Quasinormal resonances of rapidly-spinning Kerr black holes and the universal relaxation bound

The universal relaxation bound suggests that the relaxation times of perturbed thermodynamical systems is bounded from below by the simple time-times-temperature (TTT) quantum relation $\tau\times T\geq {{\hbar}\over{\pi}}$. It is known that some perturbation modes of near-extremal Kerr black holes in the regime $MT_{\text{BH}}/\hbar\ll m^{-2}$ are characterized by normalized relaxation times $\pi\tau\times T_{\text{BH}}/\hbar$ which, in the approach to the limit $MT_{\text{BH}}/\hbar\to0$, make infinitely many oscillations with a tiny constant amplitude around $1$ and therefore cannot be used directly to verify the validity of the TTT bound in the entire parameter space of the black-hole spacetime (Here $\{T_{\text{BH}},M\}$ are respectively the Bekenstein-Hawking temperature and the mass of the black hole, and $m$ is the azimuthal harmonic index of the linearized perturbation mode). In the present compact paper we explicitly prove that all rapidly-spinning Kerr black holes respect the TTT relaxation bound. In particular, using analytical techniques, it is proved that all black-hole perturbation modes in the complementary regime $m^{-1}\ll MT_{\text{BH}}/\hbar\ll1$ are characterized by relaxation times with the simple dimensionless property $\pi\tau\times T_{\text{BH}}/\hbar\geq1$.


I. INTRODUCTION
Perturbed black-hole spacetimes are usually characterized by a relaxation phase which is dominated by exponentially decaying quasinormal oscillations whose discrete complex frequencies {ω n } n=∞ n=0 encode valuable information about the fundamental physical parameters (in particular, the mass and angular momentum) of the perturbed black hole [1][2][3][4][5][6].
The timescale τ relax associated with the relaxation dynamics of a newly born black hole may be determined by the imaginary part of the fundamental (least damped) quasinormal resonant frequency: Taking cognizance of the fact that black holes have thermodynamic and quantum properties [20,21], a physically important question naturally arises: How short can the relaxation time (1) of a newly born black hole be?
A mathematically compact answer to this physically intriguing question, which is based on standard ideas from classical thermodynamics and quantum information theory, has been suggested in [22,23]: where T is the characteristic temperature of the thermodynamic physical system. Intriguingly, remembering that the semi-classical 21,24] of black holes depends linearly on the quantum Planck constant (here κ is the characteristic surface gravity of the black-hole horizon), one realizes that the time-times-temperature (TTT) quantum bound (2) provides a classical lower bound [25] on the characteristic relaxation time of a newly born black hole. In particular, taking cognizance of Eqs. (1), (2), and (3), one realizes that all dynamically formed black holes should be characterized by (at least) one relaxation mode with the simple classical property

II. RELAXATION DYNAMICS OF RAPIDLY SPINNING KERR BLACK HOLES
Interestingly, the upper bound (4) implies that newly born near-extremal black holes (with the property T BH → 0) should be characterized by extremely long relaxation times [22,26,27]. In particular, using the physically important and mathematically elegant Detweiler equation [28], which characterizes the complex resonant spectra of rapidlyspinning (near-extremal) Kerr black holes, one obtains the simple functional relation [29][30][31] for the co-rotating perturbation modes of near-extremal black holes, where C(l, m) is a constant [32], M is the blackhole mass, and the physical parameter δ(l, m) ∈ R is closely related to the characteristic eigenvalue of the angular Teukolsky equation [33,34]. Before proceeding, it is important to stress the fact that the validity of the Detweiler resonance equation [28], and thus also the validity of the analytically derived expression (5) [29,30], are restricted to the dimensionless regime [33,35,36] where m is the azimuthal harmonic index of the black-hole perturbation mode. From Eq. (5) one immediately finds that perturbation modes of rapidly-spinning Kerr black holes in the regime (6) are characterized by normalized relaxation times πτ × T BH that oscillate infinitely many times around 1 as the extremal limit M T BH → 0 is approached. One therefore deduces that, in the near-extremal M T BH ≪ 1 regime, there are finite intervals of the black-hole temperature for which a perturbation mode in the dimensionless regime (6) cannot be used to prove the general validity of the TTT relaxation bound (2) [or equivalently, the upper bound (4)] in black-hole physics [37].
The main goal of the present compact paper is to prove the physically important fact that all rapidly-spinning Kerr black holes respect the TTT relaxation bound (2). To this end, we shall study below the linearized relaxation dynamics of newly born near-extremal Kerr black holes. In particular, using analytical techniques, we shall explicitly prove that composed black-hole-field perturbation modes in the eikonal large-m regime are characterized by relaxation times that respect the TTT relaxation bound.

III. RESONANT RELAXATION SPECTRA OF RAPIDLY SPINNING KERR BLACK HOLES IN THE EIKONAL LARGE-FREQUENCY REGIME
Using a physically intuitive and mathematically elegant analysis, Mashhoon [38] has presented a simple analytical technique for calculating the quasinormal resonance spectra of black holes in the large-frequency (geometric-optics) regime. In particular, the physical idea presented in [38,39] is to relate, in the eikonal large-frequency regime, the real part of the black-hole quasinormal frequencies to the angular velocity which characterizes the motion of massless particles along the equatorial null circular geodesic of the black-hole spacetime and to relate the imaginary part of the black-hole quasinormal frequencies to the instability timescale which characterizes the gradual leakage of the perturbed massless particles (null rays) from the unstable null circular geodesic of the black-hole spacetime.
In particular, using a perturbation scheme for the instability which characterizes the dynamics of equatorial null circular geodesics in the Kerr black-hole spacetime, Mashhoon has provided the remarkably compact formula [38] ω n = mω + − i(n + 1 2 )βω + ; n = 0, 1, 2, ... , for the discrete quasinormal resonant spectra of spinning Kerr black holes. As emphasized in [38], this resonance formula is valid for perturbation modes in the eikonal large-frequency regime where {l, m} are the (spheroidal and azimuthal) angular harmonic indexes which characterize the linearized field mode. The various terms in the resonance formula (8) have the following physical interpretations [38]: (1) The term is the radius of the co-rotating equatorial null circular geodesic, where {M.a} are respectively the mass and angular momentum per unit mass [40] of the central spinning Kerr black hole.
(2) The term is the Kepler frequency which characterizes the co-rotating equatorial null circular geodesic of the black-hole spacetime.
(3) The dimensionless functional expression of β = β(M, a) is given by [38] β(a/M ) where are the (outer and inner) horizon radii of the spinning Kerr black hole. We shall henceforth focus our attention on rapidly-spinning (near-extremal) Kerr black holes, which are characterized by the simple dimensionless relation From Eqs. (13) and (14) one finds the relation Substituting (14) into Eq. (10), one obtains the near-extremal (small-ǫ) dimensionless functional expression for the radius of the co-rotating equatorial null circular geodesic. Substituting Eqs. (14), (15), and (16) into Eqs. (11) and (12), one finds and Substituting Eqs. (17) and (18) into the resonance formula (8), one obtains the dimensionless functional expression for the complex quasinormal resonant frequencies of rapidly-spinning (near-extremal) Kerr black holes in the eikonal large-frequency regime l = m >> 1 [see Eq. (9)]. It is physically interesting and mathematically convenient to express the analytically derived Kerr resonance spectrum (19) in terms of the black-hole 21] and the black-hole angular velocity Substituting (20) and (21) into Eq. (19), one finds the black-hole resonance formula

IV. SUMMARY
Motivated by the suggested time-times-temperature (TTT) universal relaxation bound (2), we have studied, using analytical techniques, the quasinormal resonance spectra of near-extremal (rapidly-spinning) Kerr black holes.
It was first noted that perturbation modes of near-extremal Kerr black holes in the regime m 2 · M T BH ≪ 1 are characterized by normalized relaxation times πτ × T BH which, in the approach to the extremal limit M T BH → 0, oscillate infinitely many times around 1 and therefore cannot be used to verify the general validity of the suggested TTT relaxation bound (2) [or equivalently, the upper bound (4)] in black-hole physics.
We have therefore analyzed the black-hole relaxation spectra in the complementary regime m −1 ≪ M T BH ≪ 1. In particular, we have explicitly proved that the equatorial black-hole-field perturbation modes in the eikonal largefrequency regime (7) are characterized by the simple near-extremal relation [It is worth noting that the functional relation (23) characterizes the fundamental (n = 0) resonant mode of the analytically derived black-hole relaxation spectrum (22)]. We therefore conclude that the eikonal relaxation modes (23), which characterize the composed black-hole-field system in the dimensionless large-frequency regime m −1 ≪ M T BH ≪ 1, guarantee that newly born near-extremal Kerr black holes respect the suggested TTT universal relaxation bound (2) [41].