MODERN COSMOLOGY

1 + 3 Covariant description: equations 117

Singularity theorem. [21, 26, 28] In a universe where the active gravitational
mass is positive at all times,

(μ+ 3 p)> 0 , (3.36)

the cosmological constant vanishes (or is negative);λ ≤ 0 , and the vorticity
and acceleration vanish; u ̇a = ωa = 0 at all times, at any instant when
H 0 =^13 ' 0 > 0 , there must have been a time t 0 < 1 /H 0 ago such that S→ 0 as
t→t 0 ; a spacetime singularity occurs there, whereμ→∞and T→∞.

The further singularity theorems of Hawking and Penrose [68,69,124] utilize
this result or its null version as an essential part of their proofs.
Closely related to this are three other results:

(1) a static universe model containing ordinary matter requiresλ>0 (Einstein’s

discovery of 1917);

(2) the Einstein static universe is unstable (Eddington’s discovery of 1930);

(3) in a universe satisfying the requirements of the singularity theorem, at each

instanttthe age of the universe is less that 1/H(t),so for example the hot

early stage of the universe takes place extremely rapidly.

Proofs follow directly from (3.35). The energy condition(μ+ 3 p)>0 will
be satisfied by all ordinary matter but will not, in general, be satisfied by a scalar
field, see (3.7).

3.3.2.2 The vorticity propagation equation

ω ̇〈a〉− 21 ηabc∇ ̃bu ̇c=−^23 'ωa+σabωb. (3.37)

If we have a barotropic perfect fluid:

qa=πab= 0 ,p=p(μ)⇒ηabc∇ ̃bu ̇c= 0 , (3.38)

thenωa=0 is involutive: i.e. the statement

ωa=0 initially⇒ ̇ω〈a〉= 0 ⇒ωa=0 at later times

follows from the vorticity conservation equation (3.37) (and it is also true in the
special casep=0). Thus non-trivial entropy dependence or an imperfect fluid is
required to create vorticity.
When the vorticity vanishes⇔ω=0:

(1) The fluid flow is hypersurface-orthogonal, and there exists a cosmic time

functiontsuch thatua =−g(xb)∇at, allowing synchronization of the

clocks of fundamental observers. If, in addition, the acceleration vanishes,

we can setg = 1 and the time function can be proper time for all of

them (whereas if the acceleration is non-zero, the coordinate timetwill

necessarily correspond to different proper times along different world-lines).