Noether charge, thermodynamics and phase transition of a black hole in the Schwarzschild–anti-de Sitter–Beltrami spacetime

We investigate the thermodynamic properties and Hawking–Page phase transition of a black hole in the Schwarzschild–anti-de Sitter–Beltrami (SAdSB) spacetime. We discuss the Beltrami, or inertial, coordinates of the anti-de Sitter (AdS) spacetime. A transformation between noninertial and inertial coordinates of the AdS spacetime is formulated in order to construct a solution of a spherical gravitating mass and other physical quantities. The Killing vector is determined and used to calculate the event horizon radius of this black hole. The SAdSB black hole entropy and temperature are determined by the Noether charge method; the temperature is shown to be bounded by the AdS radius. Similarly, the Smarr relation and the first law of black hole thermodynamics for the SAdSB spacetime are formulated. The Gibbs free energy and heat capacity of this black hole are calculated and the phase transition between small and large black holes is considered. A first-order phase transition between the thermal AdS spacetime and the large-black-hole phase is also investigated and the Hawking–Page temperature is computed and compared with that of the Schwarzschild-anti-de Sitter black hole.