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Maximal Fermi charts and geometry of inflationary universes. (English) Zbl 1272.83100

Summary: A proof is given that the maximal Fermi coordinate chart for any comoving observer in a broad class of Robertson-Walker spacetimes consists of all events within the cosmological event horizon, if there is one, or is otherwise global. Exact formulas for the metric coefficients in Fermi coordinates are derived. Sharp universal upper bounds for the proper radii of leaves of the foliation by Fermi space slices are found, i.e., for the proper radii of the spatial universe at fixed times of the comoving observer. It is proved that the radius at proper time \(\tau \) diverges to infinity for non inflationary cosmologies as \(\tau \rightarrow \infty \), but not necessarily for cosmologies with periods of inflation. It is shown that any space like geodesic orthogonal to the worldline of a comoving observer has finite proper length and terminates within the cosmological event horizon (if there is one) at the big bang. Geometric properties of inflationary versus non inflationary cosmologies are compared, and opposite inequalities for the inflationary and non inflationary cases, analogous to Hubble’s law, are obtained for the Fermi relative velocities of comoving test particles. It is proved that the Fermi relative velocities of radially moving test particles are necessarily subluminal for inflationary cosmologies in contrast to non inflationary models, where superluminal relative Fermi velocities necessarily exist.

MSC:

83F05 Relativistic cosmology
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
53Z05 Applications of differential geometry to physics
83C57 Black holes
83C75 Space-time singularities, cosmic censorship, etc.
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[1] Walker A.G.: Note on relativistic mechanics. Proc. Edin. Math. Soc. 4, 170-174 (1935) · JFM 61.1493.04 · doi:10.1017/S0013091500008166
[2] Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W. H. Freeman, San Francisco, p. 329 (1973)
[3] Chicone, C., Mashhoon, B.: Explicit Fermi coordinates and tidal dynamics in de Sitter and Gödel spacetimes. Phys. Rev. D 74, 064019 (2006). (arXiv:gr-qc/0511129) · Zbl 1096.83040
[4] Chicone, C., Mashhoon, B.: Tidal acceleration of ultrarelativistic particles. Astron. Astrophys. 437, L39-L42 (2005). (arXiv:astro-ph/0406005) · Zbl 1091.83501
[5] Ishii, M., Shibata, M., Mino, Y.: Black hole tidal problem in the Fermi normal coordinates. Phys. Rev. D 71, 044017 (2005). (arXiv:gr-qc/0501084)
[6] Pound, A.: Nonlinear gravitational self-force: Field outside a small body. Phys. Rev. D 86, 084019 (2012). (arXiv:gr-qc/1206.6538) · Zbl 0463.53016
[7] Tino, G.M., Vetrano, F.: Is it possible to detect gravitational waves with atom interferometers? Class. Quant. Grav. 24, 2167-2178 (2007). (arXiv:gr-qc/0702118) · Zbl 1114.83306
[8] Klein, D., Collas, P.: Timelike Killing fields and relativistic statistical mechanics. Class. Quantum Grav. 26, 045018 (2009). (arXiv:gr-qc/0810.1776) · Zbl 1161.83315
[9] Klein, D., Yang, W.-S.: Grand canonical ensembles in general relativity. Math. Phys. Anal. Geom. 15, 61-83 (2012). (arXiv:math-ph/1009.3846) · Zbl 1243.82025
[10] 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) · Zbl 1137.81046 · doi:10.1103/PhysRevD.74.085011
[11] Parker L.: One-electron atom as a probe of spacetime curvature. Phys. Rev. D 22, 1922-1934 (1980) · doi:10.1103/PhysRevD.22.1922
[12] Parker L., Pimentel L.O.: Gravitational perturbation of the hydrogen spectrum. Phys. Rev. D 25, 3180-3190 (1982) · Zbl 1239.70003 · doi:10.1103/PhysRevD.25.3180
[13] Rinaldi, M.: Momentum-space representation of Greens functions with modified dispersion relations on general backgrounds. Phys. Rev. D 78, 024025 (2008). (arXiv:gr-qc/0803.3684) · Zbl 0012.18007
[14] Klein, D., Collas, P.: Recessional velocities and Hubble’s Law in Schwarzschild-de Sitter space. Phys. Rev. D15, 81, 063518 (2010). (arXiv:gr-qc/1001.1875)
[15] Klein, D., Randles, E.: Fermi coordinates, simultaneity, and expanding space in Robertson-Walker cosmologies. Ann. Henri Poincaré 12 303-328 (2011). (arXiv:math-ph/1010.0588) · Zbl 1215.83062
[16] Bolós, V.J., Klein, D.: Relative velocities for radial motion in expanding Robertson-Walker spacetimes. Gen. Relativ. Gravit. 44, 1361-1391 (2012). (arXiv:gr-qc/1106.3859) · Zbl 1243.83074
[17] Bolós, V.J., Havens, S., Klein, D.: Relative velocities, geometry, and expansion of space. Recent Advances in Cosmology. Nova Science Publishers, Inc. (to appear). (arXiv:gr-qc/1210.3161)
[18] Soffel, M., et al.: The IAU 2000 resolutions for astrometry, celestial mechanics and metrology in the relativistic framework: Explanatory supplement. Astron. J. 126, 2687-2706 (2003). (arXiv:astro-ph/0303376)
[19] Lindegren, L., Dravins, D.: The fundamental definition of ‘radial velocity’. Astron. Astrophys. 401, 1185-1202 (2003). (arXiv:astro-ph/0302522)
[20] Bolós, V.J., Liern, V., Olivert, J.: Relativistic simultaneity and causality. Internat. J. Theoret. Phys. 41, 1007-1018 (2002). (arXiv:gr-qc/0503034) · Zbl 1002.83007
[21] Bolós, V.J.: Lightlike simultaneity, comoving observers and distances in general relativity. J. Geom. Phys. 56, 813-829 (2006). (arXiv:gr-qc/0501085) · Zbl 1093.53027
[22] Bolós, V.J.: Intrinsic definitions of “relative velocity” in general relativity. Commun. Math. Phys. 273, 217-236 (2007). (arXiv:gr-qc/0506032) · Zbl 1127.53060
[23] Manasse F.K., Misner C.W.: Fermi normal coordinates and some basic concepts in differential geometry. J. Math. Phys. 4, 735-745 (1963) · Zbl 0118.22903 · doi:10.1063/1.1724316
[24] Klein, D., Collas, P.: General Transformation Formulas for Fermi-Walker Coordinates. Class. Quant. Grav. 25, 145019 (2008). (arXiv:gr-qc/0712.3838) · Zbl 1145.83009
[25] Rindler W.: Visual Horizons in World-models. Mon. Not. R. Astr. Soc. 116, 662-677 (1956) · Zbl 0077.42108
[26] Rindler W.: Visual Horizons in World-models. Gen. Rel. Grav. 34, 133-153 (2002) · Zbl 0999.83078 · doi:10.1023/A:1015347106729
[27] Penrose, R.: Conformal treatment of infinity. In: DeWitt, C., DeWitt, B. (eds.) Relativity, Groups, and Topology. Les Houches, Gordon and Breach, 563-584 (1963)
[28] Griffiths J., Podolsky J.: Exact Space-Times in Einstein’s General Relativity. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2009) · Zbl 1250.83002 · doi:10.1017/CBO9780511635397
[29] Page, D.N.: How big is the universe today? Gen. Rel. Grav. 15, 181-185 (1983) · Zbl 0512.53030
[30] Rindler W.: Public and private space curvature in Robertson-Walker universes. Gen. Rel. Grav. 13, 457-461 (1981) · Zbl 0463.53016 · doi:10.1007/BF00756593
[31] Klein, D., Collas, P.: Exact Fermi coordinates for a class of spacetimes. J. Math. Phys. 51 022501 (2010). (arXiv:math-ph/0912.2779) · Zbl 1309.83015
[32] Weinberg, S.: Cosmology. Oxford University Press, New York, p. 48 (2008) · Zbl 1147.83002
[33] Zhu, Z-H., Hu, M., Alcaniz, J.S., Liu, Y.-X.: Testing power-law cosmology with galaxy clusters. Astron. Astophys. 483, 15-18 (2008). (arXiv:astro-ph/0712.3602)
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