David Klein, Evan Randles
Fermi coordinates, simultaneity, and expanding space in Robertson-Walker cosmologies
(79K, latex)
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.