Quasinormal modes and strong cosmic censorship in near-extremal Kerr-Newman-de Sitter black-hole spacetimes

The quasinormal resonant modes of massless neutral fields in near-extremal Kerr-Newman-de Sitter black-hole spacetimes are calculated in the eikonal regime. It is explicitly proved that, in the angular momentum regime ${\bar a}>\sqrt{{{1-2{\bar\Lambda}}\over{4+{\bar\Lambda}/3}}}$, the black-hole spacetimes are characterized by slowly decaying resonant modes which are described by the compact formula $\Im\omega(n)=\kappa_+\cdot(n+{1\over2})$ [here the physical parameters $\{{\bar a},\kappa_+,{\bar\Lambda},n\}$ are respectively the dimensionless angular momentum of the black hole, its characteristic surface gravity, the dimensionless cosmological constant of the spacetime, and the integer resonance parameter]. Our results support the validity of the Penrose strong cosmic censorship conjecture in these black-hole spacetimes.


I. INTRODUCTION
The dynamics of linearized matter and radiation fields in black-hole spacetimes are characterized by a discrete family of complex (decaying in time) oscillation modes. These exponentially damped quasinormal resonances, which dominate the late-time relaxation dynamics of composed black-hole-field systems, have attracted the attention of physicists and mathematicians during the last five decades (see [1][2][3] for excellent reviews and detailed lists of references).
The characteristic quasinormal resonance spectrum {ω(l, m; n)} n=∞ n=0 (here the dimensionless angular parameters {l, m} are respectively the spheroidal harmonic index and the azimuthal harmonic index which characterize the linearized perturbation modes of the composed black-hole-field systems) of an asymptotically flat composed black-holefield system is determined by the linearized Einstein-matter field equations with the physically motivated boundary conditions of purely ingoing waves at the absorbing black-hole event horizon and purely outgoing waves at spatial infinity [4,5]. The fundamental black-hole-field quasinormal resonant mode (the mode with the smallest value of ℑω) determines the characteristic timescale τ relax ≡ 1/ℑω(n = 0) (1) for the decay (relaxation) of linearized perturbation fields in the exterior regions of the curved black-hole spacetime. Due to the mathematical complexity of the linearized Einstein-matter field equations, the complex resonant spectra of most black-hole spacetimes are not known in a closed (and compact) analytical form. Instead, one is usually forced to use numerical techniques in order to solve the linearized Einstein-matter field equations with the appropriate physically motivated boundary conditions which determine the composed black-hole-field quasinormal resonance spectra.
Near-extremal (rapidly-spinning) Kerr black holes are unique in this respect. In particular, solving analytically the linearized Einstein-matter field equations, it has been explicitly proved that the fundamental (least damped) quasinormal resonant frequencies of equatorial massless perturbation modes of near-extremal Kerr black holes are characterized by the remarkably compact analytical relation [6][7][8][9][10]: where T BH is the semi-classical Bekenstein-Hawking temperature [11,12] of the black-hole spacetime, which is related to the classical surface gravity κ + of its outer (event) horizon by the simple relation [11, 12] In a very interesting work [13], it has recently been demonstrated, using semi-analytical techniques, that near-extremal Kerr-de Sitter black holes are also characterized by the simple functional relation (2). Interestingly, defining the dimensionless black-hole rotation parameter (here r + is the radius of the black-hole outer (event) horizon [see Eq. (13) below]), one finds [7] that scalar perturbation modes of near-extremal charged and spinning Kerr-Newman black-hole spacetimes with large enough angular momenta,ā are also characterized by the compact analytical relation (2) [14][15][16]. Here the critical (minimal) black-hole rotation parameterā c (l), above which the quasinormal resonant modes of the near-extremal Kerr-Newman black-hole spacetimes are characterized by the compact analytical relation (2), depends on the angular harmonic index l of the linearized perturbation modes. In particular, it has been proved analytically that [7] a KN c (l ≫ 1) = 1 2 for near-extremal Kerr-Newman black holes in the eikonal (geometric-optics) l = m ≫ 1 regime.
The main goal of the present paper is to study analytically the quasinormal resonance spectra of massless neutral perturbation fields in non-asymptotically flat charged and rotating Kerr-Newman-de Sitter (KNdS) black-hole spacetimes. It is important to note that black holes in asymptotically de Sitter spacetimes have recently attracted much attention in the context of the intriguing Penrose strong cosmic censorship (SCC) conjecture [17,18]. In particular, it has been shown (see [13,[19][20][21] and references therein) that the validity of the fundamental SCC conjecture in asymptotically de Sitter spacetimes depends on the existence of (at least) one black-hole-field perturbation mode with the property where κ − is the surface gravity which characterizes the inner (Cauchy) horizon of the black-hole spacetime [13,[19][20][21].
Using analytical techniques, we shall explicitly prove below that, in the eikonal (geometric-optics) regime l ≫ 1, the quasinormal resonant modes of the near-extremal KNdS black-hole spacetimes with large enough angular momenta, a >ā KNdS c (Λ) (hereΛ ≡ Λr 2 + > 0 is the dimensionless cosmological constant of the black-hole spacetime), are characterized by the compact functional relation (2). In particular, we shall determine the functional dependencē a KNdS c =ā KNdS c (Λ; l ≫ 1) of the critical black-hole rotation parameter above which the neutral large-l fundamental perturbation modes of the near-extremal KNdS black holes conform to the important inequality (7) [22].

II. DESCRIPTION OF THE SYSTEM
We analyze the quasinormal resonance spectra of massless neutral fields which are linearly coupled to a nonasymptotically flat Kerr-Newman-de Sitter black-hole spacetime of mass M , angular momentum J ≡ M a, electric charge Q, and cosmological constant Λ > 0. The line element of the curved black-hole spacetime can be expressed in the form [23][24][25] where the metric functions are given by [23,24] and The zeroes of the radial metric function [23,24] ∆ r (r * ) = 0 with * ∈ {−, +, c} (13) determine the horizon radii which characterize the KNdS black-hole spacetime (8). For generic KNdS black holes, there are four distinct (non-degenerate) roots r 0 < 0 < r − ≤ r + ≤ r c to the characteristic equation (13), where r − is the inner (Cauchy) horizon, r + is the outer (event) horizon of the black hole, and r c is the radius of the cosmological horizon.

III. THE QUASINORMAL RESONANCE SPECTRA OF NEAR-EXTREMAL KERR-NEWMAN-DE SITTER BLACK-HOLE SPACETIMES
In the present section we shall use analytical techniques in order to calculate the quasinormal resonance spectra of near-extremal charged and rotating KNdS black-hole spacetimes. In particular, we shall use the well established [27][28][29] relation between the black-hole quasinormal resonant frequencies in the eikonal large-l regime and the unstable null circular geodesics which characterize the corresponding black-hole spacetimes.
The quasinormal resonant modes which dominate the linearized relaxation dynamics of neutral perturbation fields in asymptotically de Sitter back-hole spacetimes are characterized by the physically motivated boundary conditions of purely ingoing waves at the outer (event) horizon of the black hole and purely outgoing waves at the cosmological horizon of the spacetime [30]: where the tortoise radial coordinate y is defined by the differential relation dy = [(r 2 + a 2 )/∆ r ]dr. As explicitly proved in [27][28][29], in the eikonal (geometric-optics) l ≫ 1 regime, the real parts of the black-hole quasinormal resonant frequencies are directly related (proportional) to the characteristic angular velocity Ω c of null particles which are trapped at the unstable null circular geodesic of the black-hole spacetime. Likewise, the imaginary parts of the complex resonant frequencies are given, in the eikonal large-l regime, by the remarkably compact relation [29] ℑω(n) = −i(n + 1 2 ) · |γ| ; n = 0, 1, 2, ... , where the integer n is the resonance parameter of the composed black-hole-field perturbation mode, and [29] is the Lyapunov exponent which characterizes the instability timescale τ = γ −1 of the null circular orbit [31] (the dot symbol˙denotes a derivative with respect to the proper time τ ). Here V r (r) [see Eq. (21) below] is an effective radial potential which determines the geodesic motions of test particles in the black-hole spacetime. It is worth emphasizing again that, as shown in [29], Eq. (17) is valid in the asymptotic large-l regime.
The geodesic motions of a test particle of proper mass m in the KNdS black-hole spacetime are characterized by three conserved quantities {E, L z , K} which are respectively related to the stationarity property of the spacetime geometry, to its axial symmetry, and to the hidden symmetry of the black-hole geometry [23,[32][33][34]. In particular, the equatorial motions of test particles in the KNdS black-hole spacetime are governed by the geodesic equations [23,[32][33][34] and where and Here the affine parameter λ is related to the proper time τ by the simple relation τ = mλ [23,[32][33][34].
We shall now use analytical techniques in order to determine the physical and mathematical properties which characterize the equatorial null circular geodesics of near-extremal [see Eq. (14)] KNdS black-hole spacetimes. To this end, it proves useful to define the dimensionless small physical parameters Substituting Eq. (29) into Eqs. (27) and (28), and using the near-horizon expansions [36] ∆ r (r = r c ) = r + ∆ ′ r (r = r + ) · x + 1 2 and ∆ ′ r (r = r c ) = ∆ ′ r (r = r + ) + r + ∆ ′′ one obtains the leading-order (with x ≪ 1 and y ≪ 1) equations and a r 2 + + a 2 · 2r 2 for the equatorial null circular geodesics of the near-extremal KNdS black-hole spacetimes. From the two coupled equations (32) and (33) one finds the two dimensionless physical parameters and which characterize the near-horizon (x ≪ 1) null circular geodesics of the near-extremal [∆ ′ r (r + )/r + ≪ 1, see Eq. (14)] KNdS black-hole spacetimes.

IV. BLACK-HOLE QUASINORMAL RESONANT FREQUENCIES AND THE PENROSE STRONG COSMIC CENSORSHIP CONJECTURE
The strong cosmic censorship conjecture, introduced by Penrose almost five decades ago [18], asserts that, starting with physically reasonable (spatially regular) initial conditions, the dynamics of self-gravitating matter and radiation fields, which are governed by the Einstein field equations, will always produce globally hyperbolic spacetimes. If true, this physically important conjecture guarantees that classical general relativity is a deterministic theory.
It is well known that eternal black-hole spacetimes which possess regular inner Cauchy horizons are not globally hyperbolic [38][39][40]. In particular, for eternal charged and rotating black holes, the inner spacetime regions which are located beyond the black-hole Cauchy horizons are characterized by the presence of past directed null geodesics that terminate on the inner timelike singularities of the eternal black-hole spacetimes [38][39][40]. Thus, the Einstein field equations may fail to determine uniquely the future dynamics of physical observers who fall into eternal black holes which contain regular inner Cauchy horizons [38][39][40].
As discussed in [19,20] (see also [13,21] and references therein), in asymptotically de Sitter black-hole spacetimes, the final fate of the fundamental Penrose SCC conjecture [18] is determined by the dimensionless ratio between the physical parameters ℑω 0 and κ − , which respectively characterize the decay rate of the linearized perturbation modes in the exterior regions of the dynamically formed black-hole spacetimes and the amplification rate of the infalling fields as they approach the inner Cauchy horizons of the corresponding black-hole spacetimes [50].
In particular, the validity of the SCC conjecture in dynamically formed non-asymptotically flat charged and spinning KNdS black-hole spacetimes depends on the existence of (at least) one black-hole-field perturbation mode which is characterized by the property [13,[19][20][21] ℑω Taking cognizance of the analytically derived functional relation (48), and using the inequality κ + ≤ κ − [40] which characterizes the surface gravities of the black-hole outer (event) and inner (Cauchy) horizons, one deduces that the fundamental (least damped, n = 0) neutral perturbation mode of the KNdS black-hole spacetimes conforms to the dimensionless relation (49). We therefore conclude that near-extremal charged and rotating KNdS blackhole spacetimes in the dimensionless physical regime (36) respect the fundamental Penrose strong cosmic censorship conjecture [18,51].

V. SUMMARY
The quasinormal resonance spectra which characterize the relaxation dynamics of linearized neutral fields in nonasymptotically flat near-extremal Kerr-Newman-de Sitter black-hole spacetimes have been studied analytically in the eikonal (geometric-optics) regime. We have proved that the characteristic relaxation rates of the composed blackhole-field systems are determined by the surface gravities [see Eqs. (17), (46), and (47) for the quasinormal resonant modes of the composed near-extremal-Kerr-Newman-de-Sitter-black-hole-linearizedneutral-field systems. Finally, we have pointed out that the fundamental (least damped, n = 0) resonant mode of the near-extremal black-hole spacetimes conforms to the inequality ℑω/κ − ≤ 1/2 [see Eq. (49)] which is imposed by the fundamental SCC conjecture [18]. The compact analysis presented in this paper therefore supports the validity of the Penrose strong cosmic censorship conjecture in these non-asymptotically flat charged and spinning Kerr-Newman-de Sitter black-hole spacetimes.