Extensivity, entropy current, area law, and Unruh effect

We present a general method to determine the entropy current of relativistic matter at local thermodynamic equilibrium in quantum statistical mechanics. Provided that the local equilibrium operator is bounded from below and its lowest lying eigenvector is non-degenerate, it is proved that, in general, the logarithm of the partition function is extensive, meaning that it can be expressed as the integral over a three-dimensional space-like hypersurface of a vector current, and that an entropy current exists. We work out a specific calculation for a nontrivial case of global thermodynamic equilibrium, namely, a system with constant comoving acceleration, whose limiting temperature is the Unruh temperature. We show that the integral of the entropy current in the right Rindler wedge is the entanglement entropy.