Linear stability of Einstein-Gauss-Bonnet static spacetimes. Part I: tensor perturbations

TL;DR

This work analyzes the linear stability of static spacetimes in Einstein-Gauss-Bonnet gravity in $D=n+2$ with spatial sections $\Sigma_\kappa^n \times \mathbb{R}^+$, focusing on tensor perturbations. By deriving the tensor perturbation equations and recasting them as a Schrödinger-type problem with a background-dependent potential, the authors employ an S-deformation approach to obtain a universal stability criterion $H \ge 0$ that is independent of the Laplacian eigenvalue $\gamma$ on $\Sigma_\kappa^n$, enabling a comprehensive stability classification across parameter regimes. They classify maximally symmetric static EGB solutions via polynomials $P(\psi)$, horizons, and singularities, and show that cosmological solutions are broadly tensor-stable, with detailed dimension- and branch-specific thresholds for black holes (including special cases for $n=3,4,5$ and a mass-critical behavior in $d=6$). The results provide a robust framework for assessing tensor stability in a wide class of EGB spacetimes and lay groundwork for vector and scalar perturbations. The methods and stability criteria have potential implications for higher-dimensional gravity theories in string-inspired setups and for holographic contexts in AdS/CFT.

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. The analysis in this paper is restricted to tensor perturbations.

Figures (4)

• Figure 1: Cases 1.1.i to 1.1.iii: a) the two branches $\psi_i(r), i=1,2$ of equation (\ref{['pp']}) 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 equation (\ref{['pp']}) in the case $\mu / \alpha < 0$, $\psi_i$ goes from $\psi_o$ to $\Lambda_i$ as $r$ goes from $r_{sing}$ 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 equation (\ref{['pp']}) 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 $r_{sing}$, $\psi_1(r)$ ($\psi_2(r)$) moves to the left (right) of $\psi_o$.