×

Critical Robertson-Walker universes. (English) Zbl 1423.53085

Summary: The integral of the mass-energy density \(\mathfrak{m}\) of a closed Robertson-Walker (RW) spacetime with source a perfect fluid and cosmological constant \({\Lambda}\) gives rise to an action functional on the space of scale functions of RW spacetime metrics. This paper studies closed RW spacetimes which are critical for this functional, subject to volume-preserving variations \((\mathfrak{m}\)-critical RW universes). A complete classification of \(\mathfrak{m}\)-critical RW universes is given and explicit solutions in terms of Weierstrass elliptic functions and their degenerate forms are computed. The standard energy conditions (weak, dominant, and strong) as well as the cyclic property of \(\mathfrak{m}\)-critical RW universes are discussed.

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
83C15 Exact solutions to problems in general relativity and gravitational theory
33E05 Elliptic functions and integrals
58E30 Variational principles in infinite-dimensional spaces
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alekseevsky, D., Lorentzian manifolds with transitive conformal group, Note Mat., 37, Suppl. 1, 35-47 (2017) · Zbl 1391.53062
[2] Angst, J., Asymptotic behavior of a relativistic diffusion in Robertson-Walker space-times, Ann. Inst. Henri Poincaré Probab. Stat., 52, 1, 376-411 (2016) · Zbl 1344.58017
[3] Banchi, L.; Caravelli, F., Geometric phases and cyclic isotropic cosmologies, Class. Quantum Gravity, 33, 10, Article 105003 pp. (2016) · Zbl 1338.83183
[4] Barbot, T.; Charette, V.; Drumm, T.; Goldman, W. M.; Melnick, K., A primer on the \((2 + 1)\)-Einstein universe, (Recent Developments in Pseudo-Riemannian Geometry. Recent Developments in Pseudo-Riemannian Geometry, ESI Lect. Math. Phys. (2008), Eur. Math. Soc.: Eur. Math. Soc. Zürich), 179-229 · Zbl 1154.53047
[5] Bochicchio, I.; Capozziello, S.; Laserra, E., The Weierstrass criterion and the Lemaître-Tolman-Bondi models with cosmological constant Λ, Int. J. Geom. Methods Mod. Phys., 8, 7, 1653-1666 (2011) · Zbl 1246.83242
[6] Bamba, K.; Yesmakhanova, K.; Yerzhanov, K.; Myrzakulov, R., Reconstruction of the equation of state for cyclic universes in homogeneous and isotropic cosmology, Cent. Eur. J. Phys., 11, 4, 397-411 (2013)
[7] Bamba, K.; Makarenko, A. N.; Myagky, A. N.; Odintsov, S. D., Bounce universe from string-inspired Gauss-Bonnet gravity, J. Cosmol. Astropart. Phys., 4, Article 001 pp. (2015)
[8] Badiale, M.; Serra, E., Semilinear Elliptic Equations for Beginners. Existence Results via the Variational Approach, Universitext (2011), Springer: Springer London · Zbl 1214.35025
[9] Cai, R.-G.; Kim, S. P., First law of thermodynamics and Friedmann equations of Friedmann-Robertson-Walker universe, J. High Energy Phys., 2, Article 050 pp. (2005)
[10] D’Ambroise, J., Applications of elliptic and theta functions to Friedmann-Robertson-Lemaître-Walker cosmology with cosmological constant, (Kirsten, K.; Williams, F. L., A Window into Zeta and Modular Physics. A Window into Zeta and Modular Physics, MSRI Publications, vol. 57 (2010), Cambridge University Press), 279-293 · Zbl 1216.83058
[11] D’Ambroise, J.; Williams, F. L., Parametric solutions of certain nonlinear differential equations in cosmology, J. Nonlinear Math. Phys., 18, 2, 269-278 (2011) · Zbl 1220.83013
[12] Dzhalilov, A.; Musso, E.; Nicolodi, L., Conformal geometry of timelike curves in the \((1 + 2)\)-Einstein universe, Nonlinear Anal., 143, 224-255 (2016) · Zbl 1342.53014
[13] Einstein, A., Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie, 142-152 (1917), Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften: Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin · JFM 46.1295.01
[14] Ferrández, A.; Giménez, A.; Lucas, P., Geometrical particle models on 3D null curves, Phys. Lett. B, 543, 3-4, 311-317 (2002) · Zbl 0997.83006
[15] Frances, C., Sur les variétés lorentziennes dont le group conforme est essentiel, Math. Ann., 332, 1, 103-119 (2005) · Zbl 1085.53061
[16] Friedmann, A., Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes, Z. Phys. A, 21, 1, 326-332 (1924) · JFM 50.0577.01
[17] Grant, J. D.; Musso, E., Coisotropic variational problems, J. Geom. Phys., 50, 303-338 (2004) · Zbl 1076.58011
[18] Gibbons, G. W.; Vyska, M., The application of Weierstrass elliptic functions to Schwarzschild null geodesics, Class. Quantum Gravity, 29, 6, Article 065016 pp. (2012) · Zbl 1470.83020
[19] Hawking, S. W.; Ellis, G. F.R., The Large Scale Structure of Space-Time (1973), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 0265.53054
[20] Jensen, G. R.; Musso, E.; Nicolodi, L., The geometric Cauchy problem for the membrane shape equation, J. Phys. A, 47, 49, Article 495201 pp. (2014) · Zbl 1314.53107
[21] Kuznetsov, Y. A.; Plyushchay, M. S., \((2 + 1)\)-dimensional models of relativistic particles with curvature and torsion, J. Math. Phys., 35, 6, 2772-2778 (1994) · Zbl 0808.70014
[22] Landau, L. D.; Lifshitz, E. M., Course of Theoretical Physics. Vol. 1. Mechanics (1976), Pergamon Press: Pergamon Press Oxford-New York-Toronto, Ont.
[23] Lawden, D. F., Elliptic Functions and Applications, Series in Applied Mathematical Science, vol. 80 (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0689.33001
[24] Lemaître, G., Expansion of the universe, A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulæ, Mon. Not. R. Astron. Soc., 91, 483-490 (1931) · JFM 57.1184.02
[25] Manzano, J. M.; Musso, E.; Nicolodi, L., Björling type problems for elastic surfaces, Rend. Semin. Mat. (Torino), 74, 1-2, 213-233 (2016) · Zbl 1440.53070
[26] Musso, E.; Nicolodi, L., On the Cauchy problem for the integrable system of Lie minimal surfaces, J. Math. Phys., 46, 11, 3509-3523 (2005) · Zbl 1111.58002
[27] Musso, E.; Nicolodi, L., Reduction for the projective arclength functional, Forum Math., 17, 569-590 (2005) · Zbl 1084.53012
[28] Musso, E.; Nicolodi, L., Closed trajectories of a particle model on null curves in anti-de Sitter 3-space, Class. Quantum Gravity, 24, 22, 5401-5411 (2007) · Zbl 1148.58008
[29] Musso, E.; Nicolodi, L., Reduction for constrained variational problems on 3-dimensional null curves, SIAM J. Control Optim., 47, 3, 1399-1414 (2008) · Zbl 1161.49041
[30] Musso, E.; Nicolodi, L., Hamiltonian flows on null curves, Nonlinearity, 23, 2117-2129 (2010) · Zbl 1203.37116
[31] Musso, E.; Nicolodi, L., Quantization of the conformal arclength functional on space curves, Commun. Anal. Geom., 25, 1, 209-242 (2017) · Zbl 1381.53030
[32] Nesterenko, V. V.; Feoli, A.; Scarpetta, G., Complete integrability for Lagrangians dependent on acceleration in a spacetime of constant curvature, Class. Quantum Gravity, 13, 1201-1211 (1996) · Zbl 0863.70019
[33] Nersessian, A.; Manvelyan, R.; Müller-Kirsten, H. J.W., Particle with torsion on 3d null-curves, Nucl. Phys. B, 88, 381-384 (2000) · Zbl 1273.83010
[34] Nersessian, A.; Ramos, E., Massive spinning particles and the geometry of null curves, Phys. Lett. B, 445, 123-128 (1998)
[35] O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Pure and Applied Mathematics, vol. 103 (1983), Academic Press, Inc.: Academic Press, Inc. New York · Zbl 0531.53051
[36] Penrose, R., Cycles of Time: An Extraordinary New View of the Universe (2010), The Bodley Head, Alfred A. Knopf, Inc.: The Bodley Head, Alfred A. Knopf, Inc. New York · Zbl 1222.00011
[37] Penrose, R., On the gravitization of quantum mechanics 2: conformal cyclic cosmology, Found. Phys., 44, 873-890 (2014) · Zbl 1302.83041
[38] Robertson, H. P., Kinematics and world-structure, Astrophys. J., 82, 284-301 (1935) · Zbl 0013.03905
[39] Rosu, H. C.; Ojeda-May, P., Supersymmetry of FRW barotropic cosmologies, Int. J. Theor. Phys., 45, 6, 873-890 (2006) · Zbl 1125.83317
[40] Ryan, M. P.; Shepley, L. C., Homogeneous Relativistic Cosmologies, Princeton Series in Physics (1975), Princeton University Press: Princeton University Press Princeton, NJ
[41] Tian, D. W.; Booth, I., Apparent horizon and gravitational thermodynamics of the Universe: solutions to the temperature and entropy confusion, and extension to modified gravity, Phys. Rev. D, 92, 2 (2015)
[42] Tod, P., Penrose’s Weyl curvature hypothesis and conformally-cyclic cosmology, J. Phys. Conf. Ser., 229, 1-5 (2010)
[43] P. Tod, Conformal methods in general relativity with application to conformal cyclic cosmology, in: A Minicourse at the IX International Meeting on Lorentzian Geometry, June 18th-22nd, 2018, Polish Acad. Sci. Inst. Math. (IMPAN), Warsaw.; P. Tod, Conformal methods in general relativity with application to conformal cyclic cosmology, in: A Minicourse at the IX International Meeting on Lorentzian Geometry, June 18th-22nd, 2018, Polish Acad. Sci. Inst. Math. (IMPAN), Warsaw.
[44] Walker, A. G., On Milne’s theory of world-structure, Proc. Lond. Math. Soc., 42, 90-127 (1937) · Zbl 0015.27904
[45] Wald, R. M., General Relativity (1984), University of Chicago Press: University of Chicago Press Chicago, IL · Zbl 0549.53001
[46] Weinberg, S., Gravitation and Cosmology (1972), John Wiley & Sons: John Wiley & Sons New York
[47] Zhu, T.; Ren, J.-R.; Li, M.-F., Corrected entropy of Friedmann-Robertson-Walker universe in tunneling method, J. Cosmol. Astropart. Phys. (2009)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.