Global existence of solutions of the semiclassical Einstein equation for cosmological spacetimes. (English) Zbl 1309.35170
Summary: We study the solutions of the semiclassical Einstein equation in flat cosmological spacetimes driven by a massive conformally coupled scalar field. In particular, we show that it is possible to give initial conditions at finite time to get a state for the quantum field which gives finite expectation values for the stress-energy tensor. Furthermore, it is possible to control this expectation value by means of a global estimate on regular cosmological spacetimes. The obtained estimates permit writing a theorem about the existence and uniqueness of the local solutions encompassing both the spacetime metric and the matter field simultaneously. Finally, we show that one can always extend local solutions up to a point where the scale factor \(a\) becomes singular or the Hubble function \(H\) reaches a critical value \(H_c=180\pi/\mathrm G\), both of which correspond to a divergence of the scalar curvature \(R\), namely a spacetime singularity.
MSC:
| 35Q76 | Einstein equations |
| 83F05 | Relativistic cosmology |
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
References:
| [1] | Anderson P.R.: Effects of quantum fields on singularities and particle horizons in the early universe. Phys. Rev. D28, 271-285 (1983) |
| [2] | Anderson P.R.: Effects of quantum fields on singularities and particle horizons in the early universe. II. Phys. Rev. D29, 615-627 (1984) |
| [3] | Anderson P.R.: Effects of quantum fields on singularities and particle horizons in the early universe. III. The conformally coupled massive scalar field. Phys. Rev. D32, 1302-1315 (1985) |
| [4] | Anderson P.R.: Effects of quantum fields on singularities and particle horizons in the early universe. IV. Initially empty universes. Phys. Rev. D33, 1567-1575 (1986) |
| [5] | Ashtekar, A., Pretorius, F., Ramazanoglu, F.M.: Surprises in the Evaporation of 2-Dimensional Black Holes. Physical Review Letters 106, 161303 (2011). arXiv:1011.6442 [gr-qc] |
| [6] | Bär, C., Ginoux, N., Pfäffle, F.: Wave equations on Lorentzian manifolds and quantization. In: ESI Lectures in Mathematical Physics. European Mathematical Society, Istanbul (2007). arXiv:0806.1036 [math.DG] · Zbl 1118.58016 |
| [7] | Brunetti, R., Fredenhagen, K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys. 208, 623-661 (2000). arXiv:math-ph/9903028 · Zbl 1040.81067 |
| [8] | Brunetti R., Fredenhagen K., Köhler M.: The microlocal spectrum condition and wick polynomials of free fields on curved spacetimes. Commun. Math. Phys. 180, 633-652 (1996) · Zbl 0923.58052 · doi:10.1007/BF02099626 |
| [9] | Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality principle—a new paradigm for local quantum field theory. Commun. Math. Phys. 237, 31-68 (2003). arXiv:math-ph/0112041 · Zbl 1047.81052 |
| [10] | Bunch T.S., Davies P.C.W.: Quantum field theory in De Sitter space: renormalization by Point-splitting. Proc. R. Soc. Lond. A. Math. Phys. Sci. 360, 117-134 (1978) · doi:10.1098/rspa.1978.0060 |
| [11] | Dappiaggi, C., Fredenhagen, K., Pinamonti, N.: Stable cosmological models driven by a free quantum scalar field. Phys. Rev. D77, 104015 (2008). arXiv:0801.2850 [gr-qc] |
| [12] | Dappiaggi, C., Moretti, V., Pinamonti, N.: Cosmological horizons and reconstruction of quantum field theories. Commun. Math. Phys. 285, 1129-1163 (2009). arXiv:0712.1770 [gr-qc] · Zbl 1171.81019 |
| [13] | 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). arXiv:0812.4033 [gr-qc] · Zbl 1216.81112 |
| [14] | Eltzner, B., Gottschalk, H.: Dynamical backreaction in Robertson-Walker spacetime. Rev. Math. Phys. 23, 531-551 (2011). arXiv:1003.3630 [math-ph] · Zbl 1219.81203 |
| [15] | Fewster, C.J., Verch, R.: Dynamical locality and covariance: what makes a physical theory the same in all spacetimes? Ann. Henri Poincaré 13, 1613-1674 (2012). arXiv:1106.4785 [math-ph] · Zbl 1280.81099 |
| [16] | Haag, R. : Local quantum physics: fields, particles, algebras, 2nd edn. Springer, Berlin (1996) · Zbl 0857.46057 · doi:10.1007/978-3-642-61458-3 |
| [17] | Hack, T.P.: On the backreaction of scalar and spinor quantum fields in curved spacetimes. Ph.D. thesis, Universität Hamburg (2010). arXiv:1008.1776 [gr-qc] · Zbl 0749.46045 |
| [18] | Hack, T.P.: The ΛCDM-model in quantum field theory on curved spacetime and dark radiation (2013). arXiv:1306.3074 [gr-qc] |
| [19] | Hamilton, R.S.: The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc. 7, 65-222 (1982) · Zbl 0499.58003 |
| [20] | Hollands, S., Wald, R.M.: Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223, 289-326 (2001). arXiv:gr-qc/0103074 · Zbl 0989.81081 |
| [21] | Junker, W., Schrohe, E.: Adiabatic vacuum states on general spacetime manifolds: definition, construction, and physical properties, construction, and physical properties. Ann. Henri Poincaré 3, 1113-1181 (2002). arXiv:math-ph/0109010 · Zbl 1038.81052 |
| [22] | Kofman L.A., Linde A.D., Starobinsky A.A.: Inflationary universe generated by the combined action of a scalar field and gravitational vacuum polarization. Phys. Lett. B157, 361-367 (1985) · doi:10.1016/0370-2693(85)90381-8 |
| [23] | Kusku, M.: A class of almost equilibrium states in Robertson-Walker spacetimes. Ph.D. thesis, Universität Hamburg (2008) |
| [24] | 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 |
| [25] | Moretti, V.: Comments on the stress-energy tensor operator in curved spacetime. Commun. Math. Phys. 232, 189-221 (2003). arXiv:gr-qc/0109048 · Zbl 1015.81044 |
| [26] | Olbermann, H.: States of low energy on Robertson-Walker spacetimes. Class. Quantum Grav. 24, 5011-5030 (2007). arXiv:0704.2986 [gr-qc] · Zbl 1206.83171 |
| [27] | Parker L.E.: Quantized fields and particle creation in expanding universes. I. Phys. Rev. 183, 1057-1068 (1969) · Zbl 0186.58603 · doi:10.1103/PhysRev.183.1057 |
| [28] | Pinamonti, N.: On the initial conditions and solutions of the semiclassical Einstein equations in a cosmological scenario. Commun. Math. Phys. 305, 563-604 (2011). arXiv:1001.0864 [gr-qc] · Zbl 1223.83056 |
| [29] | Radzikowski M.J.: Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179, 529-553 (1996) · Zbl 0858.53055 · doi:10.1007/BF02100096 |
| [30] | Sahlmann, H., Verch, R.: Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime. Rev. Math. Phys. 13, 1203-1246 (2001). arXiv:math-ph/0008029 · Zbl 1029.81053 |
| [31] | Sbierski, J.: On the existence of a maximal Cauchy development for the Einstein equations—a dezornification (2013). arXiv:1309.7591 [gr-qc] · Zbl 1335.83009 |
| [32] | Starobinsky A.A.: A new type of isotropic cosmological models without singularity. Phys. Lett. B91, 99-102 (1980) · Zbl 1371.83222 · doi:10.1016/0370-2693(80)90670-X |
| [33] | Wald, R.M.: The back reaction effect in particle creation in curved spacetime. Commun. Math. Phys. 54, 1-19 (1977) · Zbl 1173.81328 |
| [34] | Wald R.M.: Trace anomaly of a conformally invariant quantum field in curved spacetime. Phys. Rev. D17, 1477-1484 (1978) |
| [35] | Zschoche, J.: The Chaplygin gas equation of state for the quantized free scalar field on cosmological spacetimes. Ann. Henri Poincaré Online (2013). arXiv:1303.4992 [gr-qc] · Zbl 1295.83064 |
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