1 + 3 covariant description: variables 113(see equation (3.20)). The projected time and space derivatives ofUab,haband
ηabcall vanish. Finally, following [91] we use angle brackets to denote orthogonal
projections of vectors and the orthogonally projected symmetric trace-free part of
tensors:
v〈a〉=habvb, T〈ab〉=[h(achb)d−^13 habhcd]Tcd; (3.15)
for convenience the angle brackets are also used to denote othogonal projections
of covariant time derivatives alongua(‘Fermi derivatives’):
v ̇〈a〉=habv ̇b, T ̇〈ab〉=[h(achb)d−^13 habhcd]T ̇cd. (3.16)3.2.2 Kinematic quantities
The orthogonal vectoru ̇a=ub∇buais theacceleration vector, representing the
degree to which the matter moves under forces other than gravity plus inertia
(which cannot be covariantly separated from each other in general relativity). The
acceleration vanishes for matter in free fall (i.e. moving under gravity plus inertia
alone).
We split the first covariant derivative ofuainto its irreducible parts, defined
by their symmetry properties:
∇aub=−uau ̇b+∇ ̃aub=−uau ̇b+^13 'hab+σab+ωab (3.17)where the trace'=∇ ̃auais the(volume) rate of expansionof the fluid (with
H ='/3 the Hubble parameter);σab =∇ ̃〈aub〉is the trace-free symmetric
sheartensor (σab = σ(ab),σabub = 0,σaa = 0), describing the rate of
distortion of the matter flow; andωab=∇ ̃[aub]is the skew-symmetricvorticity
tensor (ωab=ω[ab],ωabub=0), describing the rotation of the matter relative
to a non-rotating (Fermi-propagated) frame. The meaning of these quantities
follows from the evolution equation for a relative position vectorηa⊥=habηb,
whereηais a deviation vector for the family of fundamental world-lines, i.e.
ub∇bηa=ηb∇bua. Writingηa⊥=δ"ea,eaea=1, we find the relative distance
δ"obeys the propagation equation
(δ").
δ"=^13 '+(σabeaeb), (3.18)(the generalized Hubble law), and the relative direction vectoreathe propagation
equation
e ̇〈a〉=(σab−(σcdeced)hab−ωab)eb, (3.19)
giving the observed rate of change of position in the sky of distant galaxies
[21, 26].
Each function f satisfies the importantcommutation relationfor the∇ ̃-
derivative [40]
∇ ̃[a∇ ̃b]f=ηabcωcf ̇. (3.20)