Instability in charged Gauss–Bonnet–de Sitter black holes

Black holes are one of the most fantastic predictions of Einstein’s theory of General Relativity and have been widely investigated both in theory and experiment. While the Einstein equation provides a good description for the evolutions of spacetime and the interaction with matter, it is intrinsically complicated and difficult to solve, making it harder to understand the properties of a black hole. A good starting point is to consider perturbation theory with the background geometry fixed. It is not only an important approach to understanding the linear stabilities or instabilities of systems, but also a useful tool for finding new black hole solutions.

One of the interesting case is the existence of a linear instability for the Reissner-Nordström de Sitter (RNdS) black hole with a positive cosmological term. While it has been proved that the RNdS background is linear-mode stable for spacetime dimension $d=4,5$ [1], the authors of [2, 3] found that the RNdS black hole can be unstable to gravitational perturbations in seven dimensions and above ($d\geqslant 7$). More recently, the authors of [4] provided strong numerical evidence for the existence of a dynamical instability triggered by the scalar-type gravitational perturbations in $d=6$. The physical origin of such dynamical instability was argued to be due to the violation of the $AdS_{2}$ BF bound in the near-horizon limit of extremal RNdS [4], shedding some light on the nature of the instability. In particular, at extremality, once there exist BF bound violating modes at a finite range of parameter space, the system is unstable in the full range of the parameter space. In some sense, the observation of [4] comes as a surprise: a local instability (the BF bound violation near the extremal horizon regime) is able to account for the dynamical instability which depends on the global geometry of the background as well as the boundary conditions.

In order to understand how general the phenomena found in [4] and to better understand the nature of the instability of charged black holes, it is valuable to classify other extensions of the analysis, in particular, for other gravity theories and spacetime dimensions. A natural generalization is the Einstein Gauss-Bonnet (GB) gravity. Higher derivative curvature terms naturally occur in many occasions, in particular in the low-energy effective action of superstring theory. Of particular interest is the GB term which is the most general quadratic expression in curvature resulting in field equations containing no more than second derivatives of the metric and thus no ghost-like mode. The thermodynamic properties of the static vacuum GB black holes in de Sitter spacetime were studied in [5]. The dynamical instability for $d\geqslant 5$ was uncovered from the time-domain analysis for these black holes at large values of the GB coupling and cosmological constant [6], which was supposed to have the same origin as the instability of higher dimensional RNdS black holes found in [2, 3]. More recently, the instability of the Gauss-Bonnet Reissner-Nordström de Sitter (GB-RNdS) black hole at large $d$ limit was studied in [7], where the effective equations at large $d$ were introduced to describe the nonlinear dynamical deformations of the black holes. Nevertheless, a comprehensive study on the instability of the GB-RNdS black hole has not been done yet. Compared with the RNdS case, in the GB-RNdS black hole there is an additional knob, i.e. the GB coefficient that can be tuned to test if the physical origin of the instability given in [4] is robust or not. In the present work, we will investigate the effect of the GB coupling on the instability of the GB-RNdS black hole.

For the GB-RNdS black hole, according to the transformation law under rotations on the sphere, the linear perturbations can be classified into tensor, vector and scalar types. Each of them can be treated independently from the others. Following [4], we shall consider the scalar type perturbations, for which the master equations are given by two coupled second order differential equations. The equations effectively behave as massive scalar fields in an $AdS_{2}$ background which is the near-horizon limit of the extremal GB-RNdS black hole. We shall adopt two instability criteria. The first one is the Durkee-Reall instability criterion based on the $AdS_{2}$ BF bound of the near-horizon limit at extremality. The Durkee-Reall criterion [8] conjectures that when the near-horizon effective mass violates the $AdS_{2}$ BF bound, then the full spacetime geometry is linear unstable. The other one is the quasi-normal mode (QNM) spectrum by solving the master equations between the event and cosmological horizons. The presence of a QNM frequency with a positive imaginary part signifies a dynamical instability and the magnitude of its imaginary part sets the timescale of the instability. Compared with the RNdS case [4], we shall uncover some novel features for the GB-RNdS black hole. We shall show that the instability of the black hole depends not only on its size and the spacetime dimension, but also significantly on the GB term. For example, we will show that while the standard RNdS black hole which is linear stable in $d=5$ dimensional spacetime [1], the RNdS black hole with GB coupling in 5 dimensions can become unstable, depending on the parameters of the black hole. Our analysis will also suggest that the physical origin of the instability for the GB-RNdS background is not solely due to the violation of the $AdS_{2}$ BF bound.

The paper is organized as follows. In Section II, we briefly review the charged Gauss-Bonnet black holes with a positive cosmological constant and give the master equations for the perturbations of the scalar type. In Section III, we query for the instabilities from the near-horizon $AdS_{2}$ geometry at extremality. The violation of the $AdS_{2}$ BF bound provides a criterion for the presence of an instability. A full numerical analysis for the dynamical instabilities is given in Section IV by searching for the QNMs. We shall mainly focus on the extremal black holes and give a brief discussion on the non-extremal case. In Section V, we prove that the unstable QNMs are purely imaginary. We conclude with some discussions in Section VI. Numerical techniques for QNMs are discussed in Appendix A.

We consider the Einstein-Gauss-Bonnet gravity with a positive cosmological constant $\Lambda$, coupled with a Maxwell field in $d=n+2$ dimensional spacetime. The action reads

where the GB term is given by

with $R$, $R_{\mu\nu}$ and $R_{\mu\nu\rho\sigma}$ the Ricci scalar, the Ricci tensor and the Riemann tensor associated with the metric $g_{\mu\nu}$, respectively. $\bar{L}$ is the de Sitter length scale and $F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$ the strength of the U(1) gauge potential $A_{\mu}$. The GB term is a topological invariant for $d=4$ and hence does not contribute to the gravitational dynamics, thus we consider $d\geqslant 5$. The constant $\bar{\alpha}$ is the GB coefficient and is positive in string theory. Therefore, we shall restrict in this paper to the case $\bar{\alpha}>0$.

The variation of the action (1) with respect to the metric yields the gravity equations

with

and

the energy-momentum tensor of the Maxwell field. The equation of motion for $A_{\mu}$ is given by

The theory allows the GB-RNdS black hole that is given as follows [9, 10].

where $d\Omega_{n}^{2}$ is the linear element of a unit radius sphere $S^{n}$, $\bar{M}$ is the black hole mass and $\bar{Q}$ is related to the charge of the black hole. By taking the limit $\bar{\alpha}\rightarrow 0$, one recovers the Einstein-Maxwell theory as well as the RNdS black hole for which

Therefore, after considering the small $\bar{\alpha}$ limit, our results below should come back to the RNdS case discussed in [4].

Besides the event horizon $\mathcal{H}^{+}$ of the GB-RNdS black hole (8), it has a cosmological horizon $\mathcal{H}_{C}$ outside $\mathcal{H}^{+}$ due to the presence of the positive cosmological constant $\Lambda$. For appropriate range parameters, there is also an inner Cauchy horizon $\mathcal{CH}$. We denote the location of the event horizon, cosmological horizon and Cauchy horizon as $\bar{r}_{+},\bar{r}_{c}$ and $\bar{r}_{-}$, respectively. They are all real positive roots of $f$ with $\bar{r}_{-}\leqslant\bar{r}_{+}\leqslant\bar{r}_{c}$. The temperature of $\mathcal{H}^{+}$ and $\mathcal{\mathcal{H}_{C}}$ are respectively given by $\bar{T}_{+}=\frac{1}{4\pi}\frac{\mathop{}\!\mathrm{d}f(\bar{r}_{+})}{\mathop{}\!\mathrm{d}\bar{r}}$ and $\bar{T}_{c}=-\frac{1}{4\pi}\frac{\mathop{}\!\mathrm{d}f(\bar{r}_{c})}{\mathop{}\!\mathrm{d}\bar{r}}$.

Note that the system is invariant under the scaling $g\rightarrow\lambda^{2}g,A\rightarrow\lambda A,\bar{L}\rightarrow\lambda\bar{L},\bar{\alpha}\rightarrow\lambda^{2}\bar{\alpha}$ with $\lambda$ a constant. In the following discussion, we shall work with dimensionless quantities in unites of $\bar{r}_{c}$. More precisely, we introduce

such that $\{M,Q,r,\alpha,L\}$ are all dimensionless quantities. For a given $n$, the parameters of the GB-RNdS black hole (8) $\{M,Q,L,r_{+},\alpha\}$ are not independent. Denoting $Q_{ext}$ as the charge for extremal black hole for which $\bar{T}_{+}=0$ and introducing $y_{+}=r_{+}/r_{c}$, one obtains three independent dimensionless parameters $\{y_{+},\alpha,q\}$ with $q\equiv Q/Q_{ext}$ the charge ratio. Other parameters can be fixed by $\{y_{+},\alpha,q\}$. In particular, one has

and

Note that $Q_{ext}$ is independent of ${\alpha}$ when $n=3$. In the following discussion, we shall mainly focus on the extremal black hole for which $Q=Q_{ext}$ ($q=1$) and the independent parameters are $\{y_{+},\alpha\}$.

The equations of motion for the linear perturbations $(\delta g_{\mu\nu},\delta A_{\mu})$ around the background (8) can be obtained by taking the variation of the equations (4) and (7). Following the main idea by the authors of [1], the perturbations of (8) can be classified into different types according to how they transform under diffeomorphisms of $S^{n}$. Master equations for scalar, vector and tensor modes were obtained for static charged Lovelock black holes in [11, 12] (the author of [11, 12] studied the instability of the static charged Lovelock black holes in asymptotic flat spacetime, i.e. $\Lambda=0$ by using the “S-deformation” approach [13, 14] ). Since we shall generalize the recent discussion of the instability in the RNdS black holes [4] to the GB-RNdS case, we consider the scalar type perturbations, for which the master equations in Fourier space are given by [12]

where

and

with

We have also introduced the tortoise coordinate $r_{*}$ which is defined as $dr_{*}=dr/f$. In contrast to the Einstein-Maxwell theory [1] where one can find one master equation in the scalar type perturbations by manipulating the Einstein and Maxwell equations, for the case with GB term (3), the gravito-electromagnetic perturbations are coupled with each other. More precisely, the variable $\phi_{\omega l}$ comes from gravitational perturbations, while $A_{\omega l}$ is obtained by a particular combination of gravitational and Maxwell perturbations. Here the subscript $\omega$ denotes the frequency measured in units of $r_{c}$, and $l=2,3,4,\dots$ is the multipole number. Modes with $l=0,1$ are pure gauge [1]. The instability region of the background black hole (8) will be obtained by solving the two coupled equations (15) with relevant boundary conditions.

The two coupled master equations (15) do not have analytic solutions, so one has to solve them numerically. There is an instability criterion conjectured by Durkee and Reall [8] which claims that when the near-horizon effective mass violates the $AdS_{2}$ BF bound, then the full spacetime geometry is linear unstable. This local criterion was proved later for asymptotically flat ($\Lambda=0$) and $AdS$ ($\Lambda<0$) black holes [15]. It was also argued to be valid in de Sitter case for which $\Lambda>0$. Recently, the authors of [4] performed a detailed analysis of the near-horizon limit of extremal RNdS black hole and provided good numerical evidence for the Durkee-Reall conjecture with $\Lambda>0$. Before doing numerics, we shall consider the near-horizon limit of the extremal charged black hole (8) and try to see if there exists any mode that violates the $AdS_{2}$ BF bound.

In this section we aim at the QNM spectrum of linear perturbations in the GB-RNdS black hole (8). See [16] for a recent review on the QNMs of black holes. More precisely, if the imaginary part of a QNM frequency, $\text{Im}[\omega]$, is positive, such mode will grow exponentially with time, signifying a dynamical instability. In particular, the value of a positive $\text{Im}[\omega]$ sets the timescale of the instability. To directly compare with the analytical near-horizon prediction in the last section, we shall mainly focus on the extremal case.

The QNMs are obtained by solving those master equations (15) between the event and cosmological horizons with the ingoing boundary condition at the event horizon and the outgoing boundary condition at the cosmological horizon. In our present case, the master equations have a cumbersome form, which makes the numerical analysis of the QNM spectrum challenging. In particular, we find that the double-precision (16 digits) arithmetics are not sufficient for our computations. Therefore, higher precision is required when solving the master equations. Numerical techniques and details are presented in Appendix A.

We now present the numeric results for the QNMs by solving two master equations (15). As we have discussed, a dynamical instability will develop if the imaginary part of a QNM frequency $\omega$ is positive. We shall mainly focus on the extremal black holes ($Q/Q_{ext}=1$) which are parametrized by $y_{+}$ and $\alpha$. In particular, we will compare the parameter space $\{y_{+},\alpha\}$ of instability from the QNMs and the one from the local $AdS_{2}$ BF bound violation. We search directly for the unstable modes in the non-extremal GB-RNdS black hole in subsection IV.6.

Before concluding, we should mention that for the QNM frequencies with positive imaginary part, i.e. Im$[\omega]>0$, we find numerically that their real part are vanishing (Re$[\omega]$=0). This is in agreement with the observation that unstable modes cannot be oscillatory when perturbing spherically symmetric static black holes [17].

The observation of only purely imaginary unstable modes can be understood as follows. We multiply the master equations (15) by the complex conjugated functions $\phi_{\omega l}^{\dagger}$ and $A_{\omega l}^{\dagger}$, and then consider the integral of the resulted equation from the event horizon ($r_{*}\rightarrow-\infty$) to the cosmological horizon ($r_{*}\rightarrow+\infty$).

Integration of the first term by parts yields

Note that we consider the ingoing boundary condition at the event horizon and the outgoing boundary condition at the cosmological horizon, (the boundary conditions are discussed in detail in Appendix A, ) from which we have

Substituting above boundary conditions into (V) and taking the imaginary part of the resulted equation, we obtain

It is clear that Im$[\omega]<0$ for non-vanishing Re$[\omega]$. Therefore, the unstable modes (Im$[\omega]>0$) should not be oscillating, i.e. Re$[\omega]=0$.

As a consequence, at the onset of instability there is a non-trivial static zero mode perturbation, suggesting the existence of a new non-spherically symmetric branch of solutions. We emphasize that the boundary conditions (29) plays an important role in above proof. While it is natural to choose ingoing boundary condition at the event horizon, it is possible to consider other kinds of boundary conditions at the cosmological horizon, e.g. the boundary condition with reflection and transmission for black hole superradiance [18]. For the later, one could have oscillating modes. On the other hand, the $AdS_{2}$ BF bound violation is local and is independent of any boundary condition.

In this work we have studied the instability of the GB-RNdS black hole (8) under gravito-electromagnetic perturbations. The scalar type perturbations are governed by two coupled second order differential equations (15). We have adopted two criteria to search for an instability. We have firstly performed a detailed analysis of the near-horizon limit of extremal GB-RNdS black holes and found that the scalar type modes behave as a massive scalar field in an $AdS_{2}$ background. According to the Durkee-Reall instability criterion [8, 15], the full extremal geometry will be unstable if the near-horizon effective mass violates the $AdS_{2}$ BF bound. Then, we have used full numerical analysis to obtain the behaviors of QNMs, from which the onset of the instability is identified by looking for a QNM frequency with a positive imaginary part. We found numerically that the real part of all unstable modes is vanishing, i.e. Re$[\omega]=0$. Moreover, a religious proof was provided to show that the unstable modes should be purely imaginary.

The instability of the black hole (8) has been found to depend not only on its size $y_{+}$ and charge ratio $q$, but also significantly on the GB coupling and the spacetime dimension. We have uncovered the gravitational instability in the GB-RNdS black hole when the spacetime dimension $d\geqslant 5$ ($n=d-2\geqslant 3$). The complete parameter space at extremality in terms of $y_{+}$ and $\alpha$ has been constructed, see Figs. 4, 6, 8 and 12. We have also provided strong numerical evidence for the Durkee-Reall instability criterion with positive cosmological constant $\Lambda>0$. While it is a sufficient criterion for the presence of instability at extremality, it is not a necessary one. By continuity, the instability extends away from extremelity. We have briefly discussed the unstable modes for the non-extremal GB-RNdS black hole. Depending on the value of GB coupling, the non-extremal solutions can be more stable or unstable than the extremal case.

Compared with the RNdS case in the Einstein-Maxwell theory with a positive cosmological constant, some novel features have been found for the GB-RNdS black hole in our present work.

• •

  While it has been found that the RNdS black holes in $d=5$ are linearly-mode stable [1], we have found that the presence of GB term makes the black holes to be unstable to gravitational perturbations in $d=5$ and higher dimensions.

• •

  For the five dimensional case, the instability occurs at $l=2$, but it becomes stronger at high multipole numbers $l$. Such eikonal instability was found to be closely related to the violation of the $AdS_{2}$ BF bound near the horizon of the extremal GB-RNdS black hole.

• •

  It has been recently shown in [4] that at extremality the gravitational instability in the RNdS black hole is present when $d\geqslant 6$ for the whole range of $y_{+}=r_{+}/r_{c}$, i.e. $0<y_{+}<1$, although typically the near-horizon $AdS_{2}$ BF bound is only violated for a finite range of $y_{+}$. It ceases to be valid for the charged black with GB term. Except for $d=5$ (see Figs. 4 and 12), the instability does not present in the full range $0<y_{+}<1$ (see e.g. Figs. 6 and 8).

• •

  The physical origin of the instability of RNdS was argued to be due to the $AdS_{2}$ BF bound violation in the near-horizon limit of extremal background. It seems not to work for our GB case. The reasons are two-fold. Firstly, the instability in general is not present in the whole range $0<y_{+}<1$. Secondly, as can be seen e.g. from Fig. 6, for some choice of $\alpha$, there exists unstable region in which the $AdS_{2}$ BF bound is no longer violated for any $y_{+}$.

By using the large $d$ expansion, the GB-RNdS black hole was found to be unstable in the $d\rightarrow\infty$ limit, when the cosmological constant is sufficiently large [7]. Our findings are qualitatively consistent with the large $d$ results for higher-dimensional cases, $n\geqslant 7$. In the large $d$ discussion [7], the local instability due to the BF bound violation near the extremal horizon regime was not considered. Moreover, compared to the large $d$ expansion, there are some novel features that only happen in lower dimensions, $n=5,6$, see Figs. 6 and 12.

While the instability has been uncovered at the linear order, it is still an open question what is the nature of the endpoint of the instability. If the onset of such instability leads to a new solution, it seems to be some yet undiscovered black holes with no spatial isometries as the modes triggering the instability are harmonics with $l\geqslant 2$. It will be quite interesting to construct these novel black holes numerically. Our numerical analysis suggests that the physical origin of the instability of the GB-RNdS black hole is not solely due to the violation of the $AdS_{2}$ BF bound. Further effort is needed to understand the physical origin of this instability.

The present work can be extended in several directions. We have mainly limited ourselves to the extremal case. It is interesting to study the QNM structure away from extremality, which is expected to exhibit intricate behaviors with bifurcations and merges as found in RNdS black holes [4]. We have focused on the instabilities triggered by the scalar type perturbations. To further understand the instability of the GB-RNdS black hole, one needs to consider the vector and tensor modes, as the black hole instability region is a combination of the instability regions of all types of perturbations. Another interesting extension is to consider the case with non-linear electrodynamics, in particular the DBI action that naturally arises in the low energy limit of string theory. Moreover, given the fact that near-extremal charge corresponds to fast rotating black holes, i.e. the charge-angular momentum analogy, it will be interesting to extend the study to the rotating case. We leave these questions to future work.

This work was supported in part by the National Key Research and Development Program of China Grant No.2020YFC2201501, by the National Natural Science Foundation of China Grants No.12122513, No.12075298, No.11991052 and No.12047503, and by the Key Research Program of the Chinese Academy of Sciences (CAS) Grant NO. XDPB15 and the CAS Project for Young Scientists in Basic Research YSBR-006.