Ruppeiner geometry and the fluctuation of the RN-AdS black hole in framework of the extensive thermodynamics

The Ruppeiner geometry and the fluctuation for a four-dimensional charged AdS black hole in the framework of the extensive thermodynamics have been investigated. By fixing the AdS radius of Vissers construction, the extensive thermodynamics of the RN-AdS black hole is established, and the central charge plays the role of particle number. With the thermodynamic variables density, we reexamine the phase transition of the black hole, and the result will hold for the black holes under different gravitational models. Moreover, we have studied the Ruppeiner geometry and the black hole fluctuation, and show that the increase of central charge will reduce the fluctuation probability.