Revisiting mass formulas of the four-dimensional Reissner-Nordstr\"{o}m-NUT-AdS solutions in a different metric form

Recently, the so-called ``consistent thermodynamics'' of the Lorentzian Reissner-Nordstr\"{o}m (RN)-NUT-AdS$_4$ spacetimes has been pursued by a lot of efforts via different means. Among these attempts, we had proposed a novel idea that ``The NUT charge is a thermodynamical multihair"to successfully tackle with the subject. In this paper, we will adopt this strategy to reconsider the mass formulas of the RN-NUT-AdS$_4$ solutions but written in an alternative form, which had not been studied before in any existing literature and might be a most appropriate ansatz for the higher dimensional multiply NUTty-charged AdS spacetimes without any constraint condition. Here, we shall discuss the Christodoulou-Ruffini-like squared mass formula and the first law as well as the Bekenstein-Smarr mass formula by introducing the secondary hair $J_n = Mn$. For the sake of generality, we have introduced a dimensionless constant $w$ into the constant factor $\Xi$ in the solution expression so that when $w = 1$, all obtained results can reproduce those delivered in our previous work.


Introduction
In recent years, there is a great deal of interest in exploring consistent thermodynamics of the Lorentzian Taub-NUT spacetimes in four dimensions [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23] and higher even dimensions [24,25].A most popular method advocated first in [3] to derive the mass formula relies on the Komar integral where the integration path is split into different patches and consisted of two Misner string tubes, so that the contribution of the Misner strings to the mass formulas is inlet via the "Ψ − N " ("Misner gravitational charge") pair [1,2,3,4,5,6,7,8].While standard method.By introducing the secondary hair J n = M n, we shall devote Sec. 3 to discussing the consistent thermodynamics by investigating the corresponding Christodoulou-Ruffini-like squared mass formula and the first law as well as the Bekenstein-Smarr mass formula.Finally, we present our conclusions and give our outlooks in Sec. 4. In the appendix Appendix B, we will establish the relation between the usual form of the solution with w = 1 and the alternative form when w = 1.In the appendix Appendix C, we present a universal method to calculate the secondary hair J n .

Alternative form of the RN-NUT-AdS 4 solution and some thermodynamical quantities
Distinct from its familiar expression of the RN-NUT-AdS 4 spacetime, here we present an alternative form for the same solution, whose metric and Abelian gauge potential are where Ξ = 1 + 3(w − 1)g 2 n 2 , and the radial function reads in which q, n and m are the electric charge, the NUT charge and the mass parameters, respectively, g is the gauge coupling constant, and w is only a numerical constant.When w = 1, the solution retains its familiar form that was adopted in our previous work [22].If w = 1, then the angular two-dimensional spherical part of the line element (1) is non-canonically normalized.The above general form of the solution had not appeared before in any literature and thus had never been studied in any previous work.In this paper, we are very interested in the case w = 0, whose higher even dimensional analogues are presented in the appendix Appendix A. But for the sake of generality, hereafter we shall keep w to be any arbitrary numerical constant, and shall not view it as a thermodynamic variable in the below discussion.In the appendix Appendix B, we will show explicitly how the above alternative form when w = 1 can be related to the usual RN-NUT-AdS 4 solution with w = 1 via some coordinate transformations and re-scaling of the solution parameters.Alternatively, we will also present the second possible choice of the time coordinate and its corresponding solution.Incidentally, we point out that the above general form of the NUT-charged AdS solution owns its relatives in the gauged supergravity, for example, the simplest static dyonic NUT AdS solution in the four-dimensional Kaluza-Klein supergravity theory must be given in a closely similar form.So it is significant to study the above general form of the solution, since it not only acts as a prototype of its higher even dimensional version, but also owns its relatives in the four-dimensional gauged supergravity theories.We begin by presenting some quantities that can be computed via the standard method.First, the Bekenstein-Hawking entropy can be straightforwardly taken as one quarter of the horizon area: where is the location of the event horizon that is the largest root r + specified by f (r + ) = 0.The Gibbons-Hawking temperature is directly proportional to the surface gravity κ on the event horizon: in which a prime, here and hereafter, denotes the partial derivative with respective to its variable.Second, as we did in our previous papers [22,23], we are only interested in the global conserved electric charge that is measured at infinity and can be computed by using the Gauss' law integral and its corresponding electrostatic potential at the event horizon simply reads because a gauge has been chosen so that the gauge potential vanishes at infinity.In the above, the timelike Killing vector χ = ∂ t is normal to the event horizon, and the surface gravity is computed via the definition: Third, one can adopt the conformal completion method [34] or the Abbott-Deser method [35] or the counter-term method [36] to compute the conserved mass M , and each technique gives the same result.Here we will follow the first one.The idea is to perform a conformal transformation on the metric (1) to remove the divergence in the integrals at the boundary (conformal infinity).After taking the r → ∞ limit in the line element ds 2 /r 2 , one obtains the boundary metric Then the conserved charges Q[ξ] associated with the Killing vector ξ can be computed by where C µ ανβ is the conformal Weyl curvature tensor, N µ = 0, −g 2 r 2 , 0, 0 is the vector normal to the boundary, and is the temporal component of the area vector in the three-dimensional conformal boundary.So the conformal mass M can be evaluated as Similarly, one can calculate the conformal dual mass by introducing the left-dual Weyl tensor as where ǫ µνρσ is the Levi-Civita totally anti-symmetric tensor.Therefore, by replacing the Weyl tensor C µ ανβ in Eq. ( 9) with the dual Weyl tensor C µ ανβ , the dual mass can be subsequently obtained as follows: Incidentally, here we would like to point out that the above conformal mass and its dual are identical to the Yano-ADM charge and its dual charge [37] associated with the Killing-Yano tensor k = db with b = (r 2 + n 2 )(dt + 2n cos θdφ/Ξ) and its Hodge dual, respectively.Finally, the NUT charge or gravitational magnetic (gravitomagnetic) charge can be computed as [38,39] (see the appendix Appendix C for the detail), Obviously, it is different from the above result (13) of the dual mass.

Mass formulas
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.[22,23,24,40] to derive a meaningful Christodoulou-Ruffini-type squared mass formula.First, substituting r + = ΞS/π − n 2 into the horizon equation: Inspired from this equation, it is clear that one can make the familiar substitutions into the rhs of Eq. ( 15): m = M Ξ, n = N Ξ = 2N/Γ, q = QΞ and g 2 = 8πP/3, where P = 3g 2 /(8π) is the generalized pressure [41], and According to the result presented in the appendix Appendix C, we now further introduce a secondary hair J n = M n = mn/Ξ, as we did before [22,23].After finishing a little algebraic manipulation, one can arrive at an useful identity: which is our new Christodoulou-Ruffini-like squared mass formula for the fourdimensional RN-NUT-AdS spacetime (1).We point out that Eq. ( 17) when w = 1 consistently reduces to the one obtained in the familiar case of the RN-NUT-AdS 4 spacetime [22].At this step, we would like to give a new interpretation for Eq. ( 17), which is very crucial to our analysis done below.Otherwise, one may doubt that there exists a mathematical inconsistency between the numbers of independent thermodynamical variables of those of the free parameters appeared in the structure function f (r) [Note that w will not be viewed as a solution parameter].According the new viewpoint suggested in our recent papers [24,25], equation ( 15) can be thought of as representing a hypersurface in the five-dimensional thermodynamical state space, the numbers of its variables (m, n, S, q, g) exactly match with those of the solution parameters that appeared in the function f (r).After introducing an extra hair J n , which is nothing but a kind of the higher-dimensional embedding mapping, it becomes a hypersurface in the six-dimensional state space, as specified by Eq. (17), which now has six variables (M, N, J n , S, Q, P ).So, our below discussions will be based upon this six-dimensional thermodynamical state space in which all its six variables could be regarded as independent.Now we are in a position to derive the differential and integral mass formulas for the generic RN-NUT-AdS 4 spacetime (1).Since the secondary hair J n will be treated as an independent variable, the above squared mass formula (17) can be regarded formally as a basic functional relation: M = M (S, N, Q, P, J n ).As we did in Refs.[22,23,24,42,43,44], 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 formulas with their conjugate thermodynamic potentials given by the common Maxwell relations.
First, differentiation of the squared mass formula (17) with respect to the entropy S leads to its conjugate Hawking temperature: which is entirely identical to that given by Eq. (5).Next, the electrostatic potential Φ and the velocity-like potential ω h , which are conjugate to Q and J n , respectively, are evaluated as Finally, the potential ψ h and the thermodynamical volume V , which are conjugate to N and P , respectively, can be computed as: It is obvious that only when w = 1, the electric charge parameter q in the expression of Eqs. ( 21) and ( 22) disappears, and the results reproduce the familiar case in our previous work [22].When w = 1, the electric charge parameter q appears apparently in the above two expressions.Now, one can check that both the first law and the Bekenstein-Smarr relation are completely fulfilled among all the aforementioned thermodynamical conjugate pairs.Note that, when w = 1 and then Ξ = 1, the derived conjugate thermodynamic volume is not equal to V = 4πr + (r 2 + + 3n 2 )/(3Ξ) as can be computed via the definition given in Ref. [41].If one prefers to match it to such a thermodynamic volume when w = 1, then the dual (magnetic) mass ( 13) can be further introduced as an additional conserved charge into the first law and the Bekenstein-Smarr relation: in which two new conjugate potentials are given by As can be seen apparently from the above expression, it is not possible to reproduce the thermodynamical volume V without the inclusion of the dual mass M .

Conclusions
In this paper, we have employed our proposal that "The NUT charge is a thermodynamical multihair" to revisit the thermodynamics of the RN-NUT-AdS 4 spacetimes written in an unusual form.After some thermodynamical quantities being evaluated via the standard method, we have analyzed the consistent thermodynamics with the inclusion of a new secondary hair J n = M n.We have obtained the consistent and reasonable Christodoulou-Ruffini-like squared mass formula, the first law and the Bekenstein-Smarr mass formulas.When w = 1, they both reproduce the previous results in the usual case [22].
Since the spacetime solution (1) is a special case for the generic even dimensional multi-NUTty charged AdS solutions presented in the appendix Appendix A, the present work acts as a warmup excises for studying the high-dimensional Lorentzian Taub-NUT (AdS) spacetimes with multiply NUT parameters [32,33] in the next step.On the other hand, we have also adopted the "Ψ − N " formalism to investigate thermodynamics of the RN-NUT-AdS 4 solution given by the metric (1) and the gauge potential (2).It is observed that one can derive a consistent Bekenstein-Smarr mass formula for arbitrary parameter w, however, the first law is only consistent for the usual w = 1 case.This is a very odd result, and we hope to report the related research soon.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix B. Solution relation between w = 1 and w = 1
Denote with a bar the case with w = 1, the usual standard form of the four-dimensional Reissner-Nordström NUT-charged AdS solution reads where the structure function is: f (r) = Ξ 2 f (r) = (r 2 −n 2 )Ξ−2mr +q 2 +g 2 r 4 + 6n 2 r 2 − 3n 4 , and Ξ = 1 + 3(w − 1)g 2 n 2 .Then, it is not difficult to find that the new form of the metric (1) and the gauge potential (2) with the radial function (3) can be obtained by further setting the time coordinate: t = t/ √ Ξ.Alternatively, it is also possible to adopt another time coordinate: t = √ Ξ t = Ξ t, so that the solution is written as This latter form can also be extended to higher even dimension AdS solutions with equal NUT-charges, but it seems inappropriate for making a generalization to the multiply NUTty cases.