Thermodynamical and dynamical stability of Einstein-Maxwell and extremal Einstein-Born-Infeld thin shells in $(2\ \mathbf{+}\ 1)$ dimensions

TL;DR

The study analyzes dynamical and thermodynamical stability of $2+1$-dimensional thin shells joining an inner AdS region to an outer BI–BTZ or Maxwell–BTZ spacetime, using the Darmois–Israel formalism and a linear equation of state. It shows that Maxwell–BTZ shells admit a wider range of dynamically stable configurations than BI–BTZ shells, and provides entropic constructions for both theories, including an analytic entropy for extremal BI shells. In the Maxwell case, there exist regions where dynamical and thermodynamical stability coincide, whereas for extremal BI shells the thermodynamical stable configurations are contained within the dynamically stable domain, with the extremal BI case allowing a closed analytic treatment via a power-law temperature. The results clarify how dynamics, thermodynamics, and nonlinear electrodynamics interplay in lower-dimensional gravity and offer a framework to map complete stability through parametric reductions, suggesting extensions to other BI shells and higher dimensions.

Abstract

We study the dynamical and thermodynamical stability of thin shells in (2+1)-dimensional spacetimes composed of an inner anti-de Sitter (AdS) region and an outer region described by a charged Bañados--Teitelboim--Zanelli (BTZ) spacetime, sourced either by Einstein--Maxwell theory (Maxwell-BTZ) or Einstein--Born--Infeld theory (BI-BTZ). Assuming a fixed charge-to-mass ratio and modeling the shell's matter with a linear equation of state, we introduce a convenient parametrization to analyze the dynamical stability configurations. We find that Maxwell-BTZ thin shells admit a wider range of dynamically stable configurations compared to BI-BTZ thin shells. We also derive the thermodynamics of the shell matter, obtaining physically meaningful entropy functions in both cases, and examine the conditions for thermodynamical stability. In the Maxwell-BTZ case, we identify regions in the parameter space where configurations are both dynamically and thermodynamically stable. In contrast, for extremal BI-BTZ thin shells, all thermodynamically stable configurations are contained within the dynamically stable ones, and shells with a linear equation of state are always dynamically stable. This work extends the understanding of thin shell configurations in lower-dimensional gravity and elucidates the interplay between dynamics, thermodynamics, and nonlinear electrodynamics.

Figures (7)

• Figure 1: (Orange region) Dynamical stability for a RN ring obeying a linear equation of state. The upper meshed region represents the region in the parameter space where the solution allows for two horizons for the outer manifold. The lower meshed region represents configurations in which the shell is glued over an overcharged spacetime (i.e., no horizons exist). The blue dashed line represents the extremal configuration where there is only one horizon.
• Figure 2: Regions of dynamical stability for a $\left({2+1}\right)$-dimensional BI thin-shell for varying $z=\lambda/\beta$. Notice that as $z$ grows the stability regions are shifted towards the domain where the outer spacetime is overcharged.
• Figure 3: (Left panel) Plot of the mass for the extremal configuration $m_\mathrm{ex}$ as a function of the charge for different values of $l$. Notice that there is a critical value in which $m_\mathrm{ex}=0$ for non-zero $Q$. (Right panel) Plot of the ansatz for the electrostatic potential in Eq. \ref{['eq:ansatzphiBI']} for different configurations of $l$. The dashed black line corresponds to the upper bound for the potential given by $Q/2$ (See Eq. \ref{['eq:boundphiBI']}). In both plots we have fixed $\beta=10$ and the dotted vertical lines represent the points where $m_\mathrm{ex}=0$.
• Figure 4: Configurations for the extremal BI-BTZ shell in the $\left({w,r_l}\right)$ plane in which the entropy in Eq. \ref{['eq:Swb']} is non-negative (colored regions), negative (gray region) and the ADM mass is negative (black region). Notice that as $\delta$ grows, the allowable configurations leading a non-negative entropy, decrease.
• Figure 5: Regions of thermodynamical and dynamical stability for the RN thin-shell with a Hawking-type entropy (green and orange regions, respectively). These conditions lead to configurations in the parameter space which correspond to the fulfillment of the two stability criteria (orange dotted regions), that can be regarded as more stable than those in which only one stability criteria holds.