Consistent mass formulas for the four-dimensional dyonic NUT-charged spacetimes
Di Wu, Shuang‐Qing Wu
Abstract
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 ($\stackrel{\texttildelow{}}{M}=N$), together with a new secondary hair (${J}_{N}=MN$), 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, ${Q}_{N}=QN$ and ${P}_{N}=PN$, 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.