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We conjecture that (when the notion of Hadamard state is suitably adapted)there is no isometry-invariant Hadamard state for the massive or masslesscovariant Klein-Gordon equation defined on the region of the Kruskal spacetimeto the left of a surface of constant Schwarzschild radius in the rightSchwarzschild wedge when Dirichlet boundary conditions are put on that surface.We also prove that, with a suitable definition for 'boost-invariant Hadamardstate' (which we call 'strongly boost-invariant globally-Hadamard') which takesinto account both the existence of the timelike boundary and the specialmassless 1+1 infra-red pathology, there is no such state for the massless waveequation on the region of 1+1 Minkowski space to the left of an eternallyuniformly accelerating mirror -- with Dirichlet boundary conditions. Thisresult is significant because such a state does exist if there is also asymmetrically placed decelerating mirror in the left wedge (and the region tothe left of this mirror is excluded). We expect a similar existence result tohold for Kruskal when there are symmetrically placed spherical boxes in bothright and left Schwarzschild wedges. Our Kruskal no-go conjecture raises basicquestions about the black holes in boxes considered in black holethermodynamics. If true, it would lend further support to the conclusion ofB.S. Kay 'Instability of enclosed horizons', Gen. Rel. Grav. 47, 1-27 (2015)(arXiv: 1310.7395) that the nearest thing to a description of a black hole inequilibrium in a box in terms of a classical spacetime with quantum fieldspropagating on it has, for the classical spacetime, the exterior Schwarzschildsolution, with the classical spacetime picture breaking down near the horizon.An appendix points out the existence of, and partially fills, a gap in theproofs of the theorems in B.S. Kay and R.M. Wald Phys. Rep. 207, 49-136 (1991).

classical and quantum gravity

Non-existence of isometry-invariant Hadamard states for a Kruskal black  hole in a box and for massless fields on 1+1 Minkowski spacetime with a  uniformly accelerating mirror