Thermodynamics of toroidal black holes (Q2738082)

The authors study the thermodynamical properties of a four-dimensional, static solution to the Einstein's equations, with a negative cosmological constant \((\Lambda)\). It represents a charged black hole with a toroidal event horizon. They follow closely the treatment first developed by York and Whiting, based on the Euclidean path integral approach [\textit{J. W. York jun.}, Phys. Rev. D (3) 33, 2092-2099 (1986) and \textit{B. F. Whiting} and \textit{J. W. York jun.}, Phys. Rev. Lett. 61, 1336 ff. (1988)]. NEWLINENEWLINENEWLINEThe black hole is charged, therefore its thermodynamical properties are most appropriately investigated in the statistical mechanical grand canonical ensemble. It means that the black hole is put in a box of fixed size: the boundary \(r_B\). There, the inverse temperature \(\beta\) is kept constant and the difference between the electrostatic potential \(\phi\) on the boundary and on the horizon is also kept constant. NEWLINENEWLINENEWLINEUnder the above conditions, the authors compute the reduced action \(I^*\) which is related to the thermodynamical grand canonical potential \(F\) by: \(I^* = \beta F\). Now, from \(F\), they evaluate the thermal energy, entropy and mean value of the charge for the toroidal black hole. They find that the entropy is given by one fourth of the area of the black hole event horizon. This result is in agreement with the Hawking-Bekenstein entropy. NEWLINENEWLINENEWLINEThe reduced action has two free parameters: the event horizon radius \(r_+\) and the electric charge \(e\). If one computes the first order variations of \(I^*\) with respect to \(r_+\) and \(e\) and sets them to zero, one finds two equations. Solving the resulting equations for \(r_+\) and \(e\), in terms of the fixed quantities, one finds the allowed black hole configurations. The authors follow this procedure and plot a two-dimensional surface representing the allowed values of \(r_+\) in terms of \(\phi\) and \(\sigma\), where \(\sigma = (\beta/4\pi)\sqrt{-\Lambda/3}\). Then, \(e\) is straightforwardly computed from an expression involving \(r_+\), \(\Lambda\) and \(\phi\). They also show that these solutions are globally stable and dominate the grand partition function. Therefore, they conclude that the toroidal black hole is more stable than the Reissner-Nordström-anti-DeSitter black hole. This is the case because, for the latter, some allowed solutions are not globally stable nor dominate the grand partition function. NEWLINENEWLINENEWLINEThe authors end the article showing two different ways to take the boundary to infinity \((r_B \to \infty)\). For each one of them they evaluate the relevant physical quantities.