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 cluster) 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.

Pre-big Bang Geometric Extensions of Inflationary Cosmologies

“…This section summarizes results from [14,15] needed in the sequel. The Robertson-Walker metric on space-time M = M k is given by the line element,…”

“…Our approach begins with the observation that cosmological time along spacelike geodesics orthogonal to the worldline of a comoving observer decreases monotonically, and the geodesics terminate in finite proper distance at the big bang (c.f. [14,15]). One should therefore be able to construct larger cosmologies by extending these geodesics further, while at the same time preserving some continuity and differentiability properties of the metric tensor.…”

“…where in this case the scale factor a(t) = t. The formulas τ = t cosh χ and ρ = t sinh χ, with τ > |ρ|, transform (t, χ) to Fermi coordinates (τ, ρ) of the comoving observer at χ = ρ = 0 [14], and the metric in (τ, ρ) coordinates becomes,…”

“…The cosmological time zero submanifold (defined by t 0 = 0) is lightlike in our extension, and parameterized by the (finite) Fermi radius of the orginal universe ρ Mτ (see Eq. (14)). In general the extension of the metric is not twice continuously differentiable, but continuity is retained along with existence and continuity of the partial derivatives of the nonvanishing leading metric coefficient g τ τ (and g ρρ is constant).…”

A note on the computation of geometrically defined relative velocities

“…Following this idea, four different geometric definitions were introduced in [3]: kinematic, Fermi, spectroscopic and astrometric relative velocities. These four concepts each have full physical sense, and have proved to be useful in the study of properties of particular spacetimes [4][5][6] (see [6] for a more detailed list of related works). Nevertheless, it is important to remark that there are other notions of relative velocity in general relativity; for example, sometimes it is desirable to consider a congruence of observers and then, local notions of velocities of a test particle with respect to this congruence can be defined geometrically through modified (scaled) Fermi-Walker derivatives (see [7]).…”