Mass Independent Area (or Entropy) and Thermodynamic Volume Products in Conformal Gravity

Abstract

In this work we investigate the thermodynamic properties of conformal gravity in four dimensions. We compute the \emph{area(or entropy) functional} relation for this black hole. We consider both de-Sitter (dS) and anti de-Sitter (AdS) cases. We derive the \emph{Cosmic-Censorship-Inequality} which is an important relation in general relativity that relates the total mass of a spacetime to the area of all the black hole horizons. Local thermodynamic stability is studied by computing the specific heat. The second order phase transition occurs at a certain condition. Various type of second order phase structure has been given for various values of $a$ and the cosmological constant $\Lambda$ in the Appendix. When $a=0$, one obtains the result of Schwarzschild-dS and Schwarzschild-AdS cases. In the limit $aM<<1$, one obtains the result of Grumiller space-time. Where $a$ is non-trivial Rindler parameter or Rindler acceleration and $M$ is the mass parameter. The \emph{thermodynamic volume functional} relation is derived in the \emph{extended phase space}, where the cosmological constant treated as a thermodynamic pressure and its conjugate variable as a thermodynamic volume. The \emph{mass-independent} area (or entropy) functional relation and thermodynamic volume functional relation that we have derived could turn out to be a \emph{universal} quantity.