Introduction 111radiation, e.g. the cosmic background radiation (‘CBR’),pr=μr/ 3 ,.However,
in more complex cases there will be further variables determiningpnandμn;for
example, in the case of a massless scalar fieldφwith potentialV(φ), on choosing
uias the unit vector normal to spacelike surfacesφ=constant, the stress tensor
takes the form (3.6) with
4 πpφ=^12 φ ̇^2 −V(φ), 4 πμφ=^12 φ ̇^2 +V(φ). (3.7)It must be noted that, in general, different matter components will each have
a different 4-velocityuin, and the total stress tensor (3.5) of perfect fluid stress
tensors (3.6) itself has the perfect fluid form if and only if the 4-velocities of all
contributing matter components are the same, i.e.uin=uifor alln; in that case,
Tij=(μ+p)uiuj+pgij,μ≡&nμn, p≡&npn (3.8)whereμis the total energy density andpthe total pressure.
The individual matter components will each separately satisfy the
conservation equation (3.4) if they are non-interacting with the other components;
however this will no longer be the case if interactions lead to exchanges of
energy and momentum between the different components. The key to a physically
realistic cosmological model is the representation of suitable matter components,
with realistic equations of state for each matter component and equations
describing the interactions between the components. For reasonable behaviour
of matter, irrespective of its constitution we require the ‘energy condition’
μ+p> 0 (3.9)on cosmological averaging scales (the vacuum caseμ+p=0 can apply only
to regions described on averaging scales less than or equal to that of clusters of
galaxies).
3.1.4 Cosmology
A key feature of cosmological models, as contrasted with general solutions of the
EFEs, is that in them, at each point a unique 4-velocityuais defined representing
the preferred motion of matter there on a cosmological scale. Whenever the matter
present is well described by the perfect fluid stress tensor (3.8), because of (3.9)
there will be a unique timelike eigenvector of this tensor that can be used to define
the vectoru,representing the average motion of the matter, and conventionally
referred to as defining thefundamental world-linesof the cosmology. Unless
stated otherwise, we will assume that observers move with this 4-velocity. At
late times, a unique frame is defined by choosing a 4-velocity such that the CBR
anisotropy dipole vanishes; the usual assumption is that this is the same frame as
defined locally by the average motion of matter [26]; indeed this assumption is
what underlies studies of large-scale motions and the ‘Great Attractor’.