Linear stability of Einstein-Gauss-Bonnet static spacetimes: Tensor perturbations

We study the stability under linear perturbations of a class of static solutions of Einstein-Gauss-Bonnet gravity in $D=n+2$ dimensions with spatial slices of the form ${\Sigma }_{\kappa }^n\times {\mathbb{R}}^{+}$, ${\Sigma }_{\kappa }^n$ an $n$ manifold of constant curvature $\kappa$. Linear perturbations for this class of spacetimes can be generally classified into tensor, vector and scalar types. The analysis in this paper is restricted to tensor perturbations. We show that the evolution equations for tensor perturbations can be cast in Schrödinger form, and obtain the exact potential. We use $S$ deformations to analyze the Hamiltonian spectrum, and find an S-deformed potential that factors in a convenient way, allowing us to draw definite conclusions about stability in every case. It is found that there is a minimal mass for a $D=6$ black hole with a positive curvature horizon to be stable. For any $D$, there is also a critical mass above which black holes with negative curvature horizons are unstable.

Figure 1

Cases 1.1.i to 1.1.iii: (a) the two branches ${\psi }_i(r),i=1,2$ of Eq. (35) in the case $\mu /\alpha >0$. ${\psi }_i\to {\Lambda }_i$ as $r\to \infty$, ${\psi }_1$ (${\psi }_2$) tends to $-\infty$ ($+\infty$) as $r\to 0^{+}$. (b) Plots of $\mu |\psi {|}^{(n+1)/2}/\alpha$ for (i) large (ii) intermediate and (iii) small positive $\mu /\alpha$.

Figure 2

Cases 1.1.v and 1.1.vi: (a) the two branches ${\psi }_i(r),i=1,2$ of Eq. (35) in the case $\mu /\alpha <0$, ${\psi }_i$ goes from ${\psi }_o$ to ${\Lambda }_i$ as $r$ goes from $r_{\mathrm{sing}}$ to $\infty$. (b) Plots of $P$ and $\mu |\psi {|}^{(n+1)/2}/\alpha$ vs $\psi$ for (v) small and (vi) large negative $\mu /\alpha$.

Figure 3

Cases 1.2.i to 1.2.vi: (a) the two branches ${\psi }_i(r),i=1,2$ of Eq. (35) in the case $\mu /\alpha >0$ together with the $\mu |\psi {|}^{(n+1)/2}/\alpha$ curve for (i) large and (ii) small positive values of $\mu /\alpha$. (b) ${\psi }_i(r),i=1,2$ in the case $\mu /\alpha <0$ together with the $\mu |\psi {|}^{(n+1)/2}/\alpha$ curve for (iv) small (v) intermediate and (vi) large negative values of $\mu /\alpha$.

Figure 4

Cases 1.3.i to 1.3.iii: $P$ and $\mu |\psi {|}^{(n+1)/2}/\alpha$ for (i) large, (ii) intermediate and (iii) small positive values of $\mu /\alpha$. $P$ has no real roots, as $r$ grows from $r_{\mathrm{sing}}$, ${\psi }_1(r)$ (${\psi }_2(r)$) moves to the left (right) of ${\psi }_o$.