Gregory-Laflamme instability of black hole in Einstein-scalar-Gauss-Bonnet theories

We investigate the stability analysis of a Schwarzschild black hole in Einstein-scalar-Gauss-Bonnet theory because the instability of the Schwarzschild black hole without scalar hair implies the Gauss-Bonnet black hole with scalar hair. The linearized scalar equation is compared to the Lichnerowicz-Ricci tensor equation in the Einstein-Weyl gravity. It turns out that the instability of the Schwarzschild black hole in Einstein-scalar-Gauss-Bonnet theory is interpreted as not the tachyonic instability but the Gregory-Laflamme instability of a black string. In the small mass regime of $1/\lambda <1.174/r_{+}$, the Schwarzschild solution becomes unstable, and a new branch of the solution with scalar hair bifurcates from the Schwarzschild one. This is very similar to finding a newly non-Schwarzschild black hole in Einstein-Weyl gravity.

Figure 1

The potentials as a function of $r\in [1,\infty )$ for horizon radius $r_{+}=1$ and $l=0$. The blue (bottom), red (middle), and green (top) curves represent the potentials $V(r)$ of the scalar for mass parameter $1/\lambda =1.095$ (sufficient condition for instability), 1.174 (threshold of instability), and 1.2 (stable case), respectively. These all have negative regions near the horizon. However, the tachyonic potential (cyan curve) $V_{\mathrm{t}}(r)$ develops a positive region near horizon while it approaches $-0.04$ as $r\to \infty$ for $m_{\mathrm{T}}=0.2$.

Figure 2

$\mathrm{\Omega }$ graphs as a function of mass parameter $1/\lambda$ for small black holes of $r_{+}=1$, 2, 3. Here, one reads off the thresholds of instability $(1/\lambda {)}^{\mathrm{th}}$ from the points at which curves of $\mathrm{\Omega }$ intersect the positive $\frac{1}{\lambda }$ axis: $(1/\lambda {)}^{\mathrm{th}}\approx 1.174$, 0.587, 0.294. The instability range decreases as the horizon radius increases.

Figure 3

Plots of unstable modes filled circle on three curves with the horizon radii $r_{+}=1$, 2, 4. The $y(x)$ axis denotes $\mathrm{\Omega }$ in $e^{\mathrm{\Omega }t}$ (mass $m$ of the massive spin-2 mode). The smallest curve represents $r_{+}=4$, the medium denotes $r_{+}=2$, and the largest one shows $r_{+}=1$. Here, the thresholds of instability are located at $m^{\mathrm{th}}\approx 0.876$, 0438, 0.219, which means that the instability region is smaller and smaller as the horizon radius is larger and larger.

Figure 4

Graphs for scalar perturbation $\delta \varphi$ as a function of $r/r_{+}$ for $r_{+}/\lambda \approx 1.174$, 0.453, 0.280, which corresponds to the number of nodes $n=0$, 1, 2 in scalar profiles.