Instability of enclosed horizons

  • Bernard S. Kay
Research Article


We point out that there are solutions to the scalar wave equation on \(1+1\) dimensional Minkowski space with finite energy tails which, if they reflect off a uniformly accelerated mirror due to (say) Dirichlet boundary conditions on it, develop an infinite stress-energy tensor on the mirror’s Rindler horizon. We also show that, in the presence of an image mirror in the opposite Rindler wedge, suitable compactly supported arbitrarily small initial data on a suitable initial surface will develop an arbitrarily large stress-energy scalar near where the two horizons cross. Also, while there is a regular Hartle–Hawking–Israel-like state for the quantum theory between these two mirrors, there are coherent states built on it for which there are similar singularities in the expectation value of the renormalized stress-energy tensor. We conjecture that in other situations with analogous enclosed horizons such as a (maximally extended) Schwarzschild black hole in equilibrium in a (stationary spherical) box or the (maximally extended) Schwarzschild-AdS spacetime, there will be similar stress-energy singularities and almost-singularities—leading to instability of the horizons when gravity is switched on and matter and gravity perturbations are allowed for. All this suggests it is incorrect to picture a black hole in equilibrium in a box or a Schwarzschild-AdS black hole as extending beyond the past and future horizons of a single Schwarzschild (/Schwarzschild-AdS) wedge. It would thus provide new evidence for ’t Hooft’s brick wall model while seeming to invalidate the picture in Maldacena’s ‘Eternal black holes in AdS’. It would thereby also support the validity of the author’s matter-gravity entanglement hypothesis and of the paper ‘Brick walls and AdS/CFT’ by the author and Ortíz.


Accelerated mirrors \(1+1\) Dimensional wave equation  Quantum field theory in curved spacetime AdS/CFT Schwarzschild  Schwarzschild Anti-de Sitter Matter-gravity entanglement hypothesis 



I wish to thank an anonymous referee of the paper [41] for asking a question which stimulated some of the work reported here. I thank Eli Hawkins, Atsushi Higuchi, Hugo Ferreira and Jorma Louko for helpful remarks and criticisms of an earlier version of this paper. I thank Umberto Lupo for a critical reading of that earlier version and also for assistance with, and checks of, many of my calculations and also for assistance with Endnote \({9}\). I also wish to thank Chris Fewster for a valuable discussion and, in particular, for a specific suggestion (indicated in a parenthetical remark above) which helped me to make the present ‘silver-plated stress-energy almost-singularity result’ considerably stronger than a previous version. I also thank Borun Chowdhury for drawing to my attention the references [48, 49] and for an interesting discussion on the connection between the work in those references and the present work.


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of YorkYorkUK

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