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/λ<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.