116 Cosmological models
One can approximate ordinary matter in this way, with 1≤γ ≤2inorder
that the causality and energy conditions are valid. Radiation corresponds to
γ=^43 ⇒μ=M/S^4 , so from Stefan’s law (μ∝T^4 )wefindthatT∝ 1 /S.
Another useful case ispressure-free matter(often described as ‘baryonic’ or ‘cold
dark matter (CDM)’); the momentum conservation: (3.30) shows that such matter
moves geodesically (as expected from the equivalence principle):
γ= 1 ⇔p= 0 ⇒ ̇ua= 0 ,μ=M/S^3. (3.32)This is the case ofpure gravitation, without fluid dynamical effects. Another
important case is that of a scalar field, see (3.7).
3.3.2 Ricci identities
The second set of equations arise from theRicci identitiesfor the vector fieldua,
i.e.
2 ∇[a∇b]uc=Rabcdud. (3.33)On substituting from (3.17), using (3.2), and separating out the parallelly
and orthogonally projected parts into a trace, symmetric trace-free and skew
symmetric part, we obtain three propagation equations and three constraint
equations. Thepropagation equationsare the Raychaudhuri equation, the
vorticity propagation equation and the shear propagation equation.
3.3.2.1 The Raychaudhuri equation
This equation
' ̇=−^1
3 '(^2) +∇
au ̇
a− 2 σ (^2) + 2 ω (^2) − 1
2 (μ+^3 p)+λ, (3.34)
thebasic equation of gravitational attraction[21, 26, 28], shows the repulsive
nature of a positive cosmological constant and leads to the identification of
(μ+ 3 p)as the active gravitational mass density. Rewriting it in terms of the
average scale factorS, this equation can be rewritten in the form
3
S ̈
S
=− 2 (σ^2 −ω^2 )+∇au ̇a−1
2
(μ+ 3 p)+λ, (3.35)showing how the curvature of the curveS(τ)along each world-line (in terms
of proper timeτalong that world-line) is determined by the shear, vorticity and
acceleration; the total energy density and pressure in terms of the combination
(μ+ 3 p)—theactive gravitational mass; and the cosmological constantλ.This
gives the basic singularity theorem.