Linear stability of Einstein-Gauss-Bonnet static spacetimes. Part II: vector and scalar perturbations

TL;DR

This paper analyzes the linear stability of static Einstein–Gauss–Bonnet spacetimes in $D=n+2$ under vector and scalar perturbations. The authors cast the perturbation equations into Schrödinger-like master equations for each perturbation type, using an $S$-deformation technique to prove vector-mode stability and a Regge–Wheeler–Zerilli formulation for scalar modes, whose general stability is parameter-dependent. They show that vector perturbations are stable across the studied parameter range, while scalar perturbations can become unstable for small-mass, spherical 5D EGB black holes, revealing a qualitative difference from GR. The analysis relies on interpolating the dimension dependence from explicit results up to $D\le 11$ and applying the framework to a Schwarzschild-like 5D EGB example, illustrating the potential for negative-energy bound states. The work provides robust tools and potentials for future investigations of quasi-normal modes, black hole uniqueness, and naked singularities in EGB gravity, with implications for higher-dimensional and string-inspired gravity scenarios.

Abstract

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 $Σ_{\k}^n \times {\mathbb R}^+$, $Σ_{\k}^n$ an $n-$manifold of constant curvature $\k$. Linear perturbations for this class of space-times 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.

Figures (6)

• 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.