Lorentzian Taub-NUT spacetimes: Misner string charges and the first law

Motivated by recent activities in Lorentzian Taub-NUT space thermodynamics, we calculate conserved charges of these spacetimes. We find additional mass, nut, angular momentum, electric and magnetic charge densities distributed along Misner string. These additional charges are needed to account for the difference between the values of the above charges at horizon and at infinity. We propose an unconstrained thermodynamical treatment for Taub-NUT spaces, where we introduce the nut charge $n$ as a relevant thermodynamic quantity with its chemical potential \phi_n. The internal energy in this treatment is M-n\phi_n rather than the mass M. This approach leads to an entropy which is a quarter of the area of the horizon and all thermodynamic quantities satisfy the first law, Gibbs-Duhem relation as well as Smarr's relation. We found a general form of the first law where the quantities depend on an arbitrary parameter. Demanding that the first law is independent of this arbitrary parameter or invariant under electric-magnetic duality leads to a unique form which depends on Misner string electric and magnetic charges. Misner string charges play an essential role in the first law, without them the first law is not satisfied.