation is only appropriate for fluctuations of massive fields (see [16] for a review of the heat kernel approach). For spacetimes with a simple symmetry structure, physicists have applied two group theoretic approaches. First, the authors of [17, 18] use characters of representations to build the explicit expression for the
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