MODERN COSMOLOGY

1 + 3 Covariant description: equations 115

3.2.5 Weyl tensor

In analogy to Fab,theWeyl conformal curvature tensor Cabcd defined by
equation (3.1) is split relative touainto ‘electric’ and ‘magnetic’Weyl curvature
parts according to

Eab=Cacbducud⇒Eaa= 0 ,Eab=E(ab),Eabub= 0 , (3.25)

Hab=^12 ηadeCdebcuc⇒Haa= 0 ,Hab=H(ab),Habub= 0. (3.26)

These influence the motion of matter and radiation through thegeodesic deviation
equationfor timelike and null vectors, see, respectively, [107] and [120].

3.3 1 +3 Covariant description: equations

There are three sets of equations to be considered, resulting from EFE (3.2) and
its associated integrability conditions.

3.3.1 Energy–momentum conservation equations

We obtain from the conservation equations (3.4), on projecting parallel and
perpendicular touaand using (3.21), the propagation equations

μ ̇+∇ ̃aqa=−'(μ+p)− 2 (u ̇aqa)−(σabπba), (3.27)

q ̇〈a〉+∇ ̃ap+∇ ̃bπab=−^43 'qa−σabqb−(μ+p)u ̇a− ̇ubπab−ηabcωbqc.

(3.28)

For perfect fluids, characterized by equation (3.8), these reduce to

μ ̇=−'(μ+p), (3.29)

theenergy conservation equation,andthemomentum conservation equation

0 =∇ ̃ap+(μ+p)u ̇a (3.30)

(which because of the perfect fluid assumption, has changed from a time-
derivative equation forqato an algebraic equation foru ̇a, and thus a time-
derivative equation forua). These equations show that(μ+p)is both the inertial
mass density and that it governs the conservation of energy. It is clear that if this
quantity is zero (the case of an effective cosmological constant) or negative, the
behaviour of matter will be anomalous; in particular velocities will be unstable
ifμ+p→ 0 ,because the acceleration generated by a given force will diverge
in this limit. If we assume a perfect fluid with a (linear)γ-law equation of state,
then (3.29) shows that

p=(γ− 1 )μ,γ ̇= 0 ⇒μ=M/S^3 γ,M ̇ = 0. (3.31)