Thermodynamics of dyonic black holes in non-linear electrodynamics

TL;DR

This work analyzes dyonic black holes in a fixed weak-field non-linear electrodynamics framework in AdS, deriving the general solutions and their extended thermodynamics. By incorporating independent couplings $a$ and $b$ for quadratic invariants and enforcing energy conditions outside the horizon, the authors reveal a richly structured phase space with up to five turning points and multiple reentrant phase transitions, going beyond RNAdS. They derive the first law and Smarr relation including $a$ and $b$ variations, and they explore both analytic (e.g., $q=0$ and $a=0$) and numerical regimes, uncovering novel features such as horizon jumps and modified extremal limits. The results illuminate how non-linear electromagnetic interactions in AdS alter black hole stability and critical phenomena, with implications for holography and the broader study of quantum gravitational effects in non-linear media.

Abstract

We investigate dyonic black holes in a weak field expansion of non-linear electrodynamics. The breadth of parameter space permits a rich thermodynamic structure, additional turning points and intricate phase phenomena. Energy conditions are employed to ensure the physical viability of solutions. Analytic special cases illustrate novel properties of black holes in non-linear electrodynamics, including modified extremal limit behaviour. Numerical solutions offer the most elaborate thermodynamic landscape, culminating in up to five turning points, and multiple reentrant phase transitions.

Figures (13)

• Figure 1: Plots of $f(r)$ against $r$ for $\Lambda = 0$, $q = 0$, for different masses $m$ and magnetic charge $p$. The left plot is for $p = 0.5$ and the right is for $p = 2$. The solid curves are the non-linear results with $a = 0.1$, and the dotted curves are their linear counterparts with $a = 0$. The horizontal thin line marks $f(r) = 0$, where the horizons occur. The vertical dashed line marks the threshold for the DEC violation discussed in Section \ref{['DECVioSec']}, so each solution shown conserves DEC outside the horizon. In the left plot, the linear theory displays an extremal limit as $m$ is varied, and yet the non-linear $f(r)$ are monotonic. In the right plot, it appears that the horizon jumps as $m$ is decreased, although to a region with DEC violation. We investigate these features further in Section \ref{['ExtremalJumpSec']}. For large $r$, all non-linear curves approach their linear counterparts, which have $f(r) \rightarrow 1$.
• Figure 2: A plot showing parameters $(a,b)$ that satisfy and violate DEC, for $p = 0.1$, $q = 0.3$, $m = 0.38$ and $\Lambda = 0$. If the event horizon $r_+$ is less than the radius from the DEC constraint, then DEC is violated outside the horizon, and the choice of parameters $(a,b)$ are excluded.
• Figure 3: A plot exhibiting a jump in black hole event horizon radius $r_+$ as $m$ is varied, for $q = 0$, $p = 1.17$, $a = 0.1$ and $\Lambda = 0$. The horizontal thin line marks $f(r) = 0$, where the horizons occur. The vertical dashed line marks the threshold for the DEC violation, so each solution shown conserves DEC outside the horizon. The black hole horizons in each case are the largest root of $f(r) = 0$, and are denoted by crosses. Each plot shown has a minimum and as the minimum crosses through $f(r) = 0$, shown for the red curve, there is a sudden jump in the horizon $r_+$. This corresponds to crossing through the local extremal limit. Note that the green and blue curves also have stationary points outside the horizon.
• Figure 4: A plot of $m(r_+)$, showing how a jump in black hole event horizon radius $r_+$ can occur as $m$ is varied, for $q = 0$, $p = 1.17$, $a = 0.1$ and $\Lambda = 0$. The vertical dashed line marks the threshold for the DEC violation, so all $r_+$ above this are valid. The green sections of the curve show the allowed horizon radii $r_+$ (modulo DEC). The red section of the curve shows the disallowed horizon radii, since for masses $m$ in this section, there is a larger valid horizon $r_+$. The horizontal dotted blue line marks a jump between valid event horizon radii.
• Figure 5: A plot showing phase diagrams of Gibbs free energy $G$ against temperature $T$, for three cases in the linear theory with RNAdS with $\Lambda = -1$. Due to the symmetry in $(p,q)$ in the linear theory, we have set $p = 0$ in these diagrams. The different cases correspond to no turning points ($q = 0.35$, blue), one turning point ($q = 1/\sqrt{12}$, green) and two turning points ($q = 0.25$, red). The curves continue in a similar fashion for larger $T$, and for smaller $T$ until $T = 0$ is reached.