LAUSR

We consider a vacuum static spacetime in a finite size cavity. On the boundary, we specify a metric and a finite constant local temperature [Formula: see text]. No spherical or any other spatial symmetry is assumed. We show that (i) inside a cavity, only a black hole or flat spacetime are possible, whereas a curved horizonless regular spacetime is excluded, (ii) in the limit when the horizon area shrinks, the Hawking temperature diverges, (iii) for the existence of a black hole, [Formula: see text] should be high enough. When [Formula: see text], a black hole phase is favorable thermodynamically. Our consideration essentially uses the coordinate system introduced by Israel in his famous proof of the uniqueness theorem.