Semiclassical Rotating AdS Black Holes with Quantum Hair in Holography

Charged and rotating black holes have garnered significant attention in both the gravity and high-energy physics communities. A key topic of interest is the stability of the Cauchy horizon, which lies deep inside the event horizon. According to the strong cosmic censorship (SCC) conjecture, proposed by Penrose [1], the evolution of spacetime should be uniquely determined by regular initial data under physically reasonable conditions. In this framework, the presence of a Cauchy horizon is generally considered unphysical, as it would indicate a breakdown of determinism in general relativity. In classical general relativity, it is well established that the Cauchy horizon is generically unstable against perturbations [2, 3]. However, the resulting singularity is weak in the sense that tidal distortions remain finite for an observer crossing the Cauchy horizon [4]. Consequently, many $C^{0}$ extensions of the metric are possible beyond the Cauchy horizon, suggesting a potential violation of the SCC in classical theory.

Thus, it is natural to ask whether quantum effects can uphold the SCC conjecture. One tractable approach is to study quantum field theory in a fixed curved spacetime background. A quantized free scalar field has been extensively analyzed in various charged and rotating black hole backgrounds [5, 6, 7, 8, 9]. Recent work, for instance, rigorously demonstrated that the vacuum expectation value of the stress-energy tensor diverges along a radial null geodesic as $\sim C\lambda^{-2}$, where $\lambda$ is the affine parameter, near the Cauchy horizon ($\lambda=0$) of a four-dimensional Reissner-Nordström-de Sitter black hole [9]. In contrast, for the near-extremal rotating BTZ black hole, a three-dimensional vacuum solution with a negative cosmological constant [10], the vacuum expectation value remains finite, as shown in [8], due to the vanishing coefficient $C=0$ (see [9] for detailed analysis). These findings indicate that the coefficient $C$, which determines the formation of a strong curvature singularity at the Cauchy horizon, depends on the specific background spacetime. Therefore, it remains unclear whether quantum field theory in a fixed curved background can consistently support the SCC conjecture. An alternative scenario supporting the SCC conjecture could involve significant quantum modifications to the background geometry itself. As observed within a classical theory [11], one might expect that the backreaction destroys the Cauchy horizon of charged or rotating black holes.

Another intriguing feature of charged or rotating black holes is the existence of extremal states with zero temperature. Motivated by the AdS/CFT duality [12], recent studies have explored static or stationary perturbations of extremal Reissner-Nordström-AdS black holes and rotating AdS black holes near their event horizons [13, 14, 15, 16]. As one approaches the horizon from the outside, the energy density or a curvature component in the parallelly propagated frame along a causal geodesic crossing the horizon diverges, despite the perturbations converging to zero and all scalar curvature invariants remaining finite. Such singularities are absent in non-extremal black holes with finite temperature. However, the curvature at the horizon grows indefinitely as the temperature decreases [15]. This behavior suggests a potential violation of the weak cosmic censorship (WCC), which posits that a distant observer near future null infinity cannot witness the formation of a naked singularity. Therefore, it becomes important to investigate whether quantum effects can prevent extremal black holes from developing curvature singularities at their event horizons.

In this paper, we investigate semiclassical rotating AdS black holes in both non-extremal and extremal cases. Given the absence of a complete theory of quantum gravity, the most tractable approach is the semiclassical procedure, where gravity is treated classically, while matter fields are treated quantum mechanically through the semiclassical Einstein equations. The source term in these equations is provided by the vacuum expectation value of the stress-energy tensor. Previous studies have been limited to lower-dimensional semiclassical black hole solutions sourced by a quantized, free, massless scalar field [17], or to four-dimensional spherically symmetric solutions under the s-wave approximation of a massless, minimally coupled scalar field [18]. To extend these works, we explore semiclassical solutions for strongly coupled quantum fields with a gravity dual via the AdS/CFT duality.

Initially proposed in Ref.[19], the semiclassical equations on a $d$-dimensional boundary spacetime can be reformulated as mixed boundary conditions on a ($d+1$)-dimensional asymptotically AdS bulk spacetime by promoting the boundary metric to a dynamical field. Following this procedure, an analytical semiclassical (static) AdS black hole solution was obtained by introducing even-parity perturbations to the BTZ solution [20]. Extending this analysis to rotating cases, we construct three- and five-dimensional semiclassical rotating AdS black holes by perturbing the rotating BTZ solution and the five-dimensional Myers-Perry AdS solution [21], respectively.

In the three-dimensional case, the non-extremal solution is obtained analytically by adding odd-parity perturbation to the static hairy solution [20] and applying a Lorentz boost. For the rotating solution, the vacuum stress-energy tensor diverges as $\sim\lambda^{-n}~{}(n=1,2)$ along a radial null geodesic with affine parameter $\lambda$ crossing the Cauchy horizon ($\lambda=0$), depending on the type of perturbation. In the five-dimensional case, we derive three coupled second-order differential equations with six independent solutions. Interestingly, while the stress-energy tensor or the parallelly propagated Riemann components along the radial null geodesic strongly diverges as $\sim\lambda^{-2}$ near the Cauchy horizon for most parameter choices, it is possible to fine-tune the parameters to construct a specific hairy solution with a $C^{0}$-regular Cauchy horizon.

In the extremal case, the semiclassical extremal solution in three-dimensions is obtained analytically. The perturbation falls off as a power law, $\sim(r-r_{0})^{\gamma}$, near the extremal horizon, $r=r_{0}$, with a very slow decay rate, i. e. , $0<\gamma<1$. This slow decay indicates the formation of a curvature singularity on the horizon, similar to the case of classical perturbations in extremal black holes [13, 14, 15, 16]. This singular behavior also leads to the divergence of the vacuum expectation value of the stress-energy tensor along a radial null geodesic transverse to the horizon. In the five-dimensional case, there are six independent mode solutions near the extremal horizon. We show that the power-law index $\gamma$ satisfies $\gamma<2$ for all parameters of the background Myers-Perry AdS solution, similar to previous semiclassical models [18]. This suggests that the formation of a curvature singularity on the extremal horizon is a generic feature of semiclassical extremal rotating black holes.

This paper is organized as follows: In section II, we briefly review the holographic set up for constructing the semiclassical solutions. In section III and IV, we construct three and five-dimensional semiclassical rotating AdS black hole solutions. Section V is devoted to summary and discussions.

We start with $d+1~{}(d=3,\,5)$-dimensional asymptotically AdS bulk spacetime with metric

where $X^{M}=(z,\,x^{\mu})$ and $L$ ($\ell$) is the bulk (boundary) AdS length and the AdS boundary is located at $z=0$ and $z=\pi$. The bulk action consists of the $d+1$-dimensional Einstein Hilbert action $S_{\text{bulk}}$, Gibbons-Hawking term $S_{\text{GH}}$, and the counter term $S_{\text{ct}}^{(d)}$ given by

where $G_{d}$ is the $d$-dimensional gravitational constant and $S^{(d)}_{\text{ct}}$ is given by

We consider $d$-dimensional spacetime with the conformal boundary metric $\mathcal{G}_{\mu\nu}:=\lim_{z\to 0}\tilde{g}_{\mu\nu}(z,x)$ satisfying the semiclassical Einstein equations [20]. According to the AdS/CFT dictionary [22, 23], the vacuum expectation value of the boundary stress-energy tensor is given by the variation of the bulk action (II) with respect to the conformal boundary metric $\mathcal{G}_{\mu\nu}$ as ¹¹1In detail derivation, see Ref. [20].

where a prime denotes differentiation with respect to $z$ and the extrinsic curvature $\tilde{K}_{\mu\nu}$ of the conformal metric $\tilde{g}_{\mu\nu}$ is defined by

and

Here, $\tilde{R}_{\mu\nu}$ is the Ricci tensor of the conformal metric $\tilde{g}_{\mu\nu}$.

The bulk geometry (II) is the solution of the $d+1$-dimensional bulk Einstein equations with a negative cosmological constant $\Lambda_{d+1}=-d(d-1)/(2L^{2})$ if the $d$-dimensional metric $\tilde{g}_{\mu\nu}=\tilde{g}_{\mu\nu}(x)$ satisfies the vacuum Einstein equations with a negative cosmological constant $\Lambda_{d}=-(d-1)(d-2)/(2\ell^{2})$. So, if the metric $\tilde{g}_{\mu\nu}$ corresponds to that of the BTZ black hole [10] or the $5$-dimensional Myers-Perry AdS black hole [21], the bulk Einstein equations are automatically satisfied. Furthermore, the vacuum expectation value of the stress-energy tensor (II) vanishes identically in both $d=3,\,5$ cases, due to the absence of the Weyl anomaly in odd dimensions. This implies that these bulk geometries are solutions to the semiclassical Einstein equations ²²2Here, for simplicity, we ignored higher curvature corrections in $d=5$ case. on the boundary:

where $\mathcal{R}_{\mu\nu}$ is the Ricci tensor of the metric $\mathcal{G}_{\mu\nu}$.

We aim to find semiclassical solutions of Eqs. (8) with non-zero expectation values, i. e. , $\left\langle~{}\mathcal{T}_{\mu\nu}~{}\right\rangle\neq 0$, by perturbing the background solution. To achieve this, we expand the metric $\tilde{g}_{\mu\nu}$ as

where $\overline{\mathcal{G}}_{\mu\nu}(x)$ is the background metric of the BTZ black hole or the $5$-dimensional Myers-Perry AdS black hole, and $\epsilon$ is an infinitesimally small parameter. Under the metric ansatz $h_{\mu\nu}=\xi(z)H_{\mu\nu}(x)$, the perturbed bulk Einstein equations can be decomposed with a separation constant $m^{2}$ as

where $\overline{\mathcal{D}}_{\mu}$ and $\overline{\mathcal{R}}_{\mu\nu\alpha\beta}$ are the covariant derivative and the Riemann tensor with respect to the background metric $\overline{\mathcal{G}}_{\mu\nu}$, respectively, and we impose the transverse and traceless condition

The AdS boundary of the bulk spacetime (II) consists of two portions: $z=0$, where the conformal boundary metric $\mathcal{G}_{\mu\nu}$ satisfying the semiclassical Einstein equations, and $z=\pi$. According to the procedure [20], we shall impose the Dirichlet boundary condition at $z=\pi$. As shown in Ref. [24], the solution of Eq. (10) is expressed by the hypergeometric function as

where $C$ is a constant and $p$ is defined by

with $\hat{m}^{2}:=m^{2}\ell^{2}$. As shown in the perturbations of the maximally symmetric AdS spacetimes [20, 24], the perturbed semiclassical Eqs. (8) reduce to an algebraic equation characterized by the dimensionless constant $\gamma_{d}$, defined by

For $d=3$, the first order perturbation of Eqs. (8) using Eqs. (II), (II), and (13) results in the algebraic equation [20]

which allows a hairly solution in the range

For $d=5$, the perturbed semiclassical Eqs. (8) in the Myers-Perry AdS black hole background reduce to the algebraic equation (77), which is identical to that derived in the perturbation of the $5$-dimensional maximally symmetric AdS solution [24]. The solution was found for a negative mass satisfying the inequality

when $\gamma_{5}$ is larger than a critical value [24]. In the following sections, we shall focus on the solutions of (11) under the inequalities, (17) and (18), ensuring that the algebraic equations (16) and (77) are also satisfied.

In this section, we investigate $3$-dimensional rotating AdS black holes by solving Eqs. (11) for a negative mass satisfying the inequality (17) in the background of the rotating BTZ solution [10]. In the non-extremal case, an analytic solution is constructed by adding an odd-parity perturbation to the static solution [20] and by applying a Lorentz boost to the perturbed solution. In the extremal case, we also obtain an analytic semiclassical solution in which $H_{\mu\nu}$ in Eqs. (11) converges to zero near the horizon. Nevertheless, we demonstrate that the expectation value of the stress-energy tensor diverges along a null geodesic crossing the event horizon.

In this section we construct $5$-dimensional semiclassical rotating AdS black hole solutions by solving Eqs. (11) and (12). In general, it is difficult to solve these equations in the background of rotating AdS black holes because the perturbations are partial differential equations. One notable exception is the odd-dimensional Myers-Perry (MP) AdS black hole [21] with equal angular momenta, which is known to be cohomogeneity-one solution of the vacuum Einstein equations with a negative cosmological constant. In this case, the equations become coupled ordinary differential equations, making it possible to analyze the singular behavior of the Cauchy horizon both analytically and numerically. Hereafter, we construct the solutions of Eqs. (11) and (12) for the $5$-dimensional MP AdS black hole with equal angular momenta in both non-extremal and extremal cases.