Key points are not available for this paper at this time.
In our previous work Phys. Rev. D 100, 101501 (R) (2019), a novel idea that the Newman-Unti-Tamburino (NUT) charge can be thought of as a thermodynamical multihair has been advocated to describe perfectly the thermodynamical character of the generic four-dimensional Taub-NUT spacetimes. According to this scheme, the Komar mass (M), the gravitomagnetic charge (N), and/or the dual (magnetic) mass (M=N), together with a new secondary hair (J₍=MN), namely, a Kerr-like conserved angular momentum, enter into the standard forms of the first law and Bekenstein-Smarr mass formula. Distinguished from other recent attempts, our consistent thermodynamic differential and integral mass formulas are both obtainable from a meaningful Christodoulou-Ruffini-type squared-mass formula of almost all of the four-dimensional NUT-charged spacetimes. As an excellent consequence, the famous Bekenstein-Hawking one-quarter area-entropy relation can be naturally restored not only in the Lorentzian sector and but also in the Euclidian counterpart of the generic Taub-NUT-type spacetimes without imposing any constraint condition. However, only purely electric-charged cases in the four-dimensional Einstein-Maxwell gravity theory with a NUT charge have been addressed there. In this paper, we shall follow the simple, systematic way proposed in that article to further investigate the dyonic NUT-charged case. It is shown that the standard thermodynamic relations continue to hold true provided that no new secondary charge is added; however, the so-obtained electrostatic and magnetostatic potentials are not coincident with those computed via the standard method. To rectify this inconsistence, a simple strategy is provided by further introducing two additional secondary hairs, Q₍=QN and P₍=PN, together with their thermodynamical conjugate potentials, so that the first law and Bekenstein-Smarr mass formula are still satisfied, where Q and P being the electric and magnetic charges, respectively.
Wu et al. (Tue,) studied this question.