Two-Horizon Sector Thermodynamics as a Diagnostic for the Bi-Hair Organization of NUT Charge

Abstract

After decades without a fully self-consistent thermodynamic formulation, recent proposals for Lorentzian Taub-NUT spacetime have yielded several inequivalent descriptions, each satisfying its own first law and Smarr relation, leaving unresolved which formulation is physically preferred. In this Letter we introduce a two-horizon diagnostic whose sector temperatures are fixed by harmonic mean relations before the thermodynamic variables are chosen. For uncharged Taub-NUT, the sum sector closes with the mass and NUT charge, whereas, within the restricted homogeneous class examined here, the difference sector requires the thermodynamic secondary hair $J_n=mn$. This result supports a bi-hair thermodynamic organization of the NUT parameter through $N=n$ and $J_n=mn$. Furthermore, this two-horizon framework offers a new route toward addressing similar thermodynamic nonuniqueness in other two-horizon systems when the first law and Smarr relation alone do not identify a physically preferred formulation.