Abstract
We consider Reissner–Nordström black holes surrounded by quintessence where both a non-extremal event horizon and a cosmological horizon exist besides an inner horizon ( $$-1\le \omega <-1/3$$ ). We determine new extreme black hole solutions that generalize the Nariai horizon to asymptotically de Sitter-like solutions for any order relation between the squares of the charge $$q^2$$ and the mass parameter $$M^2$$ provided $$q^2$$ remains smaller than some limit, which is larger than $$M^2$$ . In the limit case $$q^2=9\omega ^2 M^2/(9\omega ^2-1)$$ , we derive the general expression of the extreme cosmo-black-hole, where the three horizons merge, and we discuss some of its properties. We also show that the endpoint of the evaporation process is independent of any order relation between $$q^2$$ and $$M^2$$ . The Teitelboim energy and the Padmanabhan energy are related by a nonlinear expression and are shown to correspond to different ensembles. We also determine the enthalpy $$H$$ of the event horizon, as well as the effective thermodynamic volume which is the conjugate variable of the negative quintessential pressure, and show that in general the mass parameter and the Teitelboim energy are different from the enthalpy and internal energy; only in the cosmological case, that is, for Reissner–Nordström–de Sitter black hole we have $$H=M$$ . Generalized Smarr formulas are also derived. It is concluded that the internal energy has a universal expression for all static charged black holes, with possibly a variable mass parameter, but it is not a suitable thermodynamic potential for static-black-hole thermodynamics if $$M$$ is constant. It is also shown that the reverse isoperimetric inequality holds. We generalize the results to the case of the Reissner–Nordström–de Sitter black hole surrounded by quintessence with two physical constants yielding two thermodynamic volumes.
Highlights
The inclusion of the P–V term in the first law of thermodynamics or in its familiar equivalent laws [1,2,3,4,5,6,7,8,9,10,11] has led to the notion of the effective thermodynamic volume, which is in general different from the geometric volume excluded by, say, the event horizon
In the static case, a potential problem exists as noticed by Dolan [8]: the thermodynamic volume V is a function of the entropy, S, and so one of the two variables, S or V, is redundant
This implies that the internal energy is not a suitable thermodynamic potential for the thermodynamic description of the static de Sitter and anti-de Sitter black holes
Summary
The inclusion of the P–V term in the first law of thermodynamics or in its familiar equivalent laws [1,2,3,4,5,6,7,8,9,10,11] has led to the notion of the effective thermodynamic volume, which is in general different from the geometric volume excluded by, say, the event horizon. The thermodynamic volume is the conjugate variable, with respect to some appropriate thermodynamic potential, of the pressure exerted on the horizon attributable to the presence of a constant cosmological density, or a quintessence, or both From this point of view, much more progress has been made for anti-de Sitter black holes [12,13,14,15,16,17,18,19] thanks to the AdS/CFT correspondence, the applicability of which has ever been extended [20,21,22,23,24,25]. The inclusion of the P–V yields, on the one hand, a generalized Smarr formula preserving a scaling law between thermodynamic variables and, on the other hand, an identification of the mass parameter with the enthalpy of the event horizon These properties apply to both static and rotating black holes. The volume no longer depends on the entropy only, and so it is an independent thermodynamic variable
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