114 Cosmological models
Applying this to the energy densityμshows that ifωaμ ̇=0inanopenset
then∇ ̃aμ=0 there, so non-zero vorticity implies anisotropic number counts in
an expanding universe [61] (this is because there are then no 3-surfaces orthogonal
to the fluid flow; see [21, 26]).
3.2.2.1 Auxiliary quantities
It is useful to define some associated kinematical quantities:
- thevorticity vectorωa=^12 ηabcωbc⇒ωaua=0,ωabωb= 0 ,
- the magnitudesω^2 =^12 (ωabωab)≥0,σ^2 =^12 (σabσab)≥0, and
- theaverage length scale Sdetermined by
S ̇
S=
1
3 ', so the volume of a fluid
element varies along the fluid flow lines asS^3.3.2.3 Matter tensor
Both the total matterenergy–momentum tensor Taband each of its components
can be decomposed relative touain the form
Tab=μuaub+qaub+uaqb+phab+πab, (3.21)whereμ =(Tabuaub)is therelativistic energy densityrelative toua,qa =
−Tbcubhca is therelativistic momentum density(qaua = 0 ), which is also
the energy flux relative toua, p =^13 (Tabhab)is theisotropic pressure,and
πab=Tcdhc〈ahdb〉is the trace-freeanisotropic pressure(πaa=0,πab=π(ab),
πabub= 0 ). A different choice ofuawill result in a different splitting. The
physics of the situation is in the equations of state relating these quantities; for
example, the commonly imposed restrictions
qa=πab= 0 ⇔Tab=μuaub+phab (3.22)characterize a ‘perfect fluid’ moving with the chosen 4-velocityua as in
equation (3.8) with, in general, an equation of statep=p(μ,s)wheresis the
entropy [21, 26].
3.2.4 Electromagnetic field
TheMaxwell field tensor Fabof an electromagnetic field is split relative touainto
electric and magnetic parts by the relations (see [28])
Ea=Fabub⇒Eaua= 0 , (3.23)
Ha=^12 ηabcFbc⇒Haua= 0. (3.24)Again, a different choice ofuawill result in a different split.