Interplay of Null Energy Condition Violations and Thermodynamics in Kiselev Black Hole Evaporation

TL;DR

The paper studies a two-horizon Kiselev black hole to understand how null energy condition (NEC) violations influence thermodynamics during evaporation. By deriving the Hawking temperature $T_H$, entropy $S$, and heat capacity $C$ and identifying a phase-transition condition $N_{\mathrm{PT}}=\frac{r_+^{3\omega+1}}{9\omega^2+6\omega}$, it shows that $T_H$ can peak at a phase transition and then decay to zero, or decay monotonically depending on the heat-capacity sign; in the RN-like limit $\omega=1/3$ these results reduce to familiar RN behavior. The analysis of parameter variations $M$ and $N$ (independently and jointly) reveals rich dynamics where phase transitions can occur even when only one parameter changes, and a dynamical NEC picture with horizons and a critical $r_{\mathrm{nec}}$ explains when evaporation resembles Schwarzschild-like behavior or leads to horizon mergers. Overall, the work links thermodynamic stability, horizon dynamics, and energy-condition structure in anisotropic black holes and suggests potential observational signatures via shadow dynamics during evaporation.

Abstract

The evaporation of black holes with two horizons presents a rich thermodynamic landscape that departs fundamentally from the Schwarzschild paradigm. In this work, we analyze the Hawking temperature dynamics of the Kiselev black hole under varying mass $M$ and anisotropic fluid parameter $N$, explicitly connecting temperature behavior to phase transitions and violations of the null energy condition (NEC). We find that the temperature does not necessarily diverge during evaporation; instead, it typically falls to zero as the black hole evaporates. This cooling behavior is preceded, in certain parameter regimes, by a phase transition marked by a peak in temperature and a divergence in heat capacity. Crucially, the presence and nature of these phase transitions are dictated by the spacetime regions where the NEC is violated: global NEC violation leads to horizon merger and temperature suppression, while partial or absent violation can restore the standard evaporation picture. Our results establish a direct correspondence between thermodynamic stability, horizon dynamics, and energy condition structure in anisotropic black hole spacetimes.

Figures (6)

• Figure 1: With $N = 0.6,\, 0.65,\,0.75,\, 0.85,\, 0.95$, phase transitions are observed for alpha values between 0.7 and 1, marking a shift from black hole instability to local stability. A further phase transition occurs as alpha approaches 2, corresponding to a switch back from local stability to instability.
• Figure 2: (a) Temperature of phase transitions as a function of $\omega$. The temperature increases as $\omega$ decreases. (b) Comparison between $N_{\mathrm{PT}}$ and $N_{c}$. For $\omega= 0$, the ratio $N_{c}/N_{\mathrm{PT}}$ is approximately 2.
• Figure 3: (a) Temperature as a function of mass for increasing $M$ at fixed $N = 1$. The temperature starts from zero (extremal limit, $M = N$), rises to a maximum (phase transition), and then decreases monotonically. (b) Temperature for decreasing $M$-the mirror image of (a)-showing a monotonic rise to a critical maximum followed by a sharp drop to zero.
• Figure 4: (a) Temperature as $N$ increases at fixed $M = 1$. The curve starts at the Schwarzschild maximum ($N=0$), where the derivative vanishes-indicating a phase transition-and then decreases parabolically to zero (extremal limit, $M = N$). (b) Temperature as $N$ decreases-mirror image of (a)-showing a rise toward the Schwarzschild limit.
• Figure 5: (a) Temperature evolution when mass increases faster than the charge-like parameter $N$ (here denoted $Q$). Starting from $Q = 0.01$, both $Q$ and $M$ grow, but $M$ increases at twice the relative rate. A clear phase transition appears at a critical mass, beyond which the temperature asymptotically approaches zero. (b) Temperature behaviour when $N$ (denoted $Q$) grows faster than mass. The temperature decreases monotonically with increasing $M$, driven predominantly by the rapid growth of $N$, which suppresses the Hawking temperature.