MODERN COSMOLOGY

16 An introduction to the physics of cosmology

This defines the speed of sound to becS^2 =∂p/∂ρ. Notice that, by a fortunate
coincidence, this is exactly the same as is derived from the non-relativistic
equations, although we could not have relied upon this in advance. Thus, the
speed of sound in a radiation-dominated fluid is justc/

√

3.

2.3 The field equations

The energy–momentum tensor plausibly plays the role that the charge 4-current
Jμplays in the electromagnetic field equations,Aμ=μ 0 Jμ. The tensor on
the left-hand side of the gravitational field equations is rather more complicated.
Weinberg (1972) showed that it is only possible to make one tensor that is linear
in second derivatives of the metric, which is theRiemann tensor:

Rμαβγ =

∂μαγ

∂xβ

−

∂μαβ

∂xγ

+μσβσγα−μσγσβα.

This tensor gives a covariant description of spacetime curvature. For the field
equations, we need a second-rank tensor to matchTμν, and the Riemann tensor
may be contracted to the Ricci tensorRμν, or further to thecurvature scalar R:

Rαβ=Rμαβμ

R=Rμμ=gμνRμν.

Unfortunately, these definitions are not universally agreed, All authors, however,
agree on the definition of the Einstein tensorGμν:

Gμν=Rμν−^12 gμνR.

This tensor is what is needed, because it has zero covariant divergence. SinceTμν
also has zero covariant divergence by virtue of the conservation laws it expresses,
it therefore seems reasonable to guess that the two are proportional:

Gμν=−

8 πG

c^4

Tμν.

These are Einstein’s gravitational field equations, where the correct constant of
proportionality has been inserted. This is obtained by considering the weak-field
limit.

2.3.1 Newtonian limit

The relation between Einstein’s and Newton’s descriptions of gravity involves
taking the limit of weak gravitational fields (φ/c^2 1). We also need to consider
a classical source of gravity, withpρc^2 , so that the only non-zero component
ofTμνisT^00 =c^2 ρ. Thus, the spatial parts ofRμνmust be given by

Rij=^12 gijR.