Critical bubbles and implications for critical black strings

TL;DR

This work establishes gravitational critical phenomena in $q+3$-dimensional bubble spacetimes by performing a linear perturbation analysis of static, homogeneous bubbles with ${\bf U(1)}\times SO(q+1)$ symmetry and showing a single unstable mode with universal growth rate $\Omega$ depending only on $q$. Through Kaluza-Klein reduction and a careful treatment of the perturbations, the authors derive a self-adjoint pulsation operator whose spectrum contains a negative eigenvalue, and they obtain $\Omega$ numerically for $q$ up to $250$ with the asymptotic scaling $\Omega\sim\sqrt{q}$ for large $q$. A double analytic continuation maps these unstable bubble modes to stationary, harmonic deformations of charged black strings, yielding a critical length $L_c=2\pi R_0/\Omega$ and connecting to the Gregory-Laflamme instability, with analytic expressions for the charged case and a generalized large-$q$ mass relation $\mu$. The results provide a unified framework linking bubble critical phenomena and black-string stability across dimensions and charge, and they clarify how charge worsens stability by shortening the critical length.

Abstract

We demonstrate the existence of gravitational critical phenomena in higher dimensional electrovac bubble spacetimes. To this end, we study linear fluctuations about families of static, homogeneous spherically symmetric bubble spacetimes in Kaluza-Klein theories coupled to a Maxwell field. We prove that these solutions are linearly unstable and posses a unique unstable mode with a growth rate that is universal in the sense that it is independent of the family considered. Furthermore, by a double analytical continuation this mode can be seen to correspond to marginally stable stationary modes of perturbed black strings whose periods are integer multiples of the Gregory-Laflamme critical length. This allow us to rederive recent results about the behavior of the critical mass for large dimensions and to generalize them to the charged black string case.

Figures (3)

• Figure 1: The plot illustrates the obtained dependence of $\Omega$ on $q$ (dark circles, with a dotted joining line for guidance). The inset shows $\Omega$ versus $q$ for low values of $q$.
• Figure 2: Percentile error $PE$ of the actual value of $\Omega$ compared to the simple law $\sqrt{q}$
• Figure 3: The eigenfunction $Y = e^{\Omega\rho} X$ as a function of $U$ for three different values of $q$.