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Exploring quantum fields in rotating black holes

Christiane K. M. Klein

TL;DR

This work constructs and analyzes the Unruh state for a free scalar field on Kerr-de Sitter spacetimes under mode stability, and proves its Hadamard property beyond the previously known small-$a$ regime by a geometric analysis of the trapped set. The approach extends the Kerr result to all subextremal rotations at small cosmological constant and leverages horizon bulk-to-boundary techniques to study the stress-energy tensor near the inner horizon. It further demonstrates a universality: the leading quadratic divergences of observables near the inner horizon are state-independent up to subleading terms when a positive spectral gap $\alpha$ exists, with the difference between Hadamard states bounded by $\mathcal{O}((r-r_-)^{-\beta'})$. Collectively, the results illuminate how quantum fields interact with inner-horizon geometry and set the stage for semi-classical gravity analyses, while highlighting the need for backreaction and extensions to more complex field theories.

Abstract

In this paper, we discuss the Unruh state for a free scalar quantum field on Kerr-de Sitter under the assumption of mode stability. We summarise the proof of its Hadamard property that was previously given in [Klein:2023] for sufficiently small black-hole rotation and cosmological constant and show how it can be generalised to any subextreme black-hole angular momentum in the same range of the cosmological constant. This is done by extending a geometric analysis of the trapped set of the Kerr spacetime [Häfner, Klein:2024] to Kerr-de Sitter. Moreover, we discuss the application of this state in the numerical study of quantum effects at the inner horizon [Klein, Soltani, Casals, Hollands:2024], and describe a universality result for these effects [Hintz, Klein: 2024].

Exploring quantum fields in rotating black holes

TL;DR

This work constructs and analyzes the Unruh state for a free scalar field on Kerr-de Sitter spacetimes under mode stability, and proves its Hadamard property beyond the previously known small- regime by a geometric analysis of the trapped set. The approach extends the Kerr result to all subextremal rotations at small cosmological constant and leverages horizon bulk-to-boundary techniques to study the stress-energy tensor near the inner horizon. It further demonstrates a universality: the leading quadratic divergences of observables near the inner horizon are state-independent up to subleading terms when a positive spectral gap exists, with the difference between Hadamard states bounded by . Collectively, the results illuminate how quantum fields interact with inner-horizon geometry and set the stage for semi-classical gravity analyses, while highlighting the need for backreaction and extensions to more complex field theories.

Abstract

In this paper, we discuss the Unruh state for a free scalar quantum field on Kerr-de Sitter under the assumption of mode stability. We summarise the proof of its Hadamard property that was previously given in [Klein:2023] for sufficiently small black-hole rotation and cosmological constant and show how it can be generalised to any subextreme black-hole angular momentum in the same range of the cosmological constant. This is done by extending a geometric analysis of the trapped set of the Kerr spacetime [Häfner, Klein:2024] to Kerr-de Sitter. Moreover, we discuss the application of this state in the numerical study of quantum effects at the inner horizon [Klein, Soltani, Casals, Hollands:2024], and describe a universality result for these effects [Hintz, Klein: 2024].
Paper Structure (12 sections, 8 theorems, 66 equations, 2 figures)

This paper contains 12 sections, 8 theorems, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Kerr(-de Sitter) spacetime
  3. The Kruskal extension
  4. The null geodesics and the trapped set
  5. The Unruh state on Kerr-de Sitter
  6. The free scalar field theory
  7. The Unruh state
  8. The Hadamard property for large $a$
  9. Application: The inner horizon
  10. The stress-energy tensor in the Unruh state
  11. Universality at the inner horizon
  12. Summary

Key Result

Proposition 3.2

The bi-linear map $w$ in Definition def:Unruh is well-defined and constitutes the two-point function of a quasi-free state on ${\pazocal A}$.

Figures (2)

  • Figure 1: The subextreme parameter range of Kerr-de Sitter for $M=1$ in terms of the cosmological constant $\Lambda =3\lambda$ and the black-hole angular momentum per unit mass $a$ taken from thesis.
  • Figure 2: The region $\Omega$ in which Price's law is assumed to hold. The lower boundary indicated by the black line represents $\{r=r_2\}$, while the upper boundaries are $\{r=r_1\}$ and $\{u=u_2\}$, the latter represented by the dotted line.

Theorems & Definitions (19)

  • Definition 3.1
  • Proposition 3.2
  • Remark 3.3
  • proof
  • Remark 3.4
  • Theorem 3.5
  • proof
  • Proposition 3.6
  • Lemma 3.7
  • proof
  • ...and 9 more