On the initial conditions and solutions of the semiclassical Einstein equations in a cosmological scenario. (English) Zbl 1223.83056
Summary: In this paper we shall discuss the backreaction of a massive quantum scalar field on the curvature, the latter treated as a classical field. Furthermore, we shall deal with this problem in the realm of cosmological space-times by analyzing the Einstein equations in a semiclassical fashion. More precisely, we shall show that, at least on small intervals of time, solutions for this interacting system exist. This result will be achieved providing an iteration scheme and showing that the series, obtained starting from the massless solution, converges in the appropriate Banach space. The quantum states with good ultraviolet behavior (Hadamard property), used in order to obtain the backreaction, will be completely determined by their form on the initial surface if chosen to be light-like. Furthermore, on small intervals of time, they do not influence the behavior of the exact solution. On large intervals of time the situation is more complicated but, if the space-time is expanding, we shall show that the end-point of the evolution does not depend strongly on the quantum state, because, in this limit, the expectation values of the matter fields responsible for the backreaction do not depend on the particular homogeneous Hadamard state at all. Finally, we shall comment on the interpretation of the semiclassical Einstein equations for this kind of problems. Although the fluctuations of the expectation values of pointlike fields diverge, if the space-time and the quantum state have a large spatial symmetry and if we consider the smeared fields on regions of large spatial volume, they tend to vanish. Assuming this point of view the semiclassical Einstein equations become more reliable.
MSC:
| 83F05 | Relativistic cosmology |
| 83D05 | Relativistic gravitational theories other than Einstein’s, including asymmetric field theories |
| 83C45 | Quantization of the gravitational field |
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 35L15 | Initial value problems for second-order hyperbolic equations |
| 81V17 | Gravitational interaction in quantum theory |
| 83C15 | Exact solutions to problems in general relativity and gravitational theory |
| 83C25 | Approximation procedures, weak fields in general relativity and gravitational theory |
| 81T17 | Renormalization group methods applied to problems in quantum field theory |
References:
| [1] | Anderson P.R.: Effects Of Quantum Fields On Singularities And Particle Horizons In The Early Universe. 3. The Conformally Coupled Massive Scalar Field. Phys. Rev. D 32, 1302 (1985) · doi:10.1103/PhysRevD.32.1302 |
| [2] | Anderson P.R.: Effects Of Quantum Fields On Singularities And Particle Horizons In The Early Universe. 4. Initially Empty Universes. Phys. Rev. D 33, 1567 (1986) · doi:10.1103/PhysRevD.33.1567 |
| [3] | Anderson P.R., Eaker W.: Analytic approximation and an improved method for computing the stress-energy of quantized scalar fields in Robertson-Walker spacetimes. Phys. Rev. D 61, 024003 (2000) · doi:10.1103/PhysRevD.61.024003 |
| [4] | Bär, C., Ginoux, N., Pfäffle, F.: ”Wave equations on Lorentzian manifolds and quantization”. ESI Lectures in Mathematics and Physics, Zürich: European Math. Soc. Publishing House, 2007. · Zbl 1118.58016 |
| [5] | Brevik I., Odintsov S.D.: Quantum Annihilation of Anti-de Sitter Universe. Phys. Lett. B475, 247 (2000) · Zbl 0961.83014 |
| [6] | Brunetti R., Duetsch M., Fredenhagen K.: Perturbative Algebraic Quantum Field Theory and the Renormalization Groups. Adv. Theor. Math. Phys. 13, 1541–1599 (2009) · Zbl 1201.81090 · doi:10.4310/ATMP.2009.v13.n5.a7 |
| [7] | Brunetti R., Fredenhagen K.: Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds. Commun. Math. Phys. 208, 623 (2000) · Zbl 1040.81067 · doi:10.1007/s002200050004 |
| [8] | Brunetti, R., Fredenhagen, K.: ”Quantum Field Theory on Curved Backgrounds.” In: Lecture Notes in Physics 786, Bär, C., Fredenhagen, K., eds. Berlin-Heidelberg-New York: Springer, 2009, pp. 129–156 · Zbl 1184.81099 |
| [9] | Brunetti R., Fredenhagen K., Köhler M.: The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes. Commun. Math. Phys. 180, 633 (1996) · Zbl 0923.58052 · doi:10.1007/BF02099626 |
| [10] | Brunetti R., Fredenhagen K., Verch R.: The generally covariant locality principle: A new paradigm for local quantum physics. Commun. Math. Phys. 237, 31 (2003) · Zbl 1047.81052 |
| [11] | Bunch T.S., Davies P.C.W.: Quantum Fields theory in de Sitter space: renormalization by point-splitting. Proc. R. Soc. Lond. A 360, 117 (1978) · doi:10.1098/rspa.1978.0060 |
| [12] | Dappiaggi C., Fredenhagen K., Pinamonti N.: Stable cosmological models driven by a free quantum scalar field. Phys. Rev. D 77, 104015 (2008) · doi:10.1103/PhysRevD.77.104015 |
| [13] | Dappiaggi C., Moretti V., Pinamonti N.: Rigorous steps towards holography in asymptotically flat spacetimes. Rev. Math. Phys. 18, 349 (2006) · Zbl 1107.81040 · doi:10.1142/S0129055X0600270X |
| [14] | Dappiaggi C., Moretti V., Pinamonti N.: Cosmological horizons and reconstruction of quantum field theories. Commun. Math. Phys. 285, 1129–1163 (2009) · Zbl 1171.81019 · doi:10.1007/s00220-008-0653-8 |
| [15] | Dappiaggi C., Moretti V., Pinamonti N.: Distinguished quantum states in a class of cosmological spacetimes and their Hadamard property. J. Math. Phys. 50, 062304 (2009) · Zbl 1216.81112 · doi:10.1063/1.3122770 |
| [16] | Degner A., Verch R.: Cosmological particle creation in states of low energy. J. Math. Phys. 51, 022302 (2010) · Zbl 1309.83093 · doi:10.1063/1.3271106 |
| [17] | DeWitt B.S., Brehme R.W.: Radiation damping in a gravitational field. Ann. Phys. 9, 220 (1960) · Zbl 0092.45003 · doi:10.1016/0003-4916(60)90030-0 |
| [18] | Dimock J.: Algebras of local observables on a manifold. Commun. Math. Phys. 77, 219 (1980) · Zbl 0455.58030 · doi:10.1007/BF01269921 |
| [19] | Duistermaat J.J., Hörmander L.: Fourier integral operators II. Acta Math. 128, 183–269 (1972) · Zbl 0232.47055 · doi:10.1007/BF02392165 |
| [20] | Eltzner, B., Gottschalk, H.: ”Dynamical Backreaction in Robertson-Walker Spacetime.” http://arXiv.org/abs/1003.3630v2 [math-ph], 2010 · Zbl 1219.81203 |
| [21] | Fewster C.J.: A general worldline quantum inequality. Class. Quant. Grav. 17, 1897–1911 (2000) · Zbl 1079.81555 · doi:10.1088/0264-9381/17/9/302 |
| [22] | Flanagan E.E., Wald R.M.: Does backreaction enforce the averaged null energy condition in semiclassical gravity?. Phys. Rev. D 54, 6233 (1996) · doi:10.1103/PhysRevD.54.6233 |
| [23] | Fredenhagen K., Haag R.: On the derivation of Hawking radiation associated with the formation of a black hole. Commun. Math. Phys. 127, 273 (1990) · Zbl 0692.53040 · doi:10.1007/BF02096757 |
| [24] | Friedlander, F.G.: ”The wave equation on a curved space-time.” Cambridge: Cambridge Univeristy Press, 1975 · Zbl 0316.53021 |
| [25] | Gazzola G., Nemes M.C., Wreszinski W.F.: On the Casimir energy for a massive quantum scalar field and the cosmological constant. Ann. Phys. 324, 2095–2107 (2009) · Zbl 1176.83016 · doi:10.1016/j.aop.2009.07.001 |
| [26] | Gottlöber S., Müller V.: Vacuum polarization and scalar field effects in the early Universe. Astron. Nachr. 307, 285–287 (1986) · doi:10.1002/asna.2113070511 |
| [27] | Haag, R.: ”Local quantum physics: Fields, particles, algebras”. Second Revised and Enlarged Edition, Berlin-Heidelberg-New York: Springer, 1992 · Zbl 0777.46037 |
| [28] | Hamilton R.: The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc. 7, 65 (1982) · Zbl 0499.58003 · doi:10.1090/S0273-0979-1982-15004-2 |
| [29] | Hawking S.W.: Particle creation by black holes. Commun. Math. Phys. 43, 199 (1975) · Zbl 1378.83040 · doi:10.1007/BF02345020 |
| [30] | Hollands, S.: ”Aspects of Quantum Field Theory in Curved Spacetime”. Ph.D. Thesis, University of York, 2000, advisor B.S. Kay |
| [31] | Hollands S., Wald R.M.: Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223, 289 (2001) · Zbl 0989.81081 · doi:10.1007/s002200100540 |
| [32] | Hollands S., Wald R.M.: Existence of local covariant time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309 (2002) · Zbl 1015.81043 · doi:10.1007/s00220-002-0719-y |
| [33] | Hollands S., Wald R.M.: On the renormalization group in curved spacetime. Commun. Math. Phys. 237, 123 (2003) · Zbl 1059.81138 |
| [34] | Hollands S., Wald R.M.: Conservation of the stress tensor in interacting quantum field theory in curved spacetimes. Rev. Math. Phys. 17, 227 (2005) · Zbl 1078.81062 · doi:10.1142/S0129055X05002340 |
| [35] | Hörmander, L.: ”The Analysis of Linear Partial Differential Operators I”. Second edition, Berlin: Springer-Verlag, 1989 |
| [36] | Hu, B.L., Verdaguer, E.: ”Stochastic Gravity: Theory and Applications.” Living Rev. Rel. 11, 3 (2008); Living Rev. Rel. 7, 3 (2004) · Zbl 1070.83025 |
| [37] | Junker W., Schrohe E.: Adiabatic vacuum states on general spacetime manifolds: definition, construction, and physical properties. Ann. Henri Poincaré 3(6), 1113–1181 (2002) · Zbl 1038.81052 · doi:10.1007/s000230200001 |
| [38] | Kay B.S., Wald R.M.: Theorems On The Uniqueness And Thermal Properties Of Stationary, Nonsingular, Quasifree States On Space-Times With A Bifurcate Killing Horizon. Phys. Rept. 207, 49 (1991) · Zbl 0861.53074 · doi:10.1016/0370-1573(91)90015-E |
| [39] | Lüders C., Roberts J.E.: Local Quasiequivalence and Adiabatic Vacuum States. Commun. Math. Phys. 134, 29–63 (1990) · Zbl 0749.46045 · doi:10.1007/BF02102088 |
| [40] | Moretti V.: Comments on the stress-energy tensor operator in curved spacetime. Commun. Math. Phys. 232, 189 (2003) · Zbl 1015.81044 · doi:10.1007/s00220-002-0702-7 |
| [41] | Moretti V.: Uniqueness theorem for BMS-invariant states of scalar QFT on the null boundary of asymptotically flat spacetimes and bulk-boundary observable algebra correspondence. Commun. Math. Phys. 268, 727 (2006) · Zbl 1141.81028 · doi:10.1007/s00220-006-0107-0 |
| [42] | Moretti V.: Quantum Out-States Holographically Induced by Asymptotic Flatness: Invariance under Spacetime Symmetries, Energy Positivity and Hadamard Property. Commun. Math. Phys. 279, 3175 (2008) · Zbl 1145.83016 · doi:10.1007/s00220-008-0415-7 |
| [43] | Nojiri S., Odintsov S.D.: Effective Action for Conformal Scalars and Anti-Evaporation of Black Holes. Int. J. Mod. Phys. A14, 1293–1304 (1999) · Zbl 0940.83009 |
| [44] | Olbermann H.: States of low energy on Robertson-Walker spacetimes. Class. Quantum. Grav. 24, 5011–5030 (2007) · Zbl 1206.83171 · doi:10.1088/0264-9381/24/20/007 |
| [45] | Parker L.: Particle creation in expanding universes. Phys. Rev. Lett. 21, 562 (1968) · Zbl 0186.58603 · doi:10.1103/PhysRevLett.21.562 |
| [46] | Parker L.: Quantized Fields and Particle Creation in Expanding Universe. I. Phys. Rev. D183, 1057 (1969) · Zbl 0186.58603 · doi:10.1103/PhysRev.183.1057 |
| [47] | Parker L., Simon J.Z.: Einstein Equation with Quantum Corrections Reduced to Second Order. Phys. Rev. D 47, 1339 (1993) · doi:10.1103/PhysRevD.47.1339 |
| [48] | Parker, L., Raval, A.: Non-perturbative effects of vacuum energy on the recent expansion of the universe. Phys. Rev. D 60, 063512 (1999) [Erratum-ibid. D 67, 029901 (2003)] |
| [49] | Perez-Nadal G., Roura A., Verdaguer E.: Backreaction from non-conformal quantum fields in de Sitter spacetime. Class. Quant. Grav. 25, 154013 (2008) · Zbl 1147.83017 · doi:10.1088/0264-9381/25/15/154013 |
| [50] | Pinamonti N.: Conformal generally covariant quantum field theory: The scalar field and its Wick products. Commun. Math. Phys. 288, 1117 (2009) · Zbl 1171.81021 · doi:10.1007/s00220-009-0780-x |
| [51] | Radzikowski M.J.: Micro-Local Approach To The Hadamard Condition In Quantum Field Theory On Curved Space-Time. Commun. Math. Phys. 179, 529 (1996) · Zbl 0858.53055 · doi:10.1007/BF02100096 |
| [52] | Roura A., Verdaguer E.: Mode decomposition and renormalization in semiclassical gravity. Phys. Rev. D 60, 107503 (1999) · doi:10.1103/PhysRevD.60.107503 |
| [53] | Sahlmann H., Verch R.: Microlocal spectrum condition and Hadamard form for vector valued quantum fields in curved space-time. Rev. Math. Phys. 13, 1203 (2001) · Zbl 1029.81053 · doi:10.1142/S0129055X01001010 |
| [54] | Sanders K.: Equivalence of the (generalised) Hadamard and microlocal spectrum condition for (generalised) free fields in curved spacetime. Commun. Math. Phys. 295(2), 485–501 (2010) · Zbl 1192.53072 · doi:10.1007/s00220-009-0900-7 |
| [55] | Shapiro I.L.: Effective Action of Vacuum: Semiclassical Approach. Class. Quant. Grav. 25, 103001 (2008) · Zbl 1140.83301 · doi:10.1088/0264-9381/25/10/103001 |
| [56] | Shapiro I.L., Sola J.: Massive fields temper anomaly-induced inflation. Phys. Lett. B 530, 10 (2002) · Zbl 0992.83102 · doi:10.1016/S0370-2693(02)01355-2 |
| [57] | Starobinsky A.A.: A new type of isotropic cosmological models without singularity. Phys. Lett. B91, 99 (1980) · Zbl 1371.83222 |
| [58] | Strohmaier A., Verch R., Wollenberg M.: Microlocal analysis of quantum fields on curved space-times: analytic wavefront sets and Reeh-Schlieder theorems. J. Math. Phys. 43, 5514 (2002) · Zbl 1060.81050 · doi:10.1063/1.1506381 |
| [59] | Vilenkin A.: Classical And Quantum Cosmology Of The Starobinsky Inflationary Model. Phys. Rev. D32, 2511 (1985) |
| [60] | Wald R.M.: The Back Reaction Effect in Particle Creation in Curved Spacetime. Commun. Math. Phys. 54, 1–19 (1977) · Zbl 1173.81328 · doi:10.1007/BF01609833 |
| [61] | Wald R.M.: Axiomatic Renormalization Of Stress Tensor Of A Conformally Invariant Field In Conformally Flat Spacetimes. Ann. Phys. 110, 472 (1978) · Zbl 0366.53018 · doi:10.1016/0003-4916(78)90040-4 |
| [62] | Wald R.M.: Trace Anomaly Of A Conformally Invariant Quantum Field In Curved Space-Time. Phys. Rev. D 17, 1477 (1978) |
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