Physical Foundations of Cosmology

1.3 From Newtonian to relativistic cosmology 21

is incorporated in Einstein’s equations through the energy–momentum tensor,Tβα.
(In General Relativity the term “matter” is used for anything not the gravitational
field.) This tensor is symmetric,

Tαβ≡gβδTδα=Tβα, (1.51)

and is (almost unambiguously) determined by the condition that the equations

∂Tαβ/∂xβ= 0 (1.52)

must coincide with the equations of motion for matter in Minkowski spacetime. To
generalize to curved spacetime, the equations of motion are modified:

Tαβ;β≡

∂Tαβ

∂xβ

+αγβTγβ+γββTαγ= 0 , (1.53)

where the terms proportional toaccount for the gravitational field. Note that in
General Relativity these equations do not need to be postulated separately. They fol-
low from the Einstein equations as a consequence of the Bianchi identities satisfied
by the Einstein tensor:

Gαβ;α= 0. (1.54)

On large scales, matter can be approximated as a perfect fluid characterized by
energy densityε, pressurepand 4-velocityuα.Its energy–momentum tensor is

Tβα=(ε+p)uαuβ−pδαβ, (1.55)

where the equation of statep=p(ε)depends on the properties of matter and must
be specified. For example, if the universe is composed of ultra-relativistic gas, the
equation of state isp=ε/ 3 .In many cosmologically interesting casesp=wε,
wherewis constant.

Problem 1.11Consider a nonrelativistic, dust-like perfect fluid (u^0 ≈1,ui1,
pε) in a flat spacetime. Verify that the equationsTαβ,β=0 are equivalent to
the mass conservation law plus the Euler equations of motion.

Another important example of matter is a classical scalar fieldφwith potential
V(φ). In this case, the energy–momentum tensor is given by the expression

Tβα=φ,αφ,β−

(

1

2

φ,γφ,γ−V(φ)

)

δαβ, (1.56)

where

φ,β≡

∂φ

∂xβ

,φ,α≡gαγφ,γ.