Negative temperature for negative lapse function

Fermion dynamics distinguishes spacetimes having the same metric $g_{\mu \nu }$, but different tetrads $e_{\mu a}$, and in particular, it distinguishes a lapse with negative sign, $N<0$.[1] Here we show that the quasiequilibrium thermodynamic state may exist, in which the region with $N<0$ has negative local temperature $T(\mathbf{r})<0$, while the global Tolman temperature $T_0$ remains positive. For bosons, only $N^2$ matters. However, if bosons are composite, they may inherit the negative $T(\mathbf{r})$ from the fermions, and thus they may distinguish the spacetimes with positive and negative lapse functions via thermodynamics.