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One loop determinant in the extremal black hole from quasinormal modes

Jyotirmoy Mukherjee

TL;DR

This work computes the one-loop determinant for a scalar field in the near-horizon geometry of extremal RN black holes using the Denef–Hartnoll–Sachdev (DHS) prescription, and verifies that the logarithmic divergence matches the heat-kernel result. It then extends the DHS method to near-extremal Kerr–Newman black holes, showing that at a specific horizon angular velocity $\Omega = 2\pi T$, the finite-temperature determinant reduces to the extremal RN result in AdS$_2\times$S$^2$. For higher-spin fields, the authors identify horizon-regularity constraints that remove certain modes, leading to distinct logarithmic contributions (e.g., the vector case). Overall, the paper demonstrates a universal structure of quantum corrections governed by the near-horizon AdS$_2$ dynamics and establishes a coherent DHS-based framework for scalars, vectors, and potential future extensions to higher spins.

Abstract

In this paper, we evaluate the one-loop partition function of a scalar field in the near-horizon geometry of the extremal Reissner Nordström black hole from an infinite product over quasinormal modes using the Denef-Hartnoll-Sachdev (DHS) formula. We show that the logarithmic divergent term of the one-loop partition function computed using the DHS formula agrees with the heat kernel method. Using the same formula, we also evaluate the one-loop partition function of a scalar field in the near-extremal Kerr-Newman black hole and observe that it reduces to the same in the near-horizon $AdS_2\times S^2$ geometry of the extremal Reissner Nordström black hole when the angular velocity at the horizon is tuned to $2πT_{BH}$ value. We observe that, for higher spin fields, the mode functions are not smooth at the horizon for certain quasinormal frequencies; therefore, we remove them to obtain the one-loop determinant.

One loop determinant in the extremal black hole from quasinormal modes

TL;DR

This work computes the one-loop determinant for a scalar field in the near-horizon geometry of extremal RN black holes using the Denef–Hartnoll–Sachdev (DHS) prescription, and verifies that the logarithmic divergence matches the heat-kernel result. It then extends the DHS method to near-extremal Kerr–Newman black holes, showing that at a specific horizon angular velocity , the finite-temperature determinant reduces to the extremal RN result in AdSS. For higher-spin fields, the authors identify horizon-regularity constraints that remove certain modes, leading to distinct logarithmic contributions (e.g., the vector case). Overall, the paper demonstrates a universal structure of quantum corrections governed by the near-horizon AdS dynamics and establishes a coherent DHS-based framework for scalars, vectors, and potential future extensions to higher spins.

Abstract

In this paper, we evaluate the one-loop partition function of a scalar field in the near-horizon geometry of the extremal Reissner Nordström black hole from an infinite product over quasinormal modes using the Denef-Hartnoll-Sachdev (DHS) formula. We show that the logarithmic divergent term of the one-loop partition function computed using the DHS formula agrees with the heat kernel method. Using the same formula, we also evaluate the one-loop partition function of a scalar field in the near-extremal Kerr-Newman black hole and observe that it reduces to the same in the near-horizon geometry of the extremal Reissner Nordström black hole when the angular velocity at the horizon is tuned to value. We observe that, for higher spin fields, the mode functions are not smooth at the horizon for certain quasinormal frequencies; therefore, we remove them to obtain the one-loop determinant.
Paper Structure (17 sections, 113 equations, 2 figures)

This paper contains 17 sections, 113 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Near-horizon geometry of the extremal Reissner-Nordström black hole
  3. One-loop determinant of a scalar field in $AdS_2\times S^2$
  4. A brief review of the Denef-Hartnoll-Sachdev prescription
  5. Perturbation equation in Reissner Nordström black hole
  6. Solution in the asymptotic region
  7. Solution near the horizon
  8. Solution in the overlapping region
  9. Consistency check
  10. One-loop determinant of a scalar field
  11. Kerr-Newman solution
  12. Dudley-Finley perturbation equations
  13. One-loop determinant in Kerr-Newman black hole
  14. One-loop determinant of a scalar field using quasinormal modes
  15. One-loop determinant of the spin-1 field
  16. ...and 2 more sections

Figures (2)

  • Figure 1: The $\rm{C_{0}}$ contour in the $u$-plane for $AdS_{2}$
  • Figure 2: The contour in the $u$-plane