Abstract
It is demonstrated that the generic four-dimensional Taub-Newman-Unti-Tamburino (Taub-NUT) spacetimes can be perfectly described in terms of three or four different kinds of thermodynamic hairs: the Komar mass ($M = m$), the "angular momentum" ($J_n = mn$), the gravitomagnetic charge ($N = n$), and/or the dual (magnetic) mass ($\widetilde{M} = n$). In other words, the NUT charge is a thermodynamic multihair which means that it simultaneously has both rotation-like and electromagnetic charge-like characteristics; this is in sharp contrast with the previous knowledge that it has only one physical feature, or that it is purely a single solution parameter. To arrive at this novel result, we put forward a simple, systematic way to investigate the consistent thermodynamic first law and Bekenstein-Smarr mass formulas of all four-dimensional spacetimes that contain a nonzero NUT charge, facilitated by first deriving a meaningful Christodoulou-Ruffini-type squared-mass formula. In this way, not only can the elegant Bekenstein-Hawking one-quarter area-entropy relation be naturally restored in the Lorentzian and Euclidian sectors of generic Taub-NUT-type spacetimes without imposing any constraint condition, but also the physical meaning of the NUT parameter as a poly-facet can be completely clarified in the thermodynamic sense for the first time.
Highlights
Ever since the seminal work of Bekenstein [1] and Hawking [2], it has been well known that the (BekensteinHawking) entropy of a black hole is proportional to the area of the horizon and its Hawking temperature to the surface gravity at the horizon
When a negative cosmological constant is included for the anti– de Sitter case, the above mass formulas (2) should include a modified term ðþVdP; −2VPÞ, respectively, where V is the thermodynamic volume conjugate to the pressure P 1⁄4 3g2=ð8πÞ with g being the inverse of the cosmological radius
In this work we have presented a simple, systematic way to naturally derive the thermodynamical first law and Bekenstein-Smarr mass formula of four-dimensional TaubNUT-type spacetimes
Summary
Ever since the seminal work of Bekenstein [1] and Hawking [2], it has been well known that the (BekensteinHawking) entropy of a black hole is proportional to the area of the horizon and its Hawking temperature to the surface gravity at the horizon. In some recent attempts [26,27,28,29,30], the so-called “consistent thermodynamical first law” was pursued for the Lorentzian Taub-NUT-type spacetimes These formulas could not really represent the actual first law from our viewpoint, since the imported ψ -N pair (which was later called the “Misner gravitational charge”) does not possess the conventional characteristics of global charges that are measured at infinity; rather, it combines the contributions of the Misner strings at the horizon, contrary to common wisdom. The novelty of this new viewpoint is that it can plausibly explain many of the peculiar properties of the NUT-charged spacetime, such as why the NUT parameter has so many different names and why there are different interpretations of the physical source of Taub-NUT-type spacetimes
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