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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

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Publication details

JournalClassical and Quantum Gravity
DateAccepted/In press - 6 Sep 2016
DateE-pub ahead of print - 5 Oct 2016
DatePublished (current) - 3 Nov 2016
Issue number21
Number of pages42
Early online date5/10/16
Original languageEnglish


We conjecture that (when the notion of Hadamard state is suitably adapted to spacetimes with timelike boundaries) 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 infra-red pathology of massless fields in 1+1 dimensions, 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 at the mirror. We argue that this result is significant because, as we point out, 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 from the spacetime). 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 nature of 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. Appendix B to the paper points out the existence of, and partially fills, a gap in the proofs of the theorems in B. S. Kay and R. M. Wald, ‘Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon’, Phys. Rep. 207, 49-136 (1991).

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