Abstract
We investigate the propagation of the scalar waves in the Witten space-time called “bubble of nothing” and in its remarkable sub-manifold, the Lorentzian Hawking wormhole. Due to the global hyperbolicity, the global Cauchy problem is well-posed in the functional framework associated with the energy. We perform a complete spectral analysis that allows us to get an explicit form of the solutions in terms of special functions. If the effective mass is non zero, the profile of the waves is asymptotically almost periodic in time. In contrast, the massless case is dispersive. We develop the scattering theory, classical as well as quantum. The quantized scattering operator leaves invariant the Fock vacuum: there is no creation of particles. The resonances can be defined in the massless case and they are purely imaginary.
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Aharony, O., Fabinger, M., Horowitz, G.T., Silverstein, E.: Clean time-dependent string backgrounds from bubble baths. J. High Energy Phys. 7(007), 35 (2002)
Allen B.: Vacuum states in de Sitter space. Phys. Rev. D. 32(12), 3136–3149 (1985)
Aslanbeigi S., Buck M.: A preferred ground state for the scalar field in de Sitter space. J. High Energy Phys. 8(039), 33 (2013)
Bachelot A.: Asymptotic completeness for the Klein-Gordon equation on the Schwarzschild metric. Ann. Inst. Henri Poincaré Physique théorique 61(4), 411–441 (1994)
Bailin D., Love A.: Kaluza-Klein theories. Rep. Prog. Phys. 50, 1087–1170 (1987)
Baricz, A., Mezö, I.: On the generalization of the Lambert W function with applications in theoretical physics (2014, preprint). arXiv:1408.3999
Baskin D., Vasy A., Wunsch J.: Asymptotics of radiation fields in asymptotically Minkowski space. Am. J. Math. 137(5), 1293–1364 (2015)
Bhawal B., Vishveshwara C.V.: Scalar waves in the Witten bubble spacetime. Phys. Rev. D. (3) 42(6), 1996–2003 (1990)
Birrel N.D., Davies P.C.W.: Quantum fields in curved space. Cambridge University Press, Cambridge (1982)
Blanco-Pillado J.J., Shlaer B.: Bubbles of nothing in flux compactifications. Phys. Rev. D. 82, 086015 (2010)
Blanco-Pillado, J.J., Ramadhan, H.S., Shlaer, B.: Bubbles from nothing. J. Cosmol. Astropart. Phys. 2012(1), 45 (2012)
Blanco-Pillado J.J., Shlaer B., Sousa K., Urrestilla J.: Bubbles of nothing and supersymmetric compactifications. J. Cosmol. Astropart. Phys. 10, 002 (2016)
Bousso R., Maloney A., Strominger A.: Conformal vacua and entropy in de Sitter space. Phys. Rev. D. 65, 104039 (2002)
Brill D.R., Horowitz G.T.: Negative energy in string theory. Phys. Lett. B262, 437–443 (1991)
Brill D.R., Matlin M.D.: Geodesic motion in a Kaluza-Klein bubble spacetime. Phys. Rev. D. (3) 39(10), 3151–3154 (1989)
Bros J., Moschella U.: Two-point functions and quantum fields in de Sitter universe. Rev. Math. Phys. 8, 327–392 (1996)
Brown A.R., Dahlen A.: On “Nothing”. Phys. Rev. D. 85, 104026 (2012)
Brown A.R.: The decay of hot KK space. Phys. Rev. D. 90, 104017 (2014)
Bunch T.S., Davies P.C.W.: Quantum field theory in de Sitter space: renormalization by point-splitting. Proc. R. Soc. Lond. A 360, 117–134 (1978)
Cagnac F., Choquet-Bruhat Y.: Solution globale d’une équation non linéaire sur une variété hyperbolique. J. Math. Pures Appl. (9) 63(4), 377–390 (1984)
Choquet-Bruhat Y., Cotsakis S.: Global hyperbolicity and completeness. J. Geom. Phys. (43) 4, 345–350 (2002)
Choquet-Bruhat Y.: General Relativity and the Einstein Equations. Oxford University Press, Oxford (2009)
Culetu H.: Light dragging phenomenon and expanding wormholes. J. Korean Phys. Soc. 57(3), 419–423 (2010)
Dereziński J., Gérard C.: Mathematics of Quantization and Quantum Fields. Cambridge University Press, Cambridge (2013)
Dereziński J., Wrochna M.: Exactly solvable Schrödinger operators. Ann. Henri Poincaré. 12(2), 397–418 (2011)
Dimock J.: Quantum Mechanics end Quantum Field Theory. Cambridge University Press, Cambridge (2011)
Dowker F., Gauntlett J.P., Gibbons G.W., Horowitz G.T.: The decay of magnetic fields in Kaluza-Klein theory. Phys. Rev. D. 52, 6929–6940 (1995)
Epstein H., Moschella U.: de Sitter tachyons and related topics. Commun. Math. Phys. 336(1), 381–430 (2015)
Everitt W.N., Kalf H.: The Bessel differential equation and the Hankel transform. J. Comput. Appl. Math. 208, 3–19 (2007)
Fulling S.A.: Aspects of Quantum Field Theory in Curved Spacetime. Cambridge University Press, Cambridge (1989)
Galstian A., Yagdjian K.: Fundamental solutions for the Klein-Gordon equation in de Sitter spacetime. Commun. Math. Phys. 285, 293–344 (2009)
Gibbons, G., Hartnoll, S.A.: Gravitational instability in higher dimensions. Phys. Rev. D. 66(6):064024, 17pp (2002)
Hawking S.W.: Quantum coherence down the wormhole. Phys. Lett. B. 195(3), 337–343 (1987)
Hawking S.W.: Wormholes in spacetime. Phys. Rev. D. 37(4), 904–910 (1988)
Hirosawa H., Wirth J.: Generalised energy conservation law for wave equations with variable propagation speed. J. Math. Anal. Appl. 358(1), 56–74 (2009)
Horowitz G.T., Maeda K.: Colliding Kaluza-Klein bubbles. Class. Quant. Grav. 19, 5543–5556 (2002)
Horowitz, G.T.: Tachyon condensation and black strings. JHEP 2005(08), 91 (2005)
Kay B.S.: A uniqueness result in the Segal-Weinless approach to linear Bose fields. J. Math. Phys. 20, 1712–1713 (1979)
Kay I., Moses H.E.: Reflectionless transmission through dielectrics and scattering potentials. J. Appl. Phys. 27, 1503–1508 (1956)
Kunz J., Nedkova P.G., Stelea C.: Charged black holes on Kaluza-Klein bubbles. Nucl. Phys. B. 874, 773–791 (2013)
Leray J.: Hyperbolic differential equations. Princeton University Press, Princeton (1953)
Lions, J-L., Magenes, E.: Problèmes aux limites non homogènes et applications I. Dunod, Paris (1968)
Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)
Pearson D.B.: Quantum Scattering and Spectral Theory. Academic Press, Boston (1988)
Reed M., Simon B.: Methods of modern mathematical physics II, Fourier Analysis, Self-Adjointness. Academic Press, Boston (1975)
Sánchez, M. Recent progress on the notion of global hyperbolicity. In: Advances in Lorentzian geometry, AMS/IP Stud. Adv. Math., vol. 49, pp. 105–124 (2011)
Stotyn S., Mann R.B.: Magnetic charge can locally stabilize Kaluza-Klein bubbles. Phys. Lett. B. 705(3), 269–272 (2011)
Tanabe, H.: Functional analytic methods for partial differential equations. In: Pure and Applied Mathematics, vol. 204. Marcel Dekker New-York (1997)
Vasy A.: The wave equation on asymptotically de Sitter-like spaces. Adv. Math. 223, 49–97 (2010)
Vasy A.: Resolvents, Poisson operators and scattering matrices on asymptotically hyperbolic and de Sitter spaces. J. Spectr. Theory. 4, 643–673 (2014)
Visser M.: Lorentzian Wormholes. Springer Verlag, Berlin (1995)
Weidmann J.: Linear Operators in Hilbert Spaces, Graduate Texts in Mathematics, vol. 68. Springer-Verlag, Berlin (1980)
Weinberg S.: The cosmological constant problem. Rev. Mod. Phys. 61, 1–23 (1989)
Witten E.: Instability of the Kaluza-Klein vacuum. Nucl Phys. B. 195, 481–492 (1982)
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Communicated by P. T. Chruściel
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Bachelot, A. Waves in the Witten Bubble of Nothing and the Hawking Wormhole. Commun. Math. Phys. 351, 599–651 (2017). https://doi.org/10.1007/s00220-016-2792-7
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DOI: https://doi.org/10.1007/s00220-016-2792-7