Time’s Arrow in a Finite Universe

ABSTRACT

In this paper, a simple model for a closed multiverse as a finite probability space is analyzed. For each moment of time on a discrete time-scale, only a finite number of states are possible and hence each possible universe can be viewed as a path in a huge but finite graph. By considering very general statistical assumptions, essentially originating from Boltzmann, we make the set of all such paths (the multiverse) into a probability space, and argue that under certain assumptions, the probability for a monotonic behavior of the entropy is enormously much larger then for a behavior with low entropy at both ends. The methods used are just very simple combinatorial ones, but the conclusion suggests that we may live in a multiverse which from a global point of view is completely time-symmetric in the sense that universes with Time’s Arrow directed forwards and backwards are equally probable. However, for an observer confined to just one universe, time will still be asymmetric.

In this paper, a simple model for a closed multiverse as a finite probability space is analyzed. For each moment of time on a discrete time-scale, only a finite number of states are possible and hence each possible universe can be viewed as a path in a huge but finite graph. By considering very general statistical assumptions, essentially originating from Boltzmann, we make the set of all such paths (the multiverse) into a probability space, and argue that under certain assumptions, the probability for a monotonic behavior of the entropy is enormously much larger then for a behavior with low entropy at both ends. The methods used are just very simple combinatorial ones, but the conclusion suggests that we may live in a multiverse which from a global point of view is completely time-symmetric in the sense that universes with Time’s Arrow directed forwards and backwards are equally probable. However, for an observer confined to just one universe, time will still be asymmetric.

Cite this paper

Tamm, M. (2015) Time’s Arrow in a Finite Universe.*International Journal of Astronomy and Astrophysics*, **5**, 70-78. doi: 10.4236/ijaa.2015.52010.

Tamm, M. (2015) Time’s Arrow in a Finite Universe.

References

[1] Boltzmann, L. (1974) Theoretical Physics and Philosophical Problems. Edited by Brian McGuinness. Trans. Paul Foulkes, Reidel Publishing Co., Dordrecht.

[2] Price, H. (1996) Time’s Arrow and Archimedes’ Point. Oxford University Press, Oxford.

[3] Gold, T. (1962) The Arrow of Time. American Journal of Physics, 30, 403.

http://dx.doi.org/10.1119/1.1942052

[4] Hawking, S.W. (1985) Arrow of Time in Cosmology. Physical Review D, 32, 2489.

http://dx.doi.org/10.1103/PhysRevD.32.2489

[5] Page, D. (1985) Will the Entropy Decrease If the Universe Recollapse? Physical Review D, 32, 2496.

http://dx.doi.org/10.1103/PhysRevD.32.2496

[6] Tamm, M. (2013) Time’s Arrow from the Multiverse Point of View. Physics Essays, 26, 2.

[7] Penrose, R. (1979) Singularities and Time-Asymmetry. General Relativity: An Einstein Centenary. Cambridge University Press, Cambridge.

[8] Sakharov, A.D. (1967) ZhETF Pis’ma, 5, 32 (Sov. Phys. JEPT Lett., 6, 24).

[9] Martin, R. (2004) The St. Petersburg Paradox. In: Zalta, E.N., Ed., The Stanford Encyclopedia of Philosophy, Summer 2014 Edition.

http://plato.stanford.edu/archives/sum2014/entries/paradox-stpetersburg/

[10] DeWitt, B.S. (1967) Quantum Theory of Gravity. I. The Canonical Theory. Physical Review, 160, 1113-1148.

[11] Zeh, H.D. (2001) The Physical Basis of the Direction of Time. 4th Edition, Springer-Verlag, Berlin.

[12] Barbour, J. (1999) The End of Time. Oxford University Press, Oxford.

[13] Everett, H. (1957) “Relative State” Formulation of Quantum Mechanics. Reviews of Modern Physics, 29, 454.

http://dx.doi.org/10.1103/RevModPhys.29.454

[14] Friedman, A. (1922) über die Krümmung des Raumes. Zeitschrift für Physik, 10, 377-386.

[15] Misner, C.M., Thorne, K.S. and Wheeler, J.A. (1973) Gravitation. W. H. Freeman and Company, San Francisco.

[16] Riess, A.G., Filippenko, A.V., Challis, P., Clocchiatti, A., Diercks, A., Garnavich, P.M., et al., Supernova Search Team (1998) Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. Astronomical Journal, 116, 1009.

http://dx.doi.org/10.1086/300499

[17] Perlmutter, S., Aldering, G., Goldhaber, G., Knop, R.A., Nugent, P., Castro, P.G., et al., the Supernova Cosmology Project (1999) Measurements of Omega and Lambda from 42 High-Redshift Supernovae. Astrophysical Journal, 517, 565.

http://dx.doi.org/10.1086/307221

[18] Tamm, M. (2015) Accelerating Expansion in a Closed Universe. Journal of Modern Physics, Special Issue on “Gravitation, Astrophysics and Cosmology”, 6, 239-251.

[19] Adams, F. and Laughlin, G. (1997) A Dying Universe: The Long-Term Fate and Evolution of Astrophysical Objects. Reviews of Modern Physics, 69, 337.

http://dx.doi.org/10.1103/RevModPhys.69.337

[20] Egan, C. and Lineweaver, C. (2010) A Larger Estimate of the Entropy of the Universe. The Astrophysical Journal, 710, 1825.

http://dx.doi.org/10.1088/0004-637X/710/2/1825

[21] Barbour, J., Koslowski, T. and Mercati, F. (2014) Identification of a Gravitational Arrow of Time. Physical Review Letters, 113, Article ID: 181101.

[1] Boltzmann, L. (1974) Theoretical Physics and Philosophical Problems. Edited by Brian McGuinness. Trans. Paul Foulkes, Reidel Publishing Co., Dordrecht.

[2] Price, H. (1996) Time’s Arrow and Archimedes’ Point. Oxford University Press, Oxford.

[3] Gold, T. (1962) The Arrow of Time. American Journal of Physics, 30, 403.

http://dx.doi.org/10.1119/1.1942052

[4] Hawking, S.W. (1985) Arrow of Time in Cosmology. Physical Review D, 32, 2489.

http://dx.doi.org/10.1103/PhysRevD.32.2489

[5] Page, D. (1985) Will the Entropy Decrease If the Universe Recollapse? Physical Review D, 32, 2496.

http://dx.doi.org/10.1103/PhysRevD.32.2496

[6] Tamm, M. (2013) Time’s Arrow from the Multiverse Point of View. Physics Essays, 26, 2.

[7] Penrose, R. (1979) Singularities and Time-Asymmetry. General Relativity: An Einstein Centenary. Cambridge University Press, Cambridge.

[8] Sakharov, A.D. (1967) ZhETF Pis’ma, 5, 32 (Sov. Phys. JEPT Lett., 6, 24).

[9] Martin, R. (2004) The St. Petersburg Paradox. In: Zalta, E.N., Ed., The Stanford Encyclopedia of Philosophy, Summer 2014 Edition.

http://plato.stanford.edu/archives/sum2014/entries/paradox-stpetersburg/

[10] DeWitt, B.S. (1967) Quantum Theory of Gravity. I. The Canonical Theory. Physical Review, 160, 1113-1148.

[11] Zeh, H.D. (2001) The Physical Basis of the Direction of Time. 4th Edition, Springer-Verlag, Berlin.

[12] Barbour, J. (1999) The End of Time. Oxford University Press, Oxford.

[13] Everett, H. (1957) “Relative State” Formulation of Quantum Mechanics. Reviews of Modern Physics, 29, 454.

http://dx.doi.org/10.1103/RevModPhys.29.454

[14] Friedman, A. (1922) über die Krümmung des Raumes. Zeitschrift für Physik, 10, 377-386.

[15] Misner, C.M., Thorne, K.S. and Wheeler, J.A. (1973) Gravitation. W. H. Freeman and Company, San Francisco.

[16] Riess, A.G., Filippenko, A.V., Challis, P., Clocchiatti, A., Diercks, A., Garnavich, P.M., et al., Supernova Search Team (1998) Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. Astronomical Journal, 116, 1009.

http://dx.doi.org/10.1086/300499

[17] Perlmutter, S., Aldering, G., Goldhaber, G., Knop, R.A., Nugent, P., Castro, P.G., et al., the Supernova Cosmology Project (1999) Measurements of Omega and Lambda from 42 High-Redshift Supernovae. Astrophysical Journal, 517, 565.

http://dx.doi.org/10.1086/307221

[18] Tamm, M. (2015) Accelerating Expansion in a Closed Universe. Journal of Modern Physics, Special Issue on “Gravitation, Astrophysics and Cosmology”, 6, 239-251.

[19] Adams, F. and Laughlin, G. (1997) A Dying Universe: The Long-Term Fate and Evolution of Astrophysical Objects. Reviews of Modern Physics, 69, 337.

http://dx.doi.org/10.1103/RevModPhys.69.337

[20] Egan, C. and Lineweaver, C. (2010) A Larger Estimate of the Entropy of the Universe. The Astrophysical Journal, 710, 1825.

http://dx.doi.org/10.1088/0004-637X/710/2/1825

[21] Barbour, J., Koslowski, T. and Mercati, F. (2014) Identification of a Gravitational Arrow of Time. Physical Review Letters, 113, Article ID: 181101.