The Principle of Covariance 57The path of a test body in free fall follows from Equation (3.12) by requiring that
no forces act on the body,퐹휇=0. Making the replacement
푃휇=푚
d푥휇
d휏,
the relativistic equation of motion of the test body, itsgeodesic equation, can be written
d^2 푥휇
d휏^2+훤휎휈휇d푥휎
d휏d푥휈
d휏= 0. (3.14)
In an inertial frame the metric is flat, the metric tensor is a constant everywhere,
푔휇휈(푥)=휂휇휈, and thus the space-time derivatives of the metric tensor vanish:
휕푔휇휈(푥)
휕푥휌= 0. (3.15)
It then follows from Equation (3.13) that the affine connections also vanish, and the
covariant derivatives equal the simple space-time derivatives.
Going from an inertial frame at푥to an accelerated frame at푥+훥푥the expressions
for푔휇휈(푥)and its derivatives at푥can be obtained as the Taylor expansions
푔휇휈(푥+훥푥)=휂휇휈+^1
2
휕^2 푔휇휈(푥)
휕푥휌휕푥휎
훥푥휌훥푥휎+···
and
휕푔휇휈(푥+훥푥)
휕푥휌
=
휕^2 푔휇휈(푥)
휕푥휌휕푥휎
훥푥휎+···.
The description of a curved space-time thus involves second derivatives of푔휇휈,atleast.
(Only in a very strongly curved space-time would higher derivatives be needed.)
Recall the definition of the noncovariant Gaussian curvature퐾in Equation (2.31)
defined on a curved two-dimensional surface. In a higher-dimensional space-time,
curvature has to be defined in terms of more than just one parameter퐾. It turns
out that curvature is most conveniently defined in terms of the fourth-rankRiemann
tensor
푅훼훽훾휎=휕훤훽휎훼
휕푥훾
−
휕훤훽훾훼
휕푥휎
+훤휌훾훼훤훽휎휌−훤휌휎훼훤훽훾휌. (3.16)
In four-space this tensor has 256 components, but most of them vanish or are not inde-
pendent because of several symmetries and antisymmetries in the indices. Moreover,
an observer at rest in the comoving Robertson–Walker frame will only need to refer
to spatial curvature. In a spatial푛-manifold,푅훼훽훾훿has only푛^2 (푛^2 − 1 )∕12 nonvanish-
ing components, thus six in the spatial three-space of the Robertson–Walker metric.
On the two-sphere there is only one component, which is essentially the Gaussian
curvature퐾.
Another important tool related to curvature is the second rankRicci tensor푅훽훾,
obtained from the Riemann tensor by a summing operation over repeated indices,
calledcontraction:
푅훽훾=푅훼훽훾훼=훿휎훼푅훼훽훾휎=푔훼휎푅훼훽훾휎. (3.17)This푛^2 -component tensor is symmetric in the two indices, so it has only^12 푛(푛+ 1 )
independent components. In four-space the ten components of the Ricci tensor lead