Black holes have been accepted as thermodynamic systems ever since Hawking’s discovery of black hole radiation. The thermodynamic phase space of AdS black holes has been extended to include a P-V term in [1]. By studying the equation of state of AdS black holes in this extended phase space, many interesting phase transitions were found; a prominent example is the Van der Waals like small-to-large phase transition of charged AdS black holes which takes place below a critical pressure [2]. Recently, Visser reconsidered the situation in the context of holography [3]. While the natural unit G=1 was kept in previous studies, Visser argued that considering variations of G is necessary to achieve a duality between the bulk and boundary thermodynamic first laws. I will briefly review his argument, and rewrite the bulk first law in a new form containing both variations of P and C, the central charge of the dual CFT. This leads to a new understanding of the Van der Waals behaviour of charged AdS black holes, where the phase changes are now governed by C instead of P.
[1] D. Kastor, S. Ray and J. Traschen, Enthalpy and the Mechanics of AdS Black Holes, Class. Quant. Grav. 26 (2009) 195011, [0904.2765].
[2] D. Kubiznak and R. B. Mann, P-V criticality of charged AdS black holes, JHEP 07 (2012) 033, [1205.0559].
[3] M. Visser, Holographic Thermodynamics Requires a Chemical Potential for Color, [2101.04145].