  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).
