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Hadamard property of the Unruh state for massless fermions on Kerr spacetime : the large $a$ case

Dietrich Häfner, Christiane Klein

TL;DR

The paper resolves the Hadamard property of the Unruh state for massless Dirac fields on Kerr spacetime in the full subextreme regime $0<|a|<M$, extending prior results valid only for very small angular momentum. It achieves this by leveraging a detailed microlocal analysis of Weyl solutions, the decomposition of solution spaces into horizon and infinity data, and a careful study of the trapped geodesic set. The construction relies on the Kerr-Kruskal extension, conformal compactifications, and Selberg-type trace decompositions to propagate Hadamard singularities from asymptotic regions into the exterior. The result strengthens the physical viability of the Unruh state for rotating black holes and provides a robust framework for quantum field theory on Kerr spacetime with rapidly rotating horizons.

Abstract

In a recent paper by Gérard, Häfner, and Wrochna, the Unruh state for massless fermions on a Kerr spacetime was constructed and the authors showed its Hadmard property in the case of very slowly rotating black holes $\vert a\vert\ll M$. In this note, we extend this result to the full non extreme case $\vert a\vert<M$.

Hadamard property of the Unruh state for massless fermions on Kerr spacetime : the large $a$ case

TL;DR

The paper resolves the Hadamard property of the Unruh state for massless Dirac fields on Kerr spacetime in the full subextreme regime , extending prior results valid only for very small angular momentum. It achieves this by leveraging a detailed microlocal analysis of Weyl solutions, the decomposition of solution spaces into horizon and infinity data, and a careful study of the trapped geodesic set. The construction relies on the Kerr-Kruskal extension, conformal compactifications, and Selberg-type trace decompositions to propagate Hadamard singularities from asymptotic regions into the exterior. The result strengthens the physical viability of the Unruh state for rotating black holes and provides a robust framework for quantum field theory on Kerr spacetime with rapidly rotating horizons.

Abstract

In a recent paper by Gérard, Häfner, and Wrochna, the Unruh state for massless fermions on a Kerr spacetime was constructed and the authors showed its Hadmard property in the case of very slowly rotating black holes . In this note, we extend this result to the full non extreme case .
Paper Structure (36 sections, 14 theorems, 103 equations, 2 figures)

This paper contains 36 sections, 14 theorems, 103 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Dirac and Weyl operators
  3. Notation
  4. Dirac and Weyl operators
  5. $L^2$ solutions
  6. Action of Killing vector fields
  7. The Kerr spacetime
  8. Boyer-Lindquist blocks
  9. The ${\rm K}^{*}$ and ${}^{*}\!{\rm K}$ spacetimes
  10. The ${\rm K}^{*}$ spacetime
  11. Embedding ${\rm M}_{\rm I}$ and ${\rm M}_{\rm II}$ into ${\rm K}^{*}$
  12. The ${}^{*}\!{\rm K}$ spacetime
  13. Embedding ${\rm M}_{\rm I}$ and ${\rm M}_{\rm II}$ into ${}^{*}\!{\rm K}$
  14. Conformal extension of ${\rm M}_{\rm I}$
  15. The Kerr-Kruskal extension
  16. ...and 21 more sections

Key Result

Proposition 2.2

GHW The operator ${\rm i}^{-1}{\pazocal L}_{X}$ with domain ${\rm Sol}_{\rm sc}(M)$ is essentially self-adjoint on the Hilbert space ${\rm Sol}_{{\rm L}^{2}}(M)$.

Figures (2)

  • Figure 1: The conformal extensions of ${\rm M}_{\rm I}$.
  • Figure 2: The Kerr-Kruskal spacetime with Cauchy surface $\Sigma_{{\rm M}}$.

Theorems & Definitions (20)

  • Definition 2.1
  • Proposition 2.2
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • Proposition 4.1
  • Remark 4.2
  • Theorem 4.3
  • Remark 4.4
  • Theorem 5.1
  • ...and 10 more