Thermodynamics of Black Holes in Massive Gravity (1409.2369v1)

Abstract: We present a class of charged black hole solutions in an ($n+2)$-dimensional massive gravity with a negative cosmological constant, and study thermodynamics and phase structure of the black hole solutions both in grand canonical ensemble and canonical ensemble. The black hole horizon can have a positive, zero or negative constant curvature characterized by constant $k$. By using Hamiltonian approach, we obtain conserved charges of the solutions and find black hole entropy still obeys the area formula and the gravitational field equation at the black hole horizon can be cast into the first law form of black hole thermodynamics. In grand canonical ensemble, we find that thermodynamics and phase structure depends on the combination $k -\mu^2/4 +c_2 m^2$ in the four dimensional case, where $\mu$ is the chemical potential and $c_2m^2$ is the coefficient of the second term in the potential associated with graviton mass. When it is positive, the Hawking-Page phase transition can happen, while as it is negative, the black hole is always thermodynamically stable with a positive capacity. In canonical ensemble, the combination turns out to be $k+c_2m^2$ in the four dimensional case. When it is positive, a first order phase transition can happen between small and large black holes if the charge is less than its critical one. In higher dimensional ($n+2 \ge 5$) case, even when the charge is absent, the small/large black hole phase transition can also appear, the coefficients for the third ($c_3m^2$) and/or the fourth ($c_4m^2$) terms in the potential associated with graviton mass in the massive gravity can play the same role as the charge does in the four dimensional case.

• The paper introduces an extensive analysis of (n+2)-dimensional black hole solutions in massive gravity, uncovering phase transitions in both grand canonical and canonical ensembles.
• It employs Hamiltonian methods to compute conserved charges and shows how graviton mass terms alter the effective cosmological constant and horizon topology.
• The study draws parallels with van der Waals systems, offering new insights into quantum gravity and holography in higher-dimensional frameworks.

Thermodynamics of Black Holes in Massive Gravity

The paper "Thermodynamics of Black Holes in Massive Gravity" by Rong-Gen Cai, Ya-Peng Hu, Qi-Yuan Pan, and Yun-Long Zhang offers a thorough analysis of black hole solutions within the framework of a higher-dimensional massive gravity theory, characterized by a negative cosmological constant. The massive gravity model under consideration deviates from the standard General Relativity (GR) by incorporating a mass term for the graviton, which is realized through distinct polynomial terms in the graviton potential.

Key Findings and Methodologies

The authors establish black hole solutions in an $(n+2)$-dimensional spacetime and scrutinize black hole thermodynamics within both grand canonical and canonical ensembles. A notable feature of their solutions is the topological nature of the black hole horizons, which can assume positive, zero, or negative constant curvature. The derivation utilizes Hamiltonian methods to compute conserved charges of the black holes, ensuring compatibility with established thermodynamic relations.

• Grand Canonical Ensemble: Within a grand canonical framework, the analysis reveals that thermodynamic stability and phase transitions are dictated by a combination of geometric and potential terms, specifically $k - \mu^2/4 + c_2 m^2$ in the four-dimensional space. When positive, a Hawking-Page phase transition is possible, while a negative value indicates stability without phase transition.
• Canonical Ensemble: In a canonical setting, focusing on fixed charge, the key combination turns to $k + c_2 m^2$. The analysis shows that a first order phase transition between small and large black holes can occur when $k + c_2 m^2$ is positive, and the charge is sub-critical. This analogy to a van der Waals system highlights intricate interactions between black hole microstates akin to those seen in conventional thermodynamic systems.

Implications of the Study

The findings underscore massive graviton terms' non-trivial roles in altering black hole thermodynamics compared to GR predictions. In the four-dimensional scenarios, the addition of graviton mass modifies the effective cosmological constant, leading to novel phenomena such as topological phase transitions even in cases traditionally absent in GR. This is evident when $c_2 > 0$, allowing for phase transitions across different horizon topologies.

In higher dimensions, the terms $c_3 m^2$ and $c_4 m^2$ introduce additional dynamics, which can emulate the function of electric charge in facilitating phase transitions, extending these phenomena beyond the charge-based transitions seen in GR. This indicates a robustness of phase structures in massive gravity, potentially observable in higher-dimensional theories such as string theory scenarios.

Conclusions and Future Directions

The theoretical implications of this work suggest further explorations into massive gravity models could yield significant insights into quantum gravity and holography, particularly in AdS/CFT contexts. Moreover, the observable thermodynamic properties of these black holes may offer new avenues for understanding phase dynamics in higher dimensions and novel holographic correspondences.

While the choice of reference metrics seems pivotal, future studies could focus on varying these metric assumptions and exploring their impact on solution properties and stability. Additionally, examining the interplay between different massive gravity potentials could reveal richer structures in the space of black hole solutions and their corresponding thermodynamic systems. Overall, this work opens several paths for further research into the interplay between gravitation, topology, and thermodynamics in massive gravity frameworks.