The canonical ensemble of a self-gravitating matter thin shell in AdS

TL;DR

This work constructs the canonical ensemble of a hot self-gravitating matter thin shell in AdS via a constrained Euclidean path integral, yielding a reduced action $I_*$ whose stationary points encode mechanical and thermodynamic equilibria. In the zero-loop approximation, the partition function collapses to a one-dimensional integral over the gravitational radius $\tilde{r}_+$, with the shell radius $\alpha$ fixed by a mechanical balance; the resulting thermodynamics give the mean energy $E$, entropy $S$, and heat capacity $C$, and establish a direct link between thermodynamic stability and positive $C$. Using a barotropic EOS, the authors obtain four stationary shell solutions, two of which are mechanically stable, and analyze their entropies and phase structure, showing a first-order transition to Schwarzschild–AdS black holes for sufficiently large scale separation. The results illuminate how self-gravitating matter in AdS can mimic or compete with black hole thermodynamics, revealing a rich phase structure controlled by the EOS and by a dimensionless parameter $z$ that measures matter versus gravitational scales.

Abstract

We build the canonical ensemble of a hot self-gravitating matter thin shell in anti-de Sitter (AdS) space by finding its partition function through the Euclidean path integral approach with fixed temperature at the conformal boundary. We obtain the reduced action of the system by restricting the path integral to spherically symmetric metrics with given boundary conditions and with the Hamiltonian constraint satisfied. The stationary conditions, i.e., the mechanical equilibrium and the thermodynamic equilibrium, are obtained from minimizing the reduced action. Evaluating the perturbed reduced action at the stationary points yields the mechanical stability condition and the thermodynamic stability condition. The reduced action calculated at the stationary points gives the partition function in the zero-loop approximation and from it the thermodynamic properties of the system are acquired. Within thermodynamics alone, the only stability condition that one can establish is thermodynamic stability, which follows from the computation of the heat capacity. For given specific pressure and temperature equations of state for the shell, we obtain the solutions of the ensemble. There are four different thin shell solutions, one of them is fully stable, i.e., is stable mechanically and thermodynamically. For the equations of state given, we find a first order phase transition from the matter thermodynamic phase to the Hawking-Page black hole phase. Moreover, there is a maximum temperature above which the shell ceases to exist, presumably at these high temperatures the shell inevitably collapses to a black hole.

Figures (9)

• Figure 1: Solutions of the balance of pressure $\frac{\alpha_{\rm u}}{l}$ and $\frac{\alpha_{\rm s}}{l}$ as function of $\frac{\tilde{r}_+}{l}$.
• Figure 2: Solutions of the ensemble $\frac{\tilde{r}_{+{\rm u}1}}{l}$, $\frac{\tilde{r}_{+{\rm u}2}}{l}$, $\frac{\tilde{r}_{+{\rm s}1}}{l}$, and $\frac{\tilde{r}_{+{\rm s}2}}{l}$, as functions of ${\bar{T}}l \left(\frac{l_{\rm c}}{l}\right)^\frac{1}{4} \left(\frac{l_{\rm p}}{l}\right)^\frac{1}{2}$. Both $\frac{\tilde{r}_{+{\rm u}1}}{l}$ and $\frac{\tilde{r}_{+{\rm u}2}}{l}$ have shell radius $\alpha_{\rm u}$, while both $\frac{\tilde{r}_{+{\rm s}1}}{l}$ and $\frac{\tilde{r}_{+{\rm s}2}}{l}$ have shell radius $\alpha_{\rm s}$.
• Figure 3: Matter entropy $\left(\frac{l^3}{l_{\rm p}^2 l_{\rm c}}\right)^{\frac{1}{4}} \frac{l_{\rm p}^2}{l^2}S_{\rm m}$ in function of the gravitational radius $\frac{\tilde{r}_+}{l}$ for the two shell radius solutions $\alpha_{\rm u}(\tilde{r}_+)$ and $\alpha_{\rm s}(\tilde{r}_+)$. A fit was performed for each branch, with $\left(\frac{l^3}{l_{\rm p}^2 l_{\rm c}}\right)^{\frac{1}{4}} \frac{l_{\rm p}^2}{l^2}S_{\rm m}=1.54662 (\frac{\tilde{r}_+}{l})^{1.2323}$ for the case of $\alpha_{\rm u}(\tilde{r}_+)$ and $\left(\frac{l^3}{l_{\rm p}^2 l_{\rm c}}\right)^{\frac{1}{4}} \frac{l_{\rm p}^2}{l^2}S_{\rm m}=0.898912(\frac{\tilde{r}_+}{l})^{0.755675} +0.867397(\frac{\tilde{r}_+}{l})^{2.91424}$ for $\alpha_{\rm s}(\tilde{r}_+)$, with respective coefficients of determination $R^2 = 0.999992$ and $R^2=1$, with this last equality being approximate.
• Figure 4: Adimensional heat capacity for the solutions $\tilde{r}_{+{\rm u}1}$, $\tilde{r}_{+{\rm u}2}$, $\tilde{r}_{+{\rm s}1}$, and $\tilde{r}_{+{\rm s}2}$ as functions of ${\bar{T}}l \left(\frac{l_{\rm c}}{l}\right)^\frac{1}{4} \left(\frac{l_{\rm p}}{l}\right)^\frac{1}{2}$, where the solutions $\alpha_{\rm u}$ and $\alpha_{\rm s}$ are also assumed. The solutions $\tilde{r}_{+{\rm u}1}$ and $\tilde{r}_{+{\rm s}2}$ are thermodynamically unstable, while $\tilde{r}_{+{\rm u}2}$ and $\tilde{r}_{+{\rm s}1}$ are thermodynamically stable.
• Figure 5: Solutions of the ensemble $\frac{r_{+1}}{l}$ and $\frac{r_{+2}}{l}$ for the black hole in asymptotically AdS.