(In)stability of D-dimensional black holes in Gauss-Bonnet theory

TL;DR

This paper investigates the stability of gravitational perturbations of $D$-dimensional Gauss-Bonnet black holes across $D=5$–$11$ by performing time-domain numerical evolutions of tensor, vector, and scalar perturbations. The authors derive master wave equations with three effective potentials $V_t$, $V_v$, and $V_s$ and implement a high-precision Price-Pullin characteristic scheme to extract quasinormal modes and instability thresholds as functions of the GB coupling $α$ and multipole $\ell$. They find that instabilities occur only for $D=5$ and $D=6$ at large $α$, predominantly at high $\ell$, while for $D\ge7$ the perturbations are stable; scalar-type perturbations can exhibit damping tails due to negative potential gaps and non-monotonic mode behavior. The results place constraints on the validity of Gauss-Bonnet corrections in higher-dimensional gravity and motivate extending the analysis to charged and asymptotically AdS cases, where additional negative gaps can alter stability.

Abstract

We make an extensive study of evolution of gravitational perturbations of D-dimensional black holes in Gauss-Bonnet theory. There is an instability at higher multi-poles $\ell$ and large Gauss-Bonnet coupling $α$ for $D= 5, 6$, which is stabilized at higher $D$. Although small negative gap of the effective potential for scalar type of gravitational perturbations, exists for higher $D$ and whatever $α$, it does not lead to any instability.

Figures (6)

• Figure 1: Threshold $\alpha$ as a function of the inverse multipole number $\ell$. Tensor type of gravitational perturbations $D=6$. The points $\ell=16,20,32,40,50,64$ were fit by the line $\alpha = 2.627\ell^{-1}+1.005$. The theoretical result is $\alpha_t\approx1.006$.
• Figure 2: Threshold $\alpha$ as a function of the inverse multipole number $\ell$. Scalar type of gravitational perturbations $D=5$. The points $\ell=16,20,32,40,50$ were fit by the line $\alpha = 0.517\ell^{-1}+0.209$. The theoretical result is $\alpha_t\approx0.207$.
• Figure 3: Potential and profile for GB black hole perturbation of scalar type ($D=6$, $l=2$, $\alpha=0.3$). The negative gap does not lead to instability. It causes exponentially damping "tails" to appear just after the initial outburst. Therefore we are unable to see QN ringing.
• Figure 4: Time-domain profiles for the "region of the irregular QN-ringing" of the gravitational perturbation of scalar type $D=6$, $\ell=2$, $\alpha=0.15,0.20,0.25,0.30,0.35,0.40$ (plots from left to right). For $\alpha=0.15$ (first plot) we see usual decaying oscillations. For $\alpha=0.20$ two concurrent modes with the same damping rate. As $\alpha$ increases we observe exponentially damping tails, that do not oscillate. At higher $\alpha$ we see the oscillation behavior again, but the frequency of oscillation for $\alpha=0.40$ (last plot) differs significantly from that for $\alpha=0.15$.
• Figure 5: The picture of instability, developing at large multipole numbers: $D=6$, $\ell =8$ (red), $\ell = 12$ (green), $\ell=16$ (blue), $\alpha =1.3$. Tensor type of gravitational perturbations.