Introduces a holographic pressure and volume for static spherically symmetric black holes via quasi-local thermodynamics, showing large black holes become extensive in the large-system limit while small ones do not.
The volume of stationary black holes and the meaning of the surface gravity
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abstract
The invariant four-volume $\mathcal{V}$ of a complete black hole (the volume of the spacetime at and interior to the horizon) diverges. However, if one considers the black hole set up by the gravitational collapse of an object, and integrates only a finite time to the future of the collapse, the resultant volume is well defined and finite. In this paper we examine non-degenerate stationary black holes (and cosmological horizons) and find that $\mathcal{V}_{s} \varpropto \ln(\lambda)$ where $s$ is any shell that terminates on the horizon, $\lambda$ is the affine generator of the horizon and the constant of proportionality is the Parikh volume of $s$ divided by the surface gravity. This provides an alternative local and invariant definition of the surface gravity of a stationary black hole.
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Treating the cosmological constant as pressure in black hole thermodynamics yields an extended dictionary with enthalpy, thermodynamic volume, and chemical-like phase transitions including Van der Waals behavior, reentrant transitions, and triple points.
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Holographic pressure and volume for black holes
Introduces a holographic pressure and volume for static spherically symmetric black holes via quasi-local thermodynamics, showing large black holes become extensive in the large-system limit while small ones do not.
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Black hole chemistry: thermodynamics with Lambda
Treating the cosmological constant as pressure in black hole thermodynamics yields an extended dictionary with enthalpy, thermodynamic volume, and chemical-like phase transitions including Van der Waals behavior, reentrant transitions, and triple points.