110 Cosmological models
whereλis the cosmological constant andκthe gravitational constant. Here
Tij(with traceT =Taa)is the total energy–momentum–stress tensor for all
the matter and fields present, described at the relevant averaging scale. This
covariant equation (a set of second-order nonlinear equations for the metric tensor
components) shows that the Ricci tensor is determined pointwise by the matter
present at each point, but the Weyl tensor is not so determined; rather it is
fixed by suitable boundary conditions, together with the Bianchi identities for
the curvature tensor:
∇[eRab]cd= 0 ⇔∇[eReab]cd= 0 (3.3)(the equivalence of the full equations on the left with the first contracted equations
on the right holding only for four dimensions or less). Consequently it is this
tensor that enables gravitational ‘action at a distance’ (gravitational radiation, tidal
forces, and so on). Contracting the right-hand of equation (3.3) and substituting
into the divergence of equation (3.2) showsTijnecessarily obeys the energy–
momentum conservation equations
∇jTij= 0 (3.4)(the divergence ofλgijvanishes providedλis indeed constant, as we assume).
Thus matter determines the geometry which, in turn, determines the motion of
the matter (see e.g. [132]). We can look for exact solutions of these equations,
or approximate solutions obtained by suitable linearization of the equations; and
one can also consider how the solutions relate to Newtonian theory solutions.
Care must be exercised in the latter two cases, both because of the nonlinearity
of the theory, and because there is no fixed background spacetime available in
general relativity theory. This makes it essentially different from both Newtonian
theory and special relativity.
3.1.3 Matter description
The total stress tensorTijis the sum of theNstress tensorsTni jfor the various
matter components labelled by indexn(baryons, radiation, neutrinos, etc):
Tij=&nTni j (3.5)each component being described by suitable equations of state which encapsulate
their physics. The most common forms of matter in the cosmological context will
often to a good approximation, each have a ‘perfect fluid’ stress tensor;
Tni j=(μn+pn)uniunj+pngij (3.6)with unit 4-velocityuin(uniuin=− 1 ), energy densityμnand pressurepn, with
suitable equations of state relatingμnandpn. In simple cases, they will be related
by a barotropic relationpn=pn(μn); for example, for baryons,pb=0andfor