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

We conjecture that (when the notion of Hadamard state is suitably adapted) there is no isometry-invariant Hadamard state for the massive or massless covariant Klein-Gordon equation defined on the region of the Kruskal spacetime to the left of a surface of constant Schwarzschild radius in the right Schwarzschild wedge when Dirichlet boundary conditions are put on that surface. We also prove that, with a suitable definition for 'boost-invariant Hadamard state' (which we call 'strongly boost-invariant globally-Hadamard') which takes into account both the existence of the timelike boundary and the special massless 1+1 infra-red pathology, there is no such state for the massless wave equation on the region of 1+1 Minkowski space to the left of an eternally uniformly accelerating mirror -- with Dirichlet boundary conditions. This result is significant because such a state does exist if there is also a symmetrically placed decelerating mirror in the left wedge (and the region to the left of this mirror is excluded). We expect a similar existence result to hold for Kruskal when there are symmetrically placed spherical boxes in both right and left Schwarzschild wedges. Our Kruskal no-go conjecture raises basic questions about the black holes in boxes considered in black hole thermodynamics. If true, it would lend further support to the conclusion of B.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 in equilibrium in a box in terms of a classical spacetime with quantum fields propagating on it has, for the classical spacetime, the exterior Schwarzschild solution, with the classical spacetime picture breaking down near the horizon. An appendix points out the existence of, and partially fills, a gap in the proofs of the theorems in B.S. Kay and R.M. Wald Phys. Rep. 207, 49-136 (1991).