Fermi coordinates, simultaneity, and expanding space in Robertson-Walker cosmologies

Abstract

Explicit Fermi coordinates are given for geodesic observers comoving with the Hubble flow in expanding Robertson-Walker spacetimes, 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) comoving with the Hubble flow away from the observer. Least upper bounds of superluminal recessional Fermi velocities are given for spacetimes 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 spacetimes 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.

This revised version corrects minor typos

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• Pre-big bang geometric extensions of inflationary cosmologies
• Decelerating cosmologies are de-scramblers
• Kinematic relative velocity with respect to stationary observers in Schwarzschild spacetime
• Maximal Fermi charts and geometry of inflationary universes
• A note on the computation of geometrically defined relative velocities
• All order covariant tubular expansion
• Canonical Deformation of $N=2$ $Ad{S}_4$ SUGRA
• Electromagnetic fields and charges in expanding universes
• Multi-step Fermi normal coordinates
• Grand canonical ensembles in general relativity
• Velocity addition formulas in Robertson-Walker spacetimes
• An algorithm for computing geometric relative velocities through Fermi and observational coordinates