Analytical quasinormal modes of spherically symmetric black holes in the eikonal regime

TL;DR

This paper derives analytical expressions for high-frequency quasinormal modes of spherically symmetric black holes, including deviations from Einstein's gravity, highlighting their relation to null geodesics and potential instabilities.

Contribution

It provides a generic formula for eikonal quasinormal modes in asymptotically flat spacetimes with deviations from Schwarzschild geometry.

Findings

Analytical quasinormal modes for various gravity theories.

Corrections to Schwarzschild eikonal modes due to deviations.

A generic formula for modes in asymptotically flat metrics.

Abstract

Quasinormal modes in the high frequency (eikonal) regime can be obtained analytically as the Mashhoon-Will-Schiutz WKB formula is exact in this case. This regime is interesting because of the correspondence between eikonal quasinormal modes and null geodesics, as well as due to existence of potential eikonal instabilities in some theories of gravity. At the same time in a number of studies devoted to quasinormal modes of spherically symmetric black holes this opportunity was omitted. Here we find analytical quasinormal modes of black holes in various alternative and extended theories of gravity in the form of the Schwarzschld eikonal quasinormal modes and added corrections due to deviations from Einstein theory. We also deduce a generic formula for analytical calculations of the eikonal quasinormal modes for the class of asymptotically flat metrics in terms of small deviations from the Schwarzschild geometry.

Tables

Table 1. Analytical results for the eikonal quasinormal modes
Theory	Publication
Schwarzschild	MashhoonBlome
Schwarzschild-de Sitter	Zhidenko:2003wq ; Cardoso:2003sw
Reissner-Nordstrom	Andersson:1996xw
Dilatonic black hole	Konoplya:2001ji
Kerr black hole	Mashhoon ; Konoplya:2016pmh ; Konoplya:2017tvu
Extremal Kerr	Hod:2012bw
Higher dimensional Schwarzschild	Konoplya:2003ii
Higher dimensional, extremal Schwarzschild-de Sitter	Molina:2003ff
Higher dimensional charged Einstein-Gauss-Bonnet	Konoplya:2004xx
Einstein-dilaton-Gauss-Bonnet	Konoplya:2019hml
Einstein-Weyl	Kokkotas:2017zwt
Reissner-Nordstrom with derivative coupling to a scalar field	Konoplya:2018qov
Brane-world black holes	Kanti:2005xa ; Zhidenko:2008fp

Table 2. Corrections for eikonal quasinormal modes in terms of the small parameters of the metrics
Metric	$\alpha$	$\Delta {\omega }_R$	$\Delta {\omega }_I$
Einstein-aether I	$I$	$-\frac{8}{27}\alpha +\frac{352}{729}{\alpha }^2+𝒪\left({\alpha }^3\right)$	$-\frac{16}{27}\alpha +\frac{1664}{729}{\alpha }^2+𝒪\left({\alpha }^3\right)$
Einstein-aether II	$J$	$-\frac{1}{6}\alpha +\frac{13}{216}{\alpha }^2+𝒪\left({\alpha }^3\right)$	$\frac{1}{18}\alpha +\frac{1}{72}{\alpha }^2+𝒪\left({\alpha }^3\right)$
Horava-Lifshitz	$\frac{1}{w{M}^2}$	$\frac{1}{27}\alpha +𝒪\left({\alpha }^2\right)$	$\frac{2}{27}\alpha +𝒪\left({\alpha }^2\right)$
Hayward	$\frac{Q}{M}$	$\frac{1}{27}{\alpha }^3+𝒪\left({\alpha }^4\right)$	$\frac{2}{27}{\alpha }^3+𝒪\left({\alpha }^4\right)$
Bardeen	$\frac{Q}{M}$	$\frac{1}{6}{\alpha }^2+\frac{17}{216}{\alpha }^4+𝒪\left({\alpha }^5\right)$	$\frac{1}{9}{\alpha }^2+\frac{149}{648}{\alpha }^4+𝒪\left({\alpha }^5\right)$
ABG	$\frac{Q}{M}$	$\frac{1}{3}{\alpha }^2+𝒪\left({\alpha }^3\right)$	$\frac{1}{18}{\alpha }^2+𝒪\left({\alpha }^3\right)$

Equations

$$\omega =\frac{1}{3\sqrt{3}M}\left(\left(\ell +\frac{1}{2}\right)-i\left(n+\frac{1}{2}\right)\right)+\mathcal{O}\left(\frac{1}{\ell }\right),$$

$$d{s}^2=-f(r)d{t}^2+\frac{1}{f(r)}d{r}^2+r^2\left({\sin }^2\theta d{\varphi }^2+d{\theta }^2\right).$$

$$f(r)=1-\frac{2M}{r}-I{\left(\frac{2M}{r}\right)}^4,I=\frac{27c}{256(1-c)},$$

$$f(r)=1-\frac{2M}{r}-J{\left(\frac{M}{r}\right)}^2,J=\frac{c-d/2}{1-c},$$

$$f(r)=1+w{r}^2-\sqrt{r\left(w^2{r}^3+4wM\right)},$$

$$f(r)=1-\frac{2M{r}^2}{r^3+Q^3},$$

$$f(r)=1-\frac{2M{r}^2}{{\left(r^2+Q^2\right)}^{3/2}},$$

$$f(r)=1-\frac{2M{r}^2}{{\left(r^2+Q^2\right)}^{3/2}}+\frac{Q^2{r}^2}{{\left(r^2+Q^2\right)}^2},$$

$$\frac{d^2\Psi }{d{r}_{\ast }^2}+\left({\omega }^2-V(r)\right)\Psi =0,$$

$$d{r}_{\ast }=\frac{dr}{f(r)}.$$

$${\omega }^2=V_0+\sqrt{-2V_0^{\prime \prime }}\left(n+\frac{1}{2}\right)i,$$

$$r_{max}=r_0+r_1\alpha +r_2{\alpha }^2+r_3{\alpha }^3+\mathcal{O}({\alpha }^4),$$

$$V(r)=f(r)\left(\frac{\ell \left(\ell +1\right)}{r^2}+\mathcal{O}(1)\right),$$

$$\omega =\frac{1}{3\sqrt{3}M}\left(\ell +\frac{1}{2}\right)\left(1+\Delta {\omega }_R\right)-$$

$$-i\frac{1}{3\sqrt{3}M}\left(n+\frac{1}{2}\right)\left(1+\Delta {\omega }_I\right)+\mathcal{O}\left(\frac{1}{\ell +\frac{1}{2}}\right),$$

$$d{s}^2=-\left(1-\frac{2M}{r}+\frac{c}{r^{4+\frac{1}{{\alpha }^2}}}\right)d{t}^2+$$

$$+{\left(1-\frac{2M}{r}+\frac{c}{r^{4+\frac{1}{{\alpha }^2}}}\right)}^{-1}d{r}^2+$$

$$+\frac{r^2}{\left(1+4{\alpha }^2\right)}\left({\sin }^2\theta d{\varphi }^2+d{\theta }^2\right),$$

$$V(r)=\left(1-\frac{2M}{r}+\frac{c}{r^{4+\frac{1}{{\alpha }^2}}}\right)\frac{\ell \left(\ell +1\right)\left(1+4{\alpha }^2\right)}{r^2}$$

$$\omega =\frac{1}{3\sqrt{3}M}\left(\ell +\frac{1}{2}\right)\sqrt{1+\frac{3c}{{\left(3M\right)}^K}}-$$

$$-i\frac{1}{3\sqrt{3}M}\left(n+\frac{1}{2}\right)\sqrt{1-\frac{c{K}^2}{2{\left(3M\right)}^K}-\frac{3c^2{K}^2}{{\left(3M\right)}^{2K}}}+$$

$$+\mathcal{O}\left(\frac{1}{\ell +\frac{1}{2}}\right),$$

$$K=4+\frac{1}{{\alpha }^2},$$

$$f(r)=1-\frac{2M}{r}+\frac{{\alpha }_2}{r^2}+\frac{{\alpha }_3}{r^3}+\frac{{\alpha }_4}{r^4}+\frac{{\alpha }_5}{r^5}+\mathcal{O}\left(\frac{1}{r^6}\right),$$

$$r_{max}=r_0+r_1{\alpha }_2+r_2{\alpha }_3+r_3{\alpha }_4+r_4{\alpha }_5+\mathcal{O}({\alpha }_i{\alpha }_k).$$

$$\omega =\frac{\left(\ell +\frac{1}{2}\right)}{3\sqrt{3}M}\left(1+\frac{{\alpha }_2}{6M^2}+\frac{{\alpha }_3}{18M^3}+\frac{{\alpha }_4}{54M^4}+\frac{{\alpha }_5}{162M^5}\right)-$$

$$-i\frac{\left(n+\frac{1}{2}\right)}{3\sqrt{3}M}\left(1+\frac{{\alpha }_2}{18M^2}-\frac{{\alpha }_3}{27M^3}-\frac{{\alpha }_4}{27M^4}-\frac{11{\alpha }_5}{486M^5}\right)+$$

$$+\mathcal{O}\left(\frac{1}{\ell +\frac{1}{2}}\right).$$

Full text

I Introduction

There exist a number of alternative gravitational theories representing attempts to solve fundamental problems, such as the nature of the gravitational singularity, dark matter and dark energy problems, quantum gravitational theory, hierarchy problem. Recent measurements, either gravitational or electromagnetic, do not identify geometry of black holes or, possibly, other compact objects with sufficient accuracy Abbott:2016blz ; TheLIGOScientific:2016src ; Konoplya:2016pmh ; Goddi:2017pfy ; Akiyama:2019cqa ; Akiyama:2019bqs ; Bambi:2015kza ; Cardoso:2016rao ; Yunes:2016jcc ; Bambi:2019tjh .

That is why we consider different black-hole solutions of modified or extended Einstein equations. All these metrics have one thing in common: they have a small parameter or a number of parameters such that when they vanish, the metric tends to the Schwarzschild one. Therefore we can expand the metric in terms of these small parameters and do the same to find the quasinormal modes.

Quasinormal modes QNMreviews being the source of information about the black holes have been studied in the numerous researches, but an overwhelming part of these studies is concerned with only numerical calculation of QNMs. Nevertheless in geometrical optics approximation (with high multipole number $\ell$) an elegant analytical formula was obtained by B. Mashhoon in his seminal work MashhoonBlome :

[TABLE]

where $n$ is the overtone number. A recent review on further extensions and accuracy of the WKB approach can be found in Konoplya:2019hlu .

This formula (1) was extended to the Reissner-Nordstrom case in Andersson:1996xw , to non-extremal Schwarzschild - de Sitter in Zhidenko:2003wq and to the near extremal Schwarzschild - de Sitter case in Cardoso:2003sw . Further extensions to the case of Kerr black hole were done in a number of papers for fields of various spin Mashhoon ; Konoplya:2017tvu ; Hod:2012bw . A number of extensions of the formula (1) to higher dimensional spacetimes were suggested in Konoplya:2003ii ; Molina:2003ff ; Konoplya:2004xx ; Vanzo:2004fy . Recently eikonal regime has been studied in Einstein-dilaton-Gauss-Bonnet Konoplya:2019hml and Einstein-Weyl Kokkotas:2017zwt gravities as well as for the case of non-linear (derivative) coupling between scalar and electromagnetic fields Konoplya:2018qov . The eikonal regime for black holes in non-linear electrodynamics and regular black holes have been recently studied in Toshmatov:2015wga . However the final analytical formulae presented there are not expressed in terms of the parameters of the black holes only (such as mass, charge etc.). They also contain the maximum of the effective potential as a parameter. The summary of analytical results for the quasinormal modes emitted by the black hole in the eikonal regime are reviewed in Table 1.

The eikonal regime of quasinormal frequencies have been recently discussed because of the correspondence between eikonal quasinormal modes and null geodesics Cardoso:2008bp . The correspondence states that the real and imaginary parts of the quasinormal mode are multiples of the frequency and instability timescale of the circular null geodesics respectively. In Konoplya:2017wot it was shown that the correspondence is guaranteed for any stationary, spherically symmetric, asymptotically flat black hole provided that a) effective potential has the form of the potential barrier with a single extremum, implying two turning points and decaying at the event horizon and infinity and b) one is limited by perturbations of the test fields only, and not of the gravitational field itself or other fields, which are non-minimally coupled to gravity.

Another reason to study quasinormal modes in the eikonal regime is possibility of the so-called eikonal instability which may happen either for black branes Takahashi:2011du and holes Konoplya:2017lhs or even for wormholes Cuyubamba:2018jdl . Eikonal instability means the breakdown of the linear approximation as such.

Here we will consider a bunch of theories in which the analytical eikonal formula has not been deduced. In particular, we shall consider two different kinds of Lorentz-violating Einstein-aether theory, Horava-Lifshitz and Horndeski theories and regular BH theories: Bardeen and Ayon-Beato-Garcia black holes in theories coupled to a non-linear electrodynamics and Hayward regular black hole solution. Finally, we develop a general formula for finding eikonal quasinormal modes of asymptotically flat black holes whose geometry slightly deviates from the Schwarzschild one.

II Spherically symmetric black holes spacetimes

A static, spherically symmetric metric in the spacetimes under consideration has the form:

[TABLE]

The metric function is given by:

for the first kind Einstein aether BH

[TABLE] 2. 2.

and for the second kind Einstein aether BH

[TABLE]

where $c$, $d$ are the combinations of the coupling constants of the theory, $0\leq c<1$, $0\leq d<2$, $c\geq d/2$ Ding:2017gfw , Ding:2019tvs . Einstein aether theory is a Lorentz-violating theory endowing a spacetime with both a metric and a unit timelike vector field (aether) having a preferred time direction; 3. 3.

for the Horava-Lifshitz BH

[TABLE]

where $M$ is an integration constant, $w$ is a theory parameter, $wM^{2}\geq 1/2$ Kehagias:2009is . Quasinormal modes were analyzed in Konoplya:2009ig ; Chen:2009gsa ; 4. 4.

for the Hayward BH Hayward:2005gi

[TABLE]

where $Q$ is some real positive constant; 5. 5.

for the Bardeen BH Bardeen

[TABLE]

where $Q$ is a magnetic charge; 6. 6.

for the Ayon-Beato-Garcia (ABG) BH ABG

[TABLE]

where $Q$ is an electric (or magnetic) charge.

III Eikonal expansion

The perturbations of the black hole can be represented in the general form of the wave like equation

[TABLE]

where $r_{*}$ is the ”tortoise coordinate”, mapping the event horizon to $-\infty$,

[TABLE]

The boundary conditions for this equation are only incoming waves at the horizon ($r_{*}\rightarrow-\infty$) and only outgoing waves at the infinity($r_{*}\rightarrow+\infty$). Solving the wave equation we obtain a discrete set of complex values for the frequencies $\omega$ with the real part representing the oscillation frequency and imaginary part representing the damping rate of the oscillations in terms of the black hole parameters.

To find a solution of Eq. (9) we use the WKB approximation WillSchutz :

[TABLE]

which is accurate in the eikonal limit and therefore produces analytical results.

The effective potential has the form of the potential barrier (see, for example Fig.1) and its maximum position can be expanded in terms of a small parameter $\alpha$:

[TABLE]

where $r_{0}=3M$ corresponds to the Schwarzschild solution.

In the eikonal limit the effective potential usually does not depend on spin of the field, though there are a number of exceptions, connected either to existence of the eikonal instability or non-linear couplings. Here we will not consider such cases. Thus, the effective potential has the form:

[TABLE]

which implies applicability of the obtained analytical formulae for the quasinormal modes to all kinds of perturbations, such as gravitational, scalar, Dirac, etc., but except perturbations of non-linear electromagnetic fields in the background of charged black holes Toshmatov:2018ell ; Toshmatov:2018tyo ; Toshmatov:2019gxg .

For each metric under consideration there is a number of reasons to consider additional parameters as small: these are either requirements of good post-Newtonian behavior or other astrophysical constraints. Therefore we find the eikonal quasinormal modes in the form

[TABLE]

[TABLE]

where $\Delta\omega_{R}$ and $\Delta\omega_{I}$ are the corrections for the real and imaginary part of the quasinormal modes with regard to the Schwarzschild case. The parameters and corrections are presented in Table 2.

Unlike all the cases studied above the Horndeski theory Fang:2018vog is characterized by the metric

[TABLE]

[TABLE]

[TABLE]

which is not asimptotically flat. The effective potential for electromagnetic perturbations has the form Fang:2018vog

[TABLE]

and the formula for the eikonal quasinormal modes in this theory reads

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

$c$ is a theory constant and $\alpha$ is a small parameter, such that when it tends to zero the metric approaches that of the Schwarzschild spacetime. Then, $K$ becomes infinitely large, and $\left(3M\right)^{K}$ is large too, while $K^{2}/(3M)^{K}$ is small again. Thus when the theory parameter $\alpha$ tends to zero, the corrections in Eq. (17) vanish and we come to the same Schwarzschild case.

IV General approach for asymptotically flat metrics

The idea of expanding the position of the peak of the effective potential in terms of a small parameter in order to present it in the form of the Schwarzschild peak with added relatively small corrections can be extended onto the metric function itself. Having an asymptotically flat metric we can expand it in terms of the negative powers of $r$:

[TABLE]

where parameters $\alpha_{i}$ describe deviations of a given metric from the Schwarzschild one. The maximum of the effective potential can be expanded as follows:

[TABLE]

Using the first order WKB formula and expanding the results for $\omega$ into powers of $\alpha_{i}$, we find that

[TABLE]

[TABLE]

[TABLE]

The eikonal formulas, presented in Table 2, can then be immediately found from the above general formula by using the coefficients $\alpha_{i}$ obtained via expansion of the metric in terms of the corresponding small parameters of the system.

V Conclusions

We filled the gap in the current literature devoted to analytical calculations of quasinormal modes in the high frequency regime. For the set of the spherically symmetric metrics we obtained corrections for the real and imaginary part of the eikonal quasinormal modes in terms of the small parameters of the theories. We also deduced a general formula for eikonal quasinormal modes for the class of the asymptotically flat spacetimes in terms of small deviations from Schwarzschild geometry. In addition we reviewed publications where analytical eikonal formula was derived for various black hole spacetimes and fields. We believe that this compendium of analytical formulae for high frequency regime of black hole perturbations might be of further usage in analytic and semi-analytic treatments of black hole perturbations Hod:2012bw ; Hod:2018ifz ; Zinhailo:2018ska .

Acknowledgements.

The author acknowledges hospitality and support of Silesian University in Opava, Roman Konoplya for suggesting this problem and useful discussions and Alexander Zhidenko and Bobir Toshmatov for valuable advice.