$R^{2}$ corrections to the black string instability and the boosted black string

TL;DR

This paper investigates how $R^2$ corrections from Gauss-Bonnet gravity affect the Gregory-Laflamme instability of black strings, showing the instability persists but with modified critical wavelengths, and explores boosted black strings with corrected thermodynamics.

Contribution

It provides the first detailed analysis of $R^2$ corrections on black string stability and thermodynamics, extending previous studies to include higher curvature effects and boosted configurations.

Findings

The instability persists with $R^2$ corrections.

Critical wavelength increases with the coupling parameter $rac{ alpha}{r_+^2}$.

Thermodynamic relations are restored in an extended framework.

Abstract

We consider a perturbative Gauss-Bonnet term supplementing the Einstein-Hilbert action, and evaluate its effect on the spectrum of the scalar mode that triggers the Gregory-Laflamme instability of black strings in five dimensional General Relativity. After studying some properties of the static black string, we provide the correction to the Lichnerowicz operator up to $\mathcal{O}(\alpha^2)$. For the scalar mode of the gravitational perturbation, we find a master variable and study its spectrum, providing an analysis of the regime of validity of our scheme. We show that the instability persists under the inclusion of the $R^2$-correction, and that the critical wavelength increases with the value of $\alpha/r_+^2<<1$, providing new evidence of the phenomenon which was already observed in other approaches. We also construct the boosted black strings and compute the correction to the mass, the momentum and the tension due to the higher curvature term. The presence of the dimensionful coupling $\alpha$ spoils the validity of the Smarr relation, which is the gravitational version of the Euler relation which must hold for every homogeneous, thermodynamic system. We show that this identity can be restored by working in an extended thermodynamic setup that includes variations of the Gauss-Bonnet coupling.