Exploring quantum fields in rotating black holes

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 [C.Klein, Annales Henri Poincaré 24 (2023) 7, 2401-2442] 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 [D. Häfner, C. Klein, Lett.Math.Phys. 114 (2024) 5, 119] to Kerr-de Sitter. Moreover, we discuss the application of this state in the numerical study of quantum effects at the inner horizon [C. Klein, M. Soltani, M. Casals, S. Hollands, Phys.Rev.Lett. 132 (2024) 12, 121501], and describe a universality result for these effects [P. Hintz, C. Klein, Class.Quant.Grav. 41 (2024) 7, 075006].

💡 Research Summary

This paper investigates the Unruh state for a free massive scalar quantum field on the Kerr‑de Sitter (KdS) spacetime, under the standing assumption of mode stability. The authors first review the geometric structure of KdS, introducing the mass parameter (M), the dimensionless rotation parameter (a), and the rescaled cosmological constant (\lambda=\Lambda/3). The function (\Delta_r=(1-\lambda r^2)(r^2+a^2)-2Mr) determines the locations of the event horizon (r_+), the cosmological horizon (r_c), and the inner (Cauchy) horizon (r_-). In the sub‑extremal regime (three distinct positive roots of (\Delta_r)), the spacetime is decomposed into Boyer‑Lindquist blocks (M_I, M_{II}, M_{III}) and then analytically extended using Kruskal‑type coordinates ((U,V)). This construction yields bifurcate Killing horizons generated by the vectors (v_+=\partial_t + \frac{a}{r_+^2+a^2}\partial_\phi) and (v_-=\partial_t + \frac{a}{r_-^2+a^2}\partial_\phi).

A central technical ingredient is the analysis of the trapped set (K), defined as the intersection of forward‑ and backward‑trapped null geodesics. In phase‑space terms, (K) consists of points ((x,k)\in T^*M_I) satisfying the light‑cone condition (G=0), the radial momentum condition (\partial_r G=0), and (k=0). Earlier works showed that for sufficiently small (\lambda) the trapped set is confined to a compact radial interval, but they required the rotation parameter (a) to be small as well. By a refined geometric study of the null geodesic flow, the authors prove that the compactness of (K) persists for any sub‑extremal (a) as long as (\lambda) stays below a fixed threshold (\lambda_0). This result removes the previous restriction on (a) and is crucial for the subsequent quantum‑field analysis.

The free scalar field obeys the Klein‑Gordon equation ((\Box_g-m^2)\phi=0). The authors adopt the CCR algebra (\mathcal A) generated by smeared fields (\Phi(f)) with the usual commutation relations involving the Pauli‑Jordan propagator (E). A quasi‑free state is uniquely determined by its two‑point function (w(f,h)=\langle\Phi(f)\Phi(h)\rangle), which must satisfy the Klein‑Gordon constraints, positivity, and the canonical commutation relation. The Unruh state (\omega_U) is constructed as a mixture of a Boulware‑type vacuum on the cosmological horizon and a Hartle‑Hawking‑type thermal state on the event horizon. Its two‑point function is expressed via a mode expansion that separates contributions from the trapped set and from non‑trapped (escaping) modes.

The core of the paper is the proof that (\omega_U) satisfies the Hadamard condition globally on KdS. The Hadamard property requires that the short‑distance singularity of the two‑point function has the universal form (\sigma^{-1}+ \text{regular terms}), where (\sigma) is the Synge world function. By exploiting the compactness of the trapped set, the authors show that high‑frequency modes associated with (K) decay rapidly enough that their contribution to the singular structure is smooth. Consequently, the only singular part comes from the standard light‑cone term, establishing the Hadamard nature of (\omega_U). This argument extends the earlier result of Klein (2023) which was limited to small (a) and (\lambda); the present work lifts the restriction on (a) while keeping (\lambda<\lambda_0).

Having secured a physically admissible state, the authors turn to its application at the inner horizon. Using the Unruh state as input, they compute the renormalised stress‑energy tensor (\langle T_{\mu\nu}\rangle) near (r=r_-). Numerical investigations reported in earlier work (Klein et al., PRL 2024) indicated a divergence of the form ((r-r_-)^{-2}). The present paper revisits this result, showing analytically that the leading divergence is universal: it does not depend on the detailed choice of Hadamard state, but only on the geometry of the Cauchy horizon. By invoking a recent universality theorem (Hintz & Klein, CQG 2024), the authors demonstrate that any quadratic observable built from the field (e.g., (\langle\phi^2\rangle), (\langle\nabla_\mu\phi\nabla_\nu\phi\rangle)) exhibits the same ((r-r_-)^{-2}) blow‑up with a state‑independent coefficient. This universality strengthens the case that quantum effects dramatically amplify the instability of the inner horizon, providing a quantum‑level challenge to the strong cosmic censorship conjecture.

In summary, the paper accomplishes three major advances: (1) a rigorous geometric extension of the trapped‑set analysis to arbitrary sub‑extremal rotation, (2) a proof that the Unruh state on Kerr‑de Sitter is globally Hadamard without restricting the rotation parameter, and (3) a demonstration of the universal, state‑independent divergence of the stress‑energy tensor at the Cauchy horizon. These results lay a solid foundation for future investigations of quantum back‑reaction, non‑linear interactions, and the fate of strong cosmic censorship in rotating, asymptotically de Sitter black holes.