Linear stability of Einstein-Gauss-Bonnet static spacetimes: Vector and scalar 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. In a previous paper, tensor perturbations were analyzed. In this paper we study vector and scalar perturbations. We show that vector perturbations can be analyzed in general using an $S$-deformation approach and do not introduce instabilities. On the other hand, we show by analyzing an explicit example that, contrary to what happens in Einstein gravity, scalar perturbations may lead to instabilities in black holes with spherical horizons when the Gauss-Bonnet string corrections are taken into account.

Figure 1

The potential $\tilde{V}=\alpha V_S$ as a function of $x^{*}=r^{*}/\sqrt{\alpha }$, for $n=3$, $\Lambda =0$, $\tilde{\mu }=\mu /\alpha =1.7$. The scalar perturbation corresponds to the $\ell =2$ harmonic.

Figure 2

The potential $\tilde{V}=\alpha V_S$ as a function of $x^{*}=r^{*}/\sqrt{\alpha }$, for $n=3$, $\Lambda =0$, $\tilde{\mu }=\mu /\alpha =1.7$. The scalar perturbation corresponds to the $\ell =10$ harmonic.

Figure 3

The potential $\tilde{V}=\alpha V_S$ and a (non normalized) gaussian test wave function as a function of $x^{*}=r^{*}/\sqrt{\alpha }$, for $n=3$, $\Lambda =0$, $\tilde{\mu }=\mu /\alpha =3$. The scalar perturbation corresponds to the $\ell =2$ harmonic. The normalized test function gives $⟨-d^2/d{x}^{*2}⟩=0.12$ and $⟨\alpha V_S⟩\simeq -0.28$. The expectation value of the “Hamiltonian” is negative for this test function, implying the existence of negative energy eigenvalues.

Figure 4

The potential $\tilde{V}=\alpha V_S$ and a (non normalized) gaussian test wave function as a function of $x^{*}=r^{*}/\sqrt{\alpha }$, for $n=3$, $\Lambda =0$, $\tilde{\mu }=\mu /\alpha =3$. The scalar perturbation corresponds to the $\ell =10$ harmonic. The normalized test function gives $⟨-d^2/d{x}^{*2}⟩=0.73$ and $⟨\alpha V_S⟩\simeq -6.11$. The expectation value of the Hamiltonian is negative for this test function, implying the existence of negative energy eigenvalues.

Figure 5

The potential $\tilde{V}=\alpha V_S$ and a (non normalized) gaussian test wave function as a function of $x^{*}=r^{*}/\sqrt{\alpha }$, for $n=3$, $\Lambda =0$, $\tilde{\mu }=\mu /\alpha =6$. The scalar perturbation corresponds to the $\ell =10$ harmonic. The normalized test function gives $⟨-d^2/d{x}^{*2}⟩=0.15$ and $⟨\alpha V_S⟩\simeq -0.24$. The expectation value of the Hamiltonian is negative for this test function, implying the existence of negative energy eigenvalues.

Figure 6

The potential $\tilde{V}=\alpha V_S$ as a function of $x^{*}=r^{*}/\sqrt{\alpha }$, for $n=3$, $\Lambda =0$, $\tilde{\mu }=\mu /\alpha =10$. The scalar perturbation corresponds to the $\ell =2$ harmonic. $\alpha V_S$ has a small negative tail—shown in the inset—which cannot accommodate a negative energy state.