1.3 From Newtonian to relativistic cosmology 29time-like world-line. Special Relativity says that the speed measured using rulers
and clocks of thesameinertial coordinate system never exceeds the speed of light.
In the definition of|u|, however, we used the distance measured in the Minkowski
coordinate system and the proper time of the moving particle. This corresponds
to the spatial part of the 4-velocity, which can be arbitrarily large. The Hubble
velocity in a Milne universe is also not bounded when defined in the usual way:
vH=a ̇χ=χ.Only|v|=tanhχis well defined. Although both|u|andvHare
approximately equal to|v|forχ1, forχ≥1 they are very different and can
have no invariant meaning. In curved spacetime, the situation is even more compli-
cated. The inertial coordinate system can be introduced only locally, on scales much
smaller than the four-dimensional curvature scale, roughly 1/H. Hence, the relative
Minkowski velocity, the quantity which can never exceed the speed of light, is only
defined for particles whose separation is much less than 1/H. Any definition of
relative velocities at distances larger than the curvature scale, where the Hubble law
predicts velocities which exceed unity, cannot have an invariant meaning. These
remarks may be helpful in clarifying the notion of “superluminal expansion,” a
confusing term sometimes used in the literature to describe inflationary expansion.
The Milne solution is also useful as an illustration of the difference between
3-curvature and 4-curvature. A “spatially flat” universe (k=0) generically has
nonzero 4-curvature. For example, in the case of a dust-dominated universe with
=1, space is nonempty and the Riemann tensor is nonzero. The Milne universe
is a complementary example with nonzero spatial curvature (k=−1) but zero
4-curvature. Milne coordinates correspond to foliating the locally flat spacetime
with spatially curved homogeneous three-dimensional hypersurfaces. Hence, when-
ever the term “flat” is used in cosmology, it is important to distinguish between
3-curvature and 4-curvature.
Generally, one does not have a choice of foliation if it is to respect the homogene-
ity and isotropy of space. In particular, if the energy density is changing with time,
the appropriate foliation is hypersurfaces of constant energy density. This choice is
unique and has invariant physical meaning. Empty space, however, possesses extra
time-translational invariance, so any space-like hypersurface has uniform “energy
density” equal to zero. The other example of a homogeneous and isotropic space-
time with extra time-translational invariance is de Sitter space. In the next section
we will see that de Sitter space can be covered by three-dimensional hypersurfaces
of constant curvature with open, flat and closed geometry.
1.3.6DeSitteruniverse
The de Sitter universe is aspacetimewith positive constant 4-curvature that is
homogeneous and isotropic in both space and time. Hence, it possesses the largest
possible symmetry group, as large as the symmetry group of Minkowski spacetime