The Unruh state for bosonic Teukolsky fields on subextreme Kerr spacetimes

TL;DR

This work constructs a rigorous algebraic quantum field theory for bosonic Teukolsky scalars of spin $0$, $\pm 1$, and $\pm 2$ on subextreme Kerr spacetimes by employing an enlarged phase space that pairs spin $+s$ and spin $-s$ sectors. The authors develop a complete geometric and analytic framework on spin-weighted bundles, establish a normally hyperbolic Teukolsky operator on a doubled bundle ${\mathcal V}_s$, and prove the conservation of a charged symplectic form leading to a CCR-algebra for the theory. They implement a bulk-to-boundary construction to define the Unruh state on the physical subalgebra, prove its positivity, and demonstrate the Hadamard property on the exterior and interior up to the inner horizon. The results extend the existence of Hadamard Unruh-type states to bosonic Teukolsky fields on Kerr, providing a robust foundation for future work on linearized gravity and electromagnetism in rotating black hole spacetimes and potential numerical applications. Overall, the paper combines geometric, microlocal, and algebraic techniques to establish a physically meaningful quantum field theory for Teukolsky perturbations on Kerr backgrounds.

Abstract

We perform the quantization of Teukolsky scalars of spin $0$, $\pm 1$, and $\pm 2$ within the algebraic approach to quantum field theory. We first discuss the classical phase space, from which we subsequently construct the algebra. This sheds light on which fields are conjugates of each other. Further, we construct the Unruh state for this theory on Kerr and show that it is Hadamard on the black hole exterior and the interior up to the inner horizon. This shows not only that Hadamard states exist for this theory, but also extends the existence and Hadamard property of the Unruh state to (bosonic) Teukolsky fields on Kerr, where such a result was previously missing.

Figures (6)

• Figure 1: Penrose diagram of the Kruskal extension ${\rm M}_K$ of the Kerr spacetime. The blue and red regions represent the spacetimes ${\pazocal M}\sim {\rm M}_{{\rm I}\cup {\rm II}}^{in}$ and ${\rm M}_{{\rm I}\cup {\rm II}}^{out}$ embedded into ${\rm M}_K$, respectively. The prime on ${\rm M}_{\rm I}'$ and ${\rm M}_{\rm II}'$ denotes that they are equipped with the opposite time orientation.
• Figure 2: The black hole exterior ${\rm M}_{\rm I}$ and its conformal extension. The extension $\breve{\rm M}_{\rm I}$ based on the $K^*$-coordinates contains $\mathcal{I}_-$, while the extension $\widetilde{\rm M}_{\rm I}$ based on the ${}^*K$-coordiates contains $\mathcal{I}_+$.
• Figure 3: Structure of the semiclassical Hamiltonian flow for the original operator $\hat{T}_{s,h}^M(z_0)$ (for $z_0=1$ on the left and $z_0 = -1$ on the right), see Section \ref{['subsec:semiflow']}. Source : Millet2.
• Figure 4: Illustration of the Cauchy surface used in the proof of Corollary \ref{['cor:sympl form']} (red). The arc corresponds to the piece $\Sigma_{{\mathfrak t}_0}\cap\{x\ge \epsilon\}$.
• Figure 5: The Kruskal extension ${\rm M}_K$ of ${\pazocal M}$. The green, blue and red ellipsoids represent the sets $K$, ${\pazocal U}'$, and ${\pazocal U}$, respectively. The dashed lines in the corresponding colour delineate their causal pasts. The black dashed lines mark the region $[K,V]$. The three gray lines are the Cauchy surfaces $\Sigma$ and $\Sigma_\pm$. The cyan region marks the support of $\eta$, the darker shade is the region in which $\eta=1$. The orange region marks the support of $h_q$, the dotted region is where $h_q=1$ .

Theorems & Definitions (167)

• Theorem 1.1
• Proposition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Remark 2.6
• Remark 2.7
• Definition 2.8
• Lemma 2.9