Annales Henri Poincaré

, Volume 12, Issue 2, pp 303–328 | Cite as

Fermi Coordinates, Simultaneity, and Expanding Space in Robertson–Walker Cosmologies



Explicit Fermi coordinates are given for geodesic observers comoving with the Hubble flow in expanding Robertson–Walker space–times, along with exact expressions for the metric tensors in Fermi coordinates. For the case of non-inflationary cosmologies, it is shown that Fermi coordinate charts are global, and space–time is foliated by space slices of constant Fermi (proper) time that have finite extent. A universal upper bound for the proper radius of any leaf of the foliation, i.e., for the proper radius of the spatial universe at any fixed time of the geodesic observer, is given. A general expression is derived for the geometrically defined Fermi relative velocity of a test particle (e.g., a galaxy cluster) comoving with the Hubble flow away from the observer. Least upper bounds of superluminal recessional Fermi velocities are given for space–times whose scale factors follow power laws, including matter-dominated and radiation-dominated cosmologies. Exact expressions for the proper radius of any leaf of the foliation for this same class of space–times are given. It is shown that the radii increase linearly with proper time of the observer moving with the Hubble flow. These results are applied to particular cosmological models.


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  1. 1.
    Grøn Ø., Elgarøy Ø.: Is space expanding in the Friedmann universe models?. Am. J. Phys. 75, 151–157 (2006)ADSCrossRefGoogle Scholar
  2. 2.
    Carrera M., Giulini D.: Influence of global cosmological expansion on local dynamics and kinematics. Rev. Mod. Phys. 82, 169–208 (2010)ADSCrossRefGoogle Scholar
  3. 3.
    Cook Richard, Burns M.: Interpretation of the cosmological metric. Am. J. Phys. 77, 59–66 (2009)ADSCrossRefGoogle Scholar
  4. 4.
    Davis T., Lineweaver C.: Expanding confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe. Publ. Astron. Soc. Aust. 21, 97–109 (2004)ADSCrossRefGoogle Scholar
  5. 5.
    Walker A.G.: Note on relativistic mechanics. Proc. Edin. Math. Soc. 4, 170–174 (1935)MATHCrossRefGoogle Scholar
  6. 6.
    Rindler W.: Public and private space curvature in Robertson–Walker universes. Gen. Rel. Grav. 13, 457–461 (1981)MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    Page D.N.: How big is the universe today?. Gen. Rel. Grav. 15, 181–185 (1983)MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    Ellis G.F.R., Matravers D.R.: Spatial Homogeneity and the size of the universe. In: Dadhich, N., Rao, J.K., Narlikar, J.V., Vishveshswara, C.V. (eds) A Random Walk in Relativity and Cosmology, pp. 92–108. Wiley Eastern, Delhi (1985)Google Scholar
  9. 9.
    Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W.H. Freeman, San Francisco, p. 329 (1973)Google Scholar
  10. 10.
    Manasse F.K., Misner C.W.: Fermi normal coordinates and some basic concepts in differential geometry. J. Math. Phys. 4, 735–745 (1963)MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    Li W.Q., Ni W.T.: Expansions of the affinity, metric and geodesic equations in Fermi normal coordinates about a geodesic. J. Math. Phys. 20, 1925–1929 (1979)ADSMATHCrossRefGoogle Scholar
  12. 12.
    O’Neill B.: Semi-Riemannian geometry with applications to relativity. Academic Press, New York (1983)MATHGoogle Scholar
  13. 13.
    Klein, D., Collas, P.: General transformation formulas for Fermi-Walker coordinates. Class. Quant. Grav. 25, (17pp) 145019 (2008), doi:10.1088/0264-9381/25/14/145019, [gr-qc]
  14. 14.
    Chicone C., Mashhoon B.: Explicit Fermi coordinates and tidal dynamics in de Sitter and Gödel spacetimes. Phys. Rev. D 74, 064019 (2006)ADSCrossRefGoogle Scholar
  15. 15.
    Klein, D., Collas, P.: Exact Fermi coordinates for a class of spacetimes. J. Math. Phys. 51, (10pp) 022501(2010), doi:10.1063/1.3298684, arXiv:0912.2779v1 [math-ph]
  16. 16.
    Ishii M., Shibata M., Mino Y.: Black hole tidal problem in the Fermi normal coordinates. Phys. Rev. D 71, 044017 (2005)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Marzlin K.-P.: Fermi coordinates for weak gravitational fields. Phys. Rev. D 50, 888–891 (1994)ADSCrossRefGoogle Scholar
  18. 18.
    Fortini P.L., Gualdi C.: Fermi normal co-ordinate system and electromagnetic detectors of gravitational waves. I - Calculation of the metric. Nuovo Cimento B 71, 37–54 (1982)ADSCrossRefGoogle Scholar
  19. 19.
    Klein, D., Collas, P.: Timelike Killing fields and relativistic statistical mechanics. Class. Quantum Grav. 26, (16 pp) 045018 (2009), arXiv:0810.1776v2 [gr-qc]Google Scholar
  20. 20.
    Klein, D., Yang, W-S.: Grand canonical ensembles in general relativity. arXiv:1009.3846v1 [math-ph] (2010)Google Scholar
  21. 21.
    Bimonte G., Calloni E., Esposito G., Rosa L.: Energy-momentum tensor for a Casimir apparatus in a weak gravitational field. Phys. Rev. D 74, 085011 (2006)ADSCrossRefGoogle Scholar
  22. 22.
    Parker L.: One-electron atom as a probe of spacetime curvature. Phys. Rev. D 22, 1922–1934 (1980)ADSCrossRefGoogle Scholar
  23. 23.
    Parker L., Pimentel L.O.: Gravitational perturbation of the hydrogen spectrum. Phys. Rev. D 25, 3180–3190 (1982)ADSCrossRefGoogle Scholar
  24. 24.
    Griffiths J., Podolsky J.: Exact Space–Times in Einstein’s General Relativity, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, UK (2009)CrossRefGoogle Scholar
  25. 25.
    Bolós V.: Lightlike simultaneity, comoving observers and distances in general relativity. J. Geom. Phys. 56, 813–829 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  26. 26.
    Bolós V.: Intrinsic definitions of “relative velocity” in general relativity. Comm. Math. Phys. 273, 217–236 (2007)MathSciNetADSMATHCrossRefGoogle Scholar
  27. 27.
    Klein D., Collas P.: Recessional velocities and Hubble’s Law in Schwarzschild-de Sitter space. Phy. Rev. D15 81, 063518 (2010)CrossRefGoogle Scholar
  28. 28.
    Gibbons G.W., Hawking S.W.: Cosmological event horizons, thermodynamics, and particle creation. Phys. Rev. D 15, 2738–2751 (1977)MathSciNetADSCrossRefGoogle Scholar

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© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Interdisciplinary, Research Institute for the SciencesCalifornia State University NorthridgeNorthridgeUSA
  2. 2.Department of MathematicsCalifornia State University NorthridgeNorthridgeUSA

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