Thermodynamic Volume and the Extended Smarr Relation

Understand

We continue to explore the scaling transformation in the reduced action formalism of gravity models.

• As an extension of our construction, we consider the extended forms of the Smarr relation for various black holes, adopting the cosmological constant as the bulk pressure as in some literatures on black holes.
• Firstly, by using the quasi-local formalism for charges, we show that, in a general theory of gravity, the volume in the black hole thermodynamics could be defined as the thermodynamic conjugate variable to the bulk pressure in such a way that the first law can be extended consistently.
• This, so called, thermodynamic volume can be expressed explicitly in terms of the metric and field variables.