Classical and quantum Reissner-Nordström black hole thermodynamics and first order phase transition

Abstract

First we consider classical Reissner-Nordström black hole (CRNBH) metric which is obtained by solving Einstein-Maxwell metric equation for a point electric charge $e$ inside of a spherical static body with mass $M$. It has 2 interior and exterior horizons. Using Bekenstein-Hawking entropy theorem we calculate interior and exterior entropy, temperature, Gibbs free energy and heat capacity at constant electric charge. We calculate first derivative of the Gibbs free energy with respect to temperature which become a singular function having a singularity at critical point $M_{c}=\frac{2|e|}{\sqrt{3}}$ with corresponding temperature $T_{c}=\frac{1}{24\pi\sqrt{3}|e|}$. Hence we claim first order phase transition is happened there. Temperature same as Gibbs free energy takes absolutely positive (negative) values on the exterior (interior) horizon. The Gibbs free energy takes two different positive values synchronously for $0< T< T_{c}$ but not for negative values which means the system is made from two subsystem. For negative temperatures entropy reaches to zero value at $T\to-\infty$ and so takes Bose-Einstein condensation single state. Entropy increases monotonically in case $0< T< T_{c}$. Regarding results of the work presented at Wang and Huang (Phys. Rev. D 63:124014, 2001) we calculate again the mentioned thermodynamical variables for remnant stable final state of evaporating quantum Reissner-Nordström black hole (QRNBH) and obtained results same as one in case of the CRNBH. Finally, we solve mass loss equation of QRNBH against advance Eddington-Finkelstein time coordinate and derive luminosity function. We obtain switching off of QRNBH evaporation before than the mass completely vanishes. It reaches to a could Lukewarm type of RN black hole which its final remnant mass is $m_{final}=|e|$ in geometrical units. Its temperature and luminosity vanish but not in Schwarzschild case of evaporation. Our calculations can be take some acceptable statements about information loss paradox (ILP).