# The Reissner–Nordström black hole with the fastest relaxation rate

## Abstract

Numerous *numerical* investigations of the quasinormal resonant spectra of Kerr-Newman black holes have revealed the interesting fact that the characteristic relaxation times \(\tau ({\bar{a}},{\bar{Q}})\) of these canonical black-hole spacetimes can be described by a two-dimensional function \({{\bar{\tau }}}\equiv \tau /M\) which increases monotonically with increasing values of the dimensionless angular-momentum parameter \({\bar{a}}\equiv J/M^2\) and, in addition, is characterized by a non-trivial (*non*-monotonic) functional dependence on the dimensionless charge parameter \({\bar{Q}}\equiv Q/M\). In particular, previous numerical investigations have indicated that, within the family of spherically symmetric charged Reissner–Nordström spacetimes, the black hole with \({\bar{Q}}\simeq 0.7\) has the *fastest* relaxation rate. In the present paper we use *analytical* techniques in order to investigate this intriguing non-monotonic functional dependence of the Reissner–Nordström black-hole relaxation rates on the dimensionless physical parameter \({\bar{Q}}\). In particular, it is proved that, in the eikonal (geometric-optics) regime, the black hole with \({\bar{Q}}={{\sqrt{51-3\sqrt{33}}}\over {8}}\simeq 0.73\) is characterized by the *fastest* relaxation rate (the smallest dimensionless relaxation time \({{\bar{\tau }}}\)) within the family of charged Reissner–Nordström black-hole spacetimes.

## 1 Introduction

The mathematically elegant uniqueness theorems of Israel, Carter, and Hawking [1, 2, 3, 4] (see also [5, 6, 7]) have revealed the physically important fact that all asymptotically flat stationary black-hole solutions of the non-linearly coupled Einstein-Maxwell field equations belong to the Kerr-Newman family of curved spacetimes [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. The three-dimensional phase-space of these canonical black-hole spacetimes is described by conserved physical parameters that can be measured by asymptotic observers: the black-hole mass *M*, the black-hole electric charge *Q*, and the black-hole angular momentum \(J\equiv Ma\).

In accord with the results of the uniqueness theorems [1, 3, 4, 5, 6, 7], the linearized dynamics of massless gravitational and electromagnetic fields in perturbed Kerr-Newman black-hole spacetimes are characterized by damped quasinormal resonant modes with the physically motivated boundary conditions of purely ingoing waves at the black-hole outer horizon and purely outgoing waves at spatial infinity [42]. These exponentially decaying black-hole-field oscillation modes describe the gradual dissipation of the external massless perturbation fields which are radiated into the black hole and to spatial infinity. As a result of this dissipation process, the perturbed spacetime gradually returns into a stationary Kerr-Newman black-hole solution of the Einstein-Maxwell field equations.

*not*unique. In particular, previous analytical [50, 51, 52, 53, 54, 55] and numerical [56, 57, 58, 59, 60, 61] studies of the Kerr and Kerr-Newman quasinormal resonance spectra have explicitly proved that the characteristic black-hole relaxation times grow unboundedly as the extremal limit \({{\bar{T}}_{\text {BH}}}\rightarrow 0\) is approached [62]:

In the present paper we shall focus on the opposite regime of Reissner–Nordström black holes which *minimize* the dimensionless relaxation time (1). In particular, we here raise the following physically interesting question: Within the family of spherically symmetric charged Reissner–Nordström black holes, which black hole has the *fastest* relaxation rate [that is, the shortest relaxation time \({{\bar{\tau }}_{\text {relax}}}({\bar{Q}})\)]? Intriguingly, below we shall explicitly prove that, as opposed to the case of near-extremal spacetimes which maximize the black-hole relaxation times [see Eq. (2)], the answer to the black-hole minimization question is *unique*.

## 2 Review of former analytical and numerical studies

Former analytical and numerical studies of the Kerr-Newman resonance spectra [50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61] have revealed the fact that, for a given value of the black-hole electric charge \({\bar{Q}}\), the characteristic black-hole relaxation time \({{\bar{\tau }}_{\text {relax}}}({\bar{Q}},{\bar{a}})\) is a monotonically increasing function of the rotation parameter \({\bar{a}}\). Hence, within the canonical family of Kerr-Newman black-hole spacetimes, the relaxation rate \({{\bar{\tau }}_{\text {relax}}}^{-1}({\bar{Q}},{\bar{a}})\) can be maximized by considering non-spinning (\({\bar{a}}=0\)) black holes [66]. In the present paper we shall therefore focus our attention on spherically symmetric charged Reissner–Nordström black-hole spacetimes.

*non*-monotonic) functional dependence on the dimensionless physical parameter \({\bar{Q}}\). In particular, detailed numerical computations [57, 58, 69] indicate that, in the physically interesting case of coupled gravitational-electromagnetic quadrupole perturbations fields with \(l=2\), the black-hole relaxation time (1) is minimized for

*analytical*treatment of the black-hole minimization problem, we shall study in the present paper the eikonal (geometric-optics) regime of the quasinormal resonance spectra which characterize the composed black-hole-field system. Interestingly, below we shall explicitly demonstrate that the analytical results obtained in the eikonal \(l\gg 1\) regime agree remarkably well with the numerical results [57, 58, 69] of the quadrupole \(l=2\) perturbation fields.

## 3 The composed black-hole-field quasinormal resonance spectra in the eikonal (geometric-optics) \(l\gg 1\) regime

*M*and electric charge

*Q*which is characterized by the curved line element [8, 43]

*y*is defined by the differential relation

*l*regime. To this end, we shall use the results of the elegant WKB analysis presented in [70, 71], according to which the quasinormal resonance spectrum which characterizes the Schrödinger-like ordinary differential equation (7) with the boundary conditions (10) is determined, in the eikonal \(l\gg 1\) regime, by the simple leading-order resonance equation [70, 71]

*l*regime. Substituting (13) into the WKB resonance equation (11) and using the differential relation (8), one obtains the (rather cumbersome) functional expressions

*minimized*for the particular value

Interestingly, one finds that the analytically derived expression (16) for the dimensionless charge parameter \({\bar{Q}}_{\text {min}}\), which characterizes the Reissner–Nordström black hole with the fastest relaxation rate (the shortest relaxation time) in the eikonal large-*l* regime, agrees remarkably well with the numerically computed value \({\bar{Q}}^{\text {numerical}}_{\text {min}}\simeq 0.7\) [56, 57, 58, 59, 60, 61, 69] [see Eq. (3)] for the quadrupole \(l=2\) perturbation fields.

## 4 Summary and discussion

The characteristic quasinormal resonance spectra of black holes have attracted the attention of physicists and mathematicians during the last five decades [46, 47, 48]. In the present paper we have raised the following physically interesting questions: Within the canonical family of charged and rotating Kerr-Newman black holes, which black hole has the slowest relaxation rate and which black hole has the *fastest* relaxation rate?

Using analytical techniques we have shown that, while the answer to the first question is not unique [see the characteristic relation (2) for the family of near-extremal black holes with diverging relaxation times in the \({{\bar{T}}_{\text {BH}}}\rightarrow 0\) limit], the answer to the second question is unique within the phase space of spherically symmetric charged Reissner–Nordström black-hole spacetimes. In particular, based on previous analytical and numerical studies of the black-hole resonance spectra [50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61] which revealed the interesting fact that, for a given value of the dimensionless charge parameter \({\bar{Q}}\), the characteristic black-hole relaxation time \({{\bar{\tau }}}({\bar{a}},{\bar{Q}})\) is a monotonically increasing function of the dimensionless angular-momentum parameter \({\bar{a}}\), we have focused our attention on the relaxation properties of non-spinning (\({\bar{a}}=0\)) charged Reissner–Nordström black-hole spacetimes. In addition, in order to facilitate a fully *analytical* exploration of the intriguing non-monotonic functional dependence of the black-hole relaxation rates \({{\bar{\tau }}}^{-1}({\bar{Q}})\) on the dimensionless physical parameter \({\bar{Q}}\), we have considered in the present paper the eikonal (geometric-optics) regime of the composed black-hole-field resonance spectrum.

*analytically*calculated black-hole charge parameter [see Eq. (16)]

*numerically*computed charge parameter \({\bar{Q}}^{\text {numerical}}_{\text {min}}\simeq 0.7\) [see Eq. (3)] which minimizes the black-hole relaxation time in the canonical case of coupled gravitational-electromagnetic quadrupole perturbation fields with \(l=2\) [57, 58, 69].

## Notes

### Acknowledgements

This research is supported by the Carmel Science Foundation. I would like to thank Yael Oren, Arbel M. Ongo, Ayelet B. Lata, and Alona B. Tea for helpful discussions.

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