Abstract
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.
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References
Maldacena, J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998). [also published as, Int. J. Theor. Phys. 38, 1113 (1999)]. arXiv:hep-th/9711200
Maldacena, J.M.: Eternal black holes in anti-de Sitter. JHEP 4, 021 (2003). arXiv:hep-th/0106112
’t Hooft, G.: On the quantum structure of a black hole. Nucl. Phys. B 256, 727 (1985)
Mukohyama, S., Israel, W.: Black holes, brick walls and the Boulware state. Phys. Rev. D 58, 104005 (1998). arXiv:gr-qc/9806012
Press, W., Teukolsky, S.: Floating orbits, superradiant scattering and the black-hole bomb. Nature 238, 211 (1972)
Eardley, D.M.: Death of white holes in the early universe. Phys. Rev. Lett. 33, 442 (1974)
Blau, S.K., Guth, A.H.: The stability of the white hole horizon. (1989) (manuscript submitted to the Gravity Research Foundation). http://gravityresearchfoundation.org/pdf/awarded/1989/blau_guth
Blau, S.K.: Dray ’t Hooft geometries and the death of white holes. Phys. Rev. D 39, 2901 (1989)
Lake, K.: Reissner-Nordstrm-de Sitter metric, the third law, and cosmic censorship. Phys. Rev. D 19, 421 (1979)
Wald, R.M., Ramaswamy, S.: Particle production by white holes. Phys. Rev. D 21, 2736 (1980)
Hawking, S.W.: Black holes and thermodynamics. Phys. Rev. D 13, 191 (1976)
Hawking, S.W., Page, D.: Thermodynamics of black holes in anti de-Sitter space. Commun. Math. Phys. 87, 577 (1973)
Davies, P.C.W.: Quantum vacuum friction. J. Opt. B Quantum Semiclass. Opt. 7, S40–S46 (2005)
Wang, Q., Unruh, W.G.: Motion of a mirror under infinitely fluctuating quantum vacuum stress. Phys. Rev. D 89, 085009 (2014). arXiv:1312.4591
Hartle, J.B., Hawking, S.W.: Path-integral derivation of black-hole radiance. Phys. Rev. D 13, 2188 (1976)
Israel, W.: Thermofield dynamics of black holes. Phys. Lett. A 57, 107 (1976)
Wightman, A.S.: Introduction to some aspects of the relativistic dynamics of quantum fields. In: Lévy, M. (ed.) 1964 Cargèse Lectures in theoretical physics: high energy electromagnetic interactions and field theory. Gordon and Breach, New York (1967)
Fulling, S.A., Ruijsenaars, S.N.M.: Temperature, periodicity and horizons. Phys. Rep. 152, 135 (1987)
Kay, B.S.: Application of linear hyperbolic PDE to linear quantum fields in curved spacetimes: especially black holes, time machines and a new semi-local vacuum concept. Journes Equations aux Drives Partielles IX-1 (2000). arXiv:gr-qc/0103056
Kay, B.S., Wald, R.M.: 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). Note that the uniqueness result in this paper was later strengthened in Kay B.S.: Sufficient conditions for quasifree states and an zimproved uniqueness theorem for quantum fields on spacetimes with horizons. J. Math. Phys. 34, 4519 (1993)
Kay, B.S.: Quantum Fields in Time-Dependent Backgrounds and in Curved Space-times. University of London PhD thesis (1977)
Jaffe, A., Ritter, G.: Reflection postivity and monotonicity. J. Math. Phys. 49, 052301 (2008). arXiv:0705.0712
Birrell, N.D., Davies, P.C.W.: Quantum Fields in Curved Space. Cambridge University Press, Cambridge (1982)
Kay, B.S.: The Casimir effect in quantum field theory. (Original title The Casimir effect without magic). Phys. Rev. D 20, 3052 (1979)
Rindler, W.: Kruskal space and the uniformly accelerated frame. Am. J. Phys. 34, 1174 (1966)
Simpson, M., Penrose, R.: Internal instability in a Reissner-Nordström black hole. Int. J. Theor. Phys. 7, 183 (1973)
Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973)
Chandrasekhar, S., Hartle, J.B.: On crossing the Cauchy horizon of a Reissner-Nordström black-hole. Proc. R. Soc. A 384, 301 (1982)
Hiscock, W.A.: Stress-energy tensor near a charged, rotating, evaporating black hole. Phys. Rev. D 15, 3054 (1977)
Dafermos, M.: Stability and instability of the Cauchy horizon for the spherically symmetric Einstein–Maxwell-scalar field equations. Ann. Math. 158, 875 (2003)
Dafermos, M.: Stability and Instability of the Reissner-Nordstrom Cauchy horizon and the problem of uniqueness in general relativity. Contemp. Math. 350, 99 (2004). arXiv:gr-qc/0209052
Kay, B.S., Lupo, U.: 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 (to appear)
Sewell, G.L.: Quantum fields on manifolds: PCT and gravitationally induced thermal states. Ann. Phys. 141, 201 (1982)
Hawking, S.W., Penrose, R.: The Nature of Space and Time. Princeton University Press, Princeton (1996, 2010)
Kay, B.S.: Entropy defined, entropy increase and decoherence understood, and some black-hole puzzles solved (1998). arXiv:hep-th/9802172
Kay, B.S.: Decoherence of macroscopic closed systems within Newtonian quantum gravity. Class. Quantum Gravit. 15, L89–L98 (1998). arXiv:hep-th/9810077
Kay, B.S., Abyaneh, V.: Expectation values, experimental predictions, events and entropy in quantum gravitationally decohered quantum mechanics (2007). arXiv:0710.0992
Kay, B.S. On the origin of thermality (2012). arXiv:1209.5125
Kay, B.S.: Modern foundations for thermodynamics and the stringy limit of black hole equilibria (2012). arXiv:1209.5085
Kay, B.S.: More about the stringy limit of black hole equilibria (2012). arXiv:1209.5110
Kay, B.S., Ortíz, L.: Brick walls and AdS/CFT. J. Gen. Relativ. Gravit. 46, 1727 (2014). arXiv:1111.6429
Arnsdorf, M., Smolin, L.: The Maldacena conjecture and Rehren duality (2001). arXiv:hep-th/0106073
Rehren, K.-H.: Algebraic holography. Ann. Henri Poin caré 1, 607 (2000). arXiv:hep-th/9905179
Czech, B., Karczmarek, J.L., Nogueira, F., Van Raamsdonk, M.: Rindler quantum gravity. Class. Quantum Gravit. 29, 235025 (2012). arXiv:1206.1323
Parikh, M., Samantray, P.: Rindler-AdS/CFT (2012). arXiv:1211.7370
Casini, H., Huerta, M., Myers, R.C.: Towards a derivation of holographic entanglement entropy. JHEP 05, 036 (2011). arXiv:1102.0440
de la Fuente A., Sundrum R.: Holography of the BTZ black hole, inside and out (2013). arXiv:1307.7738
Avery S.G., Chowdhury B.D.: No holography for eternal AdS black holes (2013). arXiv:1312.3346
Mathur S.: What is the dual of two entangled CFTs? (2014). arXiv:1402.6378
Chowdhury B.D.: Limitations of holography (2014). arXiv:1405.4292
Chowdhury B.D., Parikh M.K.: When UV and IR Collide: Inequivalent CFTs From Different Foliations Of AdS (2014). arXiv:1407.4467
Acknowledgments
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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Kay, B.S. Instability of enclosed horizons. Gen Relativ Gravit 47, 31 (2015). https://doi.org/10.1007/s10714-015-1858-8
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DOI: https://doi.org/10.1007/s10714-015-1858-8