Quantum scalar field theory on equal-angular-momenta Myers-Perry-AdS black holes

TL;DR

This paper analyzes the canonical quantization of a massive scalar field on a five-dimensional, equal-angular-momenta Myers–Perry–AdS black hole, where a timelike Killing vector outside the horizon allows a Hartle-Hawking equilibrium state in the absence of a speed-of-light surface. By exploiting enhanced symmetry, the authors separate the Klein-Gordon equation into angular and radial parts, use both Schrödinger-like and Heun formulations for the radial function, and construct Boulware- and Hartle-Hawking-like quantum states. They compute the differences in vacuum polarization and stress-energy tensor between these states, employing Hadamard-state regularity to avoid renormalization and performing extensive numerical analysis of mode sums with angular-addition theorems. The results show horizon divergences in the Boulware-Hawking difference, boundary decay toward AdS infinity, and a thermally rotating distribution near the horizon, with explicit dependence on the scalar-curvature coupling $\xi$ and effective mass $\nu$. These findings open avenues for renormalized SET calculations and interior-region extensions in higher-dimensional rotating AdS backgrounds.

Abstract

We study the canonical quantization of a massive scalar field on a five dimensional, rotating black hole space-time. We focus on the case where the space-time is asymptotically anti-de Sitter and the black hole's two angular momentum parameters are equal. In this situation the geometry possesses additional symmetries which simplify both the mode solutions of the scalar field equation and the stress-energy tensor. When the angular momentum of the black hole is sufficiently small that there is no speed-of-light surface, there exists a Killing vector which is time-like in the region exterior to the event horizon. In this case classical superradiance is absent and we construct analogues of the usual Boulware and Hartle-Hawking quantum states for the quantum scalar field. We compute the differences in expectation values of the square of the quantum scalar field operator and the stress-energy tensor operator between these two quantum states.

Figures (12)

• Figure 1: Penrose diagram of an equal-angular-momenta, asymptotically-anti-de Sitter, Myers-Perry black hole AlBalushi:2020rqe. Dotted lines denote horizons. The event horizon is located at $r=r_{+}$, and the inner horizon at $r=r_{-}$. Null infinity is the time-like surface at $r=\infty$, while there is a curvature singularity at $r=0$ (denoted by the dashed lines).
• Figure 2: Locations of the event horizon $r_{+}$ (\ref{['eq:horizon']}), stationary limit surface $r_{{\rm {S}}}$ (\ref{['eq:rs']}) and speed-of-light surface $r_{\rm {L}}$ (\ref{['eq:rL']}) as functions of the spin parameter $a$ for fixed mass parameter $M=10L^{2}$. We use units in which $L=1$.
• Figure 3: Location of the speed-of-light surface $r_{\rm {L}}$ (\ref{['eq:rL']}) as a function of the spin parameter $a$ for fixed mass parameter $M=10L^{2}$. The vertical lines give the values of $a_{\rm {max}}$ (\ref{['eq:amax']}) and $a_{\rm {ext}}$ (\ref{['eq:aext']}). We use units in which $L=1$.
• Figure 4: Typical integrand in (\ref{['eq:integral']}) as a function of the shifted frequency ${\widetilde{\omega }}$ (\ref{['eq:tildeomega']}). We have fixed the space-time parameters to be $M=10$, $L=1$, $a=1/2$ and the scalar field effective mass (\ref{['eq:nu']}) to be $\nu = 1/100$. The integrand is shown for radial coordinate $z=1/10$ (\ref{['eq:zdef']}), for the scalar field mode with $p=5$ and $\ell = 5/2$.
• Figure 5: Summand $S_{\ell}(r)$ (\ref{['eq:Sigma']}) as a function of $2\ell$ for a selection of values of $z$ (\ref{['eq:zdef']}). The space-time and scalar field parameters are as in Fig. \ref{['fig:integrand']}.