120 Cosmological models
3.3.4 Implications
Altogether, we have six propagation equations and six constraint equations;
considered as a set of evolution equations for the 1+3 covariant variables, they are
a first-order system of equations. This set is determinate once the fluid equations
of state are given; together they then form a dynamical system (the set closes up,
but is essentially an infinite dimensional dynamical system because of the spatial
derivatives that occur).
Thekey issuethat arises is consistency of the constraints with the evolution
equations. It is believed that they aregenerally consistent for physically
reasonable and well-defined equations of state, i.e. they are consistent if no
restrictions are placed on their evolution other than those implied by the constraint
equations and the equations of state (this has been shown for irrotational dust
[91]). It is this that makes consistent the overall hyperbolic nature of the equations
with the ‘instantaneous’ action at a distance implicit in the Gauss-like equations
(specifically, the(divE)-equation), the point being that the ‘action at a distance’
nature of the solutions to these equations is built into the initial data, which must
be chosen so that the constraints are satisfied initially, and they then remain
satisfied thereafter because the time evolution preserves these constraints (cf
[49]).
3.3.5 Shear-free dust
One must be very cautious with imposing simplifying assumptions in order to
obtain solutions: this can lead to major restrictions on the possible flows, and
one can be badly misled if their consistency is not investigated carefully. A case
of particular interest isshear-free dust, that is perfect-fluid solutions for which
σab= 0 ,p = 0 ⇒ ̇ua =0. In this case, careful study of the consistency
conditions between all the equations [25] shows that necessarilyω'=0: the
solutions either do not rotate, or do not expand. This conclusion is of considerable
importance, because if it were not true, there would be shear-free expanding and
rotating solutions which would violate the Hawking–Penrose singularity theorems
for cosmology [68,69] (integrating the vorticity equation along the fluid flow lines
(3.37) givesω=ω 0 /S^2 ;substituting in the Raychaudhuri equation (3.34) and
integrating, using the conservation equation (3.29), gives a first integral which is a
generalized Friedmann equation, in which vorticity dominates expansion at early
times and allows a bounce and singularity avoidance). The interesting point then
is thatthis result does not hold in Newtonian theory[113], in which case there
do indeed exist such solutions when suitable boundary conditions are imposed.
If one uses these solutions as an argument against the singularity theorems, the
argument is invalid; what they really do is point out the dangers of the Newtonian
limit of cosmological equations.