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Universality of the quantum energy flux at the inner horizon of asymptotically de Sitter black holes

Peter Hintz, Christiane Klein

Abstract

Recently, it was found that the energy flux of a free scalar quantum field on a Reissner-Nordström-de Sitter spacetime has a quadratic divergence towards the inner horizon of the black hole. Moreover, the leading divergence was found to be state independent as long as the spectral gap of the wave equation on the spacetime is sufficiently large. In this work, we show that the latter result can be extended to all subextremal Reissner-Nordström-de Sitter and subextremal Kerr-de Sitter spacetimes with a positive spectral gap.

Universality of the quantum energy flux at the inner horizon of asymptotically de Sitter black holes

Abstract

Recently, it was found that the energy flux of a free scalar quantum field on a Reissner-Nordström-de Sitter spacetime has a quadratic divergence towards the inner horizon of the black hole. Moreover, the leading divergence was found to be state independent as long as the spectral gap of the wave equation on the spacetime is sufficiently large. In this work, we show that the latter result can be extended to all subextremal Reissner-Nordström-de Sitter and subextremal Kerr-de Sitter spacetimes with a positive spectral gap.
Paper Structure (7 sections, 6 theorems, 53 equations, 4 figures)

This paper contains 7 sections, 6 theorems, 53 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Setup
  3. The RNdS and KdS spacetimes
  4. The free scalar field theory
  5. Expansion of classical solutions
  6. Bounding the state-dependence
  7. Conclusion

Key Result

Theorem 3.1

Let $r_-\in(r_1,r_2)$ and $r_+>r_3$. Write $e^{-\alpha t_*}H^s$ for the space of functions $\psi=\psi(t_*,x)$ (where $x\in\mathbb{R}^3$ denotes Cartesian coordinates on $(r_-,r_+)\times\mathbb{S}^2\subset\mathbb{R}^3$) with support in $t_*\geq 0$ so that where $\|\cdot\|_{L^2}$ is the spacetime $L^2$-norm. Then there exists $\alpha_1>0$ so that the following holds. Let $s>\frac{1}{2}+\alpha_1\max

Figures (4)

  • Figure 1: The Penrose diagram of the subextremal RNdS spacetime, or the Carter--Penrose diagram for the subextremal KdS spacetime. The gray region indicates our physical spacetime $\mathcal{M}$, while the diagram shows also the analytic extension across $\mathcal{CH}$.
  • Figure 2: Illustration of the domain $\Omega$, in which the results of decay towards $i^+$ are propagated and converted to regularity results at $\mathcal{CH}$.
  • Figure 3: Illustration of the coordinates $U,V$, and the domain $\Omega'$. Upon blowing up $i^+$, the radial coordinate $V=r-r_1$ becomes a smooth local coordinate function down to blown-up $i^+$ (where $U=0$ in the region $r<r_2$ of validity of the coordinates $U,V$).
  • Figure 4: Illustration of the construction for the proof of Lemma \ref{['lem: series expansion']}. The orange ellipse represents the set $\mathcal{U}$. $J^-(\mathcal{U}_{\mathcal{M}})$ is indicated by the dashed orange lines. The blue and red hypersurfaces represent $\Sigma_\pm$ and $\sigma_\pm$, respectively. The filled blue region is the compact set $G$, the red region is $\mathop{\mathrm{supp}}\nolimits(B)$.

Theorems & Definitions (9)

  • Theorem 3.1: Petersen:2021
  • Theorem 3.2
  • Proposition 3.3
  • Corollary 3.4: Pointwise bounds near the Cauchy horizon of KdS
  • proof : Proof of Proposition \ref{['prop:wave-CH']}
  • Proposition 4.1
  • Lemma 4.2
  • proof
  • proof : Proof of Proposition \ref{['prop:t mu nu bound']}: