Consistent mass formulas for the four-dimensional dyonic NUT-charged spacetimes

Revisiting mass formulae of the four-dimensional Reissner-Nordström-NUT-AdS solutions in a different metric form

“…In order to obtain the first law which is reasonable and consistent in both physical and mathematical sense, we shall adopt the method used in Refs. [20,21,22,38] to derive a meaningful Christodoulou-Ruffini-type squared-mass formula. First, substituting r + = Ξ S/π − n 2 into the horizon equation:…”

“…As did in Refs. [20,21,22,40,41,42], differentiating it with respect to the thermodynamical variables (S, N, Q, P, J n ) yields their conjugate quantities, respectively, subsequently we can arrive at the differential and integral mass formulae with their conjugate thermodynamic potentials given by the common Maxwell relations.…”

“…By contrast, a novel viewpoint that "NUT charge as a multi-hair" was put forward in our previous papers [20,21], where the Christodoulou-Ruffini-type squared-mass formula [26,27] is derived from the beginning and acts as the starting point of the whole work, then the first law and the Bekenstein-Smarr mass formula of almost all of the four-dimensional dyonic NUT-charged spacetimes are consistently deduced by simply applying the ordinary Maxwell relation. It is emphasized there that the NUT charge exhibits the two-fold facial features: rotation-like and electric-charge-like characters.…”

“…In recent years, there is a great deal of interest in exploring consistent thermodynamics of the Lorentzian Taub-NUT spacetimes in four dimensions [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21] and higher even-dimensions [22,23]. A most popular method advocated first in [3] to derive the mass formula relies on the Komar integration where the integral path is split into different patches and consisted of two Misner string tubes, so that the contribution of the Misner strings to the mass formulae is inlet via the Ψ − N ("Misner gravitational charge") pair [1,2,3,4,5,6,7,8].…”