(In)stability of black holes in the $4D$ Einstein–Gauss–Bonnet and Einstein–Lovelock gravities

Abstract

A $(3+1)$-dimensional Einstein–Gauss–Bonnet effective description of gravity has been recently formulated as the $D\to 4$ limit of the higher dimensional field equations after the rescaling of the coupling constant. This approach has been recently extended to the four-dimensional Einstein–Lovelock gravity. Although validity of the regularization procedure has not been shown for the general case, but only for a wide class of metrics, the black-hole solution obtained as a result of such a regularization is also an exact solution in the well defined $4D$ Einstein–Gauss–Bonnet theory suggested by Aoki et al. (2005) and in the scalar–tensor effective classical theories. Here we study the eikonal gravitational instability of asymptotically flat, de Sitter and anti-de Sitter black holes in the four dimensional Einstein–Gauss–Bonnet and Einstein–Lovelock theories. We find parametric regions of the eikonal instability for various orders of the Lovelock gravity, values of coupling and cosmological constants, and share the code which allows one to construct the instability region for an arbitrary set of parameters. For the four-dimensional Gauss–Bonnet black holes we obtain the region of stability in analytic form. Unlike the higher dimensional Einstein–Lovelock case, the eikonal instability serves as an effective cut-off of higher curvature Lovelock terms for the $4D$ black holes.

Introduction

Stability of a black-hole metric against small (linear) perturbations of spacetime is the necessary condition for viability of the black-hole model under consideration. Therefore, a number of black-hole solutions in various alternative theories of gravity were tested for stability [1]. One of the most promising approaches to construction of alternative theories of gravity is related to the modification of the gravitational sector via adding higher curvature corrections to the Einstein action. This is well motivated by the low energy limit of string theory. Among higher curvature corrections, the Gauss–Bonnet term (quadratic in curvature) and its natural generalization to higher orders of curvature in the Lovelock form [2], [3] play an important role.

The Lovelock theorem states that only metric tensor and the Einstein tensor are divergence free, symmetric, and concomitant of the metric tensor and its derivatives in four dimensions [2], [3]. Therefore, it was concluded that the appropriate vacuum equations in $D=4$ are the Einstein equations (with the cosmological term). In $D>4$ the theory of gravity is generalized by adding higher curvature Lovelock terms to the Einstein action.

Black hole in the $D>4$ Einstein–Gauss–Bonnet gravity and its Lovelock generalization were extensively studied and various peculiar properties were observed. For example, the life-time of the black hole whose geometry is only slightly corrected by the Gauss–Bonnet term is characterized by a much longer lifetime and a few orders smaller evaporation rate [4]. The eikonal quasinormal modes in the gravitational channel break down the correspondence between the eikonal quasinormal modes and null geodesics [5], [6]. However, apparently the most interesting feature of higher curvature corrected black hole is the gravitational instability: When the coupling constants are not small enough, the black holes are unstable and the instability develops at high multipoles numbers [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. Therefore, it was called the eikonal instability [13].

Recently, it was claimed that there was found the way to bypass the Lovelock’s theorem [17] by performing a kind of dimensional regularization of the Gauss–Bonnet equations and obtaining of a four-dimensional metric theory of gravity with diffeomorphism invariance and second order equations of motion. The approach was first formulated in $D>4$ dimensions and then, the four-dimensional theory is defined as the limit $D\to 4$ of the higher-dimensional theory after the rescaling of the coupling constant $\alpha \to \alpha ∕(D-4)$. The properties of black holes in this theory, such as (in)stability, quasinormal modes and shadows, were considered in [18], while the innermost circular orbits were analyzed in [19]. The generalization to the charged black holes and an asymptotically anti-de Sitter and de Sitter cases in the $4D$ Einstein–Gauss–Bonnet theory was considered in [20] and to the higher curvature corrections, that is, the $4D$ Einstein–Lovelock theory, in [21], [22]. Some further properties of black holes for this novel theory, such as axial symmetry and thermodynamics, were considered in [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57].

It should be pointed out that such a “naïve” formulation of the dimensional regularization has faced some criticism, starting from a straightforward observation of the lack of the tensorial description for the corresponding theory [58]. It has been found further that in some cases different ways for regularization lead to nonuniqueness of some solutions, such as Taub-NUT black holes [59]. It was pointed out that in four dimensions there is no four-point graviton scattering tree amplitudes other than those leading to the Einstein theory, so that additional degrees of freedom, for instance, a scalar field ${\left(\partial \varphi \right)}^4$, should be added for consistency [60]. In addition, the nonlinear perturbations of the metric cannot be regularized by taking the limit $D\to 4$ due to divergent terms appearing in the corresponding equations of the Gauss–Bonnet theory [61].

In order to solve the above problems additional scalar degrees of freedom were proposed in [62], [63] through a Kaluza–Klein reduction of a $D$-dimensional theory, which in the limit $D\to 4$ leads to a particular subclass of the Horndeski theory with a scalar field ${\left(\partial \varphi \right)}^4$. An alternative approach for introducing the scalar field, which does not exploit a particular assumption on the extra-dimensional geometry, leading to the same scalar–tensor theory (when the internal Kaluza–Klein space is flat), has been proposed in [64], [65]. The theory admits two vacua, one corresponding to the Einstein gravity and the other one — to the regularized Gauss–Bonnet case, and do not have an additional propagating degree of freedom associated with the scalar field [66]. Thus, in order to study gravitational dynamics, we need to take into account only gravitational degrees of freedom. However, it turns out that the gravitational degrees of freedom in such a scalar–tensor theory are infinitely strong coupled due to lack of the quadratic kinetic term of the scalar field [63].

A consistent description for the theory has been given in [67], where, using the ADM decomposition, it was shown that the regularization either

• breaks the diffeomorphism invariance, leading to a particular vacuum and implying no scalar-field degree of freedom, or,

• introduces an extra degree of freedom given by a scalar field, which is in agreement with the Lovelock theorem.

Here we study the linear stability of asymptotically flat, de Sitter and anti-de Sitter $4D$ Einstein–Lovelock black holes. We show that not only maximally symmetric spacetimes, but even much less symmetric time-dependent linear perturbations can be studied in four dimensions with arbitrary Lovelock couplings by taking the limit $D\to 4$ without encountering divergences. We find the parametric regions of the eikonal instability for the $4D$ Einstein–Gauss–Bonnet-(anti-)de Sitter black holes and various examples for its Lovelock extension. We show that the positive values of coupling constants are bounded by the instability region. Thus, small black holes are not allowed in the $4D$ Einstein–Lovelock description with positive couplings: When the coupling constants are sufficiently large compared to the black-hole size, such black holes are always eikonally unstable. This situation is qualitatively different from the higher dimensional Einstein–Lovelock theory, where the effect of higher curvature terms cannot be discarded, because there the instability still allows for large values of the coupling constants. Inequalities determining the instability region for the four-dimensional Gauss–Bonnet black holes are derived.

Our paper is organized as follows. In Section 2 we briefly describe the static $4D$ Einstein–Lovelock black hole solution. Section 3 is devoted to the gravitational perturbations of the black holes, and Section 4 discusses their eikonal instability. Finally, in Conclusions, we summarize the obtained results.