118 Cosmological models
(2) The metric of the orthogonal 3-spacest =constant formed by meshing
together the tangent spaces orthogonal touaishab.
(3) From the Gauss equation and the Ricci identities forua, the Ricci tensor of
these 3-spaces is given by [21, 26](^3) R
ab=− ̇σ〈ab〉−'σab+∇ ̃〈au ̇b〉+ ̇u〈au ̇b〉+πab+
1
3 hab
(^3) R, (3.39)
and their Ricci scalar is given by
(^3) R= 2 μ− 2
3 '
(^2) + 2 σ (^2) + 2 λ, (3.40)
which is a generalized Friedmann equation, showing how the matter tensor
determines the 3-space average curvature. These equations fully determine
the curvature tensor^3 Rabcdof the orthogonal 3-spaces, and so show how the
EFEs result inspatialcurvature (as well as spacetime curvature) [21, 26].
3.3.2.3 The shear propagation equation
σ ̇〈ab〉−∇ ̃〈au ̇b〉=−^23 'σab+ ̇u〈au ̇b〉−σ〈acσb〉c−ω〈aωb〉−(Eab−^12 πab). (3.41)
This shows how the tidal gravitational fieldEabdirectly induces shear (which
then feeds into the Raychaudhuri and vorticity propagation equations, thereby
changing the nature of the fluid flow), and that the anisotropic pressure termπab
also generates shear in an imperfect fluid situation. Shear-free solutions are very
special solutions, because (in contrast to the case of vorticity) a conspiracy of
terms is required to maintain the shear zero if it is zero at any initial time (see
later for a specific example).
Theconstraint equationsare as follows:
(1) The( 0 α)-equation
0 =(C 1 )a=∇ ̃bσab−^23 ∇ ̃a'+ηabc[∇ ̃bωc+ 2 u ̇bωc]+qa, (3.42)
shows how the momentum fluxqa(zero for a comoving perfect fluid) relates
to the spatial inhomogeneity of the expansion.
(2) Thevorticity divergence identity
0 =(C 2 )=∇ ̃aωa−(u ̇aωa), (3.43)
follows becauseωais a curl.
(3) TheHab-equation
0 =(C 3 )ab=Hab+ 2 u ̇〈aωb〉+(curlσ)ωab−(curlσ)ab, (3.44)
characterizes the magnetic part of the Weyl tensor as being constructed from
the ‘curls’ of the vorticity and shear tensors:(curlω)ab =ηcd〈a∇ ̃cωb〉d,
(curlσ)ab=ηcd〈a∇ ̃cσb〉d.