Advertisement

An extension of cosmological dynamics with York time

  • Philipp Roser
Research Article
  • 73 Downloads

Abstract

It has been suggested that the York parameter T (effectively the scalar extrinsic curvature of a spatial hypersurface) may play the role of a fundamental time parameter. In a flat, forever expanding cosmology the York parameter remains always negative, taking values \(T=-\infty \) at the big bang and approaching some finite non-positive value as \(t\rightarrow \infty \), t being the usual cosmological time coordinate. Based on previous results concerning a simple, spatially flat cosmological model with a scalar field, we provide a temporal extension of this model to include ‘times’ \(T>0\), an epoch not covered by the cosmological time coordinate t, and discuss the dynamics of this ‘other side’ and its significance. We argue that the extension is necessary if a consistent quantisation scheme is to exist. Furthermore, we investigate which types of potentials lead to smooth transitions, paying particular attention to currently favoured inflaton potentials.

Keywords

York time CMC slicing Cosmological extension  Scalar-field cosmology 

Notes

Acknowledgments

The author would like to thank Antony Valentini for helpful discussions and comments. This work was in part supported by the Foundational Questions Institute (fqxi.org).

References

  1. 1.
    Arnowitt, R., Deser, S., Misner, C.W.: The dynamics of general relativity. In: Witten, L. (ed.) Gravitation: An Introduction to Current Research. Wiley, Hoboken (1962)Google Scholar
  2. 2.
    Barbour, J., Koslowski, T., Mercati, F.: The Solution to the Problem of Time in Shape Dynamics (2013). arXiv:1302.6264 [Gr-qc]
  3. 3.
    Brill, D.R., Cavallo, J.M., Isenberg, J.A.: K-surfaces in the Schwarzschild spacetime and the construction of lattice cosmologies. J. Math. Phys. 21, 2789 (1980)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Choquet-Bruhat, Y., York, J.: The cauchy problem. In: Held, A. (ed.) General Relativity and Gravitation I. Plenum, New York (1980)Google Scholar
  5. 5.
    Eddington, A.S.: A comparison of Whitehead’s and Einstein’s formulae [Letters to the Editor]. Nature 113, 192 (1924)ADSCrossRefGoogle Scholar
  6. 6.
    Ferreira, P., Joyce, M.: Cosmology with a primordial scaling field. Phys. Rev. D 58, 023,503 (1998)CrossRefGoogle Scholar
  7. 7.
    Giambò, R., Miritzis, J., Tzanni, K.: Negative potentials and collapsing universes (2014). arXiv:1411.0218 [Gr-qc]
  8. 8.
    Gomes, H.: A Birkhoff theorem for shape dynamics. Class. Quantum Gravity 31, 085,008 (2014). arXiv:1305.0310 [Gr-qc]MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gomes, H., Gryb, S., Koslowski, T.: Einstein gravity as a 3d conformally invariant theory. Class. Quantum Gravity 28, 045,004 (2011)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gomes, H., Herczeg, G.: A rotating black hole solution for shape dynamics. Class. Quantum Gravity 31, 175,014 (2014). arXiv:1310.6095v4 [Gr-qc]MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Halliwell, J.: Scalar fields in cosmology with an exponential potential. Phys. Lett. B 185, 341 (1987)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Lemaître, G.: L’univers en expansion. Ann. Soc. Sci. Brux. A 53, 51–85 (1933)MATHGoogle Scholar
  13. 13.
    Marsden, J., Tipler, F.: Maximal hypersurfaces and foliations of constant mean curvature in general relativity. Phys. Rep. 66, 109–139 (1980)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Martin, J., Ringeval, C., Trotta, R., Vennin, V.: Best inflationary models after Planck. J. Cosmol. Astropart. Phys. 1403, 039 (2014)ADSMathSciNetGoogle Scholar
  15. 15.
    Martin, J., Ringeval, C., Vennin, V.: Encyclopaedia inflationaris. Phys. Dark Universe 5–6, 75–235 (2014). arXiv:1303.3787 [Astro-ph]ADSCrossRefGoogle Scholar
  16. 16.
    Mercati, F.: A shape dynamics tutorial (2014). arXiv:1409.0105v1 [Gr-qc]
  17. 17.
    Misner, C.: Absolute zero of time. Phys. Rev. 186, 1328–1333 (1969)ADSCrossRefMATHGoogle Scholar
  18. 18.
    Misner, C., Thorne, K., Wheeler, J.: Gravitation. W.H. Freeman, San Francisco (1973)Google Scholar
  19. 19.
    Planck Collaboration: Planck 2015 results. I. Overview of products and scientific results (2015). arXiv:1502.01582v1 [Astro-ph]
  20. 20.
    Qadir, A., Wheeler, J.: From \(SU(3)\) to Gravity. Cambridge University Press, Cambridge (1985)Google Scholar
  21. 21.
    Roser, P., Valentini, A.: Classical and quantum cosmology with York time. Class. Quantum Gravity 31, 245,001 (2014). arXiv:1406.2036 [Gr-qc]MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Spergel, D.N., et al.: Three-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: implications for cosmology. Astrophys. J. Suppl. Ser. 107, 377 (2007). arXiv:astro-ph/0603449 ADSCrossRefGoogle Scholar
  23. 23.
    Synge, J.L.: The gravitational field of a particle. Proc. R. Ir. Acad. A 53, 83–114 (1950)MathSciNetMATHGoogle Scholar
  24. 24.
    Valentini, A.: Pilot-wave theory of fields, gravitation and cosmology. In: Cushing, J.T., Fine, A., Goldstein, S. (eds.) Bohmian Mechanics and Quantum Theory: An Appraisal. Kluwer, Dordrecht (1996)Google Scholar
  25. 25.
    Valentini, A.: On Galilean and Lorentz invariance in pilot-wave dynamics. Phys. Lett. A 228, 215–222 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Valentini, A.: Hidden variables and the large-scale structure of spacetime. In: Craig, W.L., Smith, Q. (eds.) Einstein, relativity and absolute simultaneity. Routledge, London (2008)Google Scholar
  27. 27.
    York, J.: Role of conformal three-geometry in the dynamics of gravitation. Phys. Rev. Lett. 28, 1082–1085 (1972)ADSCrossRefGoogle Scholar
  28. 28.
    York, J.: Conformally invariant orthogonal decomposition of symmetric tensors on riemannian manifolds and the initial-value problem of general relativity. J. Math. Phys. 14, 456–464 (1973)ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Kinard Laboratory, Department of Physics and AstronomyClemson UniversityClemsonUSA

Personalised recommendations