The Physical Basis of The Direction of Time (The Frontiers ...

154 5 The Time Arrow of Spacetime Geometry
describes the proper time for objects which are at rest in these coordinates
(‘comoving clocks’). This metric may remain valid close to the big bang (for
a = 0) in accordance with the Weyl tensor hypothesis. It can be generalized
by means of a multipole expansion on the Friedmann sphere (see Halliwell and
Hawking 1985, and Sect. 5.4). This general-relativistic form has the advantages
of not requiring a special ‘center at rest’, and of allowing a finite universe
without a boundary (for positive curvature).
The exact FRW metric (5.20) depends only on the expansion parameter
a(t). The latter’s dynamics, derived from the Einstein equations (5.7) with
an additional cosmological constant, assumes the form of an ‘energy integral’
with a fixed vanishing value of the energy:
( ) 2
1 1 da
= 1 2 a dt 2
( ) 2 dα
= −V (α) . (5.21)
dt
The logarithm of spatial extension, α =lna, which formally sends the big bang
to minus infinity, will prove convenient on several occasions. The Friedmann
potential V (α) is given by the energy density of matter ρ(a), the cosmological
constant Λ, and the spatial curvature k/a 2 , in the form
V (α) =− 4πρ(eα )
3
− Λ 3 + ke−2α . (5.22)
One would have obtained essentially the same equation (without curvature
term and cosmological constant, but with variable energy) from Newton’s
dynamics for the radius of a gravitating homogeneous sphere of matter.
The energy density ρ may depend on a in various ways. In the matterdominated
epoch it is proportional to the inverse density, a −3 . During the
radiation era – less than 10 −4 of the present age of the universe – it decreased
according to a −4 , since all wavelengths expand with a. Much earlier (for extremely
high matter density), quite novel phenomena must be expected to
have affected the relativistic equation of state, here described by ρ(a). According
to some theories, for example, the vacuum state of matter passed
through one or several phase transitions (see Sect. 6.1). Similar to a condensation
process, this situation may be characterized by a constant function of
state, ρ(a) =ρ 0 . The matter term in the potential V would then simulate
a cosmological constant – albeit only for a limited time (see Fig. 5.6). In the
‘Planck era’, that is, for values of a of order unity, quantum gravity must
become essential (see Sect. 6.2).
Different eras, described by such analytic equations of state ρ(a), possess
different solutions a(t). For example, a dominating (fundamental or simulated)
cosmological constant would lead to a ‘de Sitter era’ with a(t) =ce ±Ht and
a ‘Hubble constant’ H =ȧ/a =˙α. For a matter- or radiation-dominated universe,
one has a(t) =c ′ t 2/3 or a(t) =c ′′ t 1/2 , respectively, while for low matter
densities the curvature term may dominate. Recent observations indicate that
our Universe is approximately flat (negligible curvature term), while an effec-