56 General Relativity
described by the Robertson–Walker metric the approach to general covariance is to
find appropriate invariants in terms of tensors which have the desired properties.
Since vectors are rank 1 tensors, vector equations may already be covariant. How-
ever, dynamical laws contain many other quantities that are not tensors, in particular
space-time derivatives such as d∕d휏in Equation (3.11). Space-time derivatives are
not invariants because they imply transporting d푠along some curve and that makes
them coordinate dependent. Therefore we have to start by redefining derivatives and
replacing them with newcovariant derivatives, which are tensor quantities.
To make the space-time derivative of a vector generally covariant one has to take
into account that the direction of a parallel-transported vector changes in terms of
the local coordinates along the curve as shown in Figure 3.4. The change is certainly
some function of the space-time derivatives of the curved space that is described by
the metric tensor.
The covariant derivative operator with respect to the proper time휏is denoted D∕D휏
(for a detailed derivation see, e.g., references [1] and [3]). Operating with it on the
momentum four-vector푃휇results in another four-vector:
퐹휇=D푃휇
D휏≡d푃휇
d휏+훤휎휈휇푃휎d푥휈
d휏. (3.12)
The second term contains the changes this vector undergoes when it is parallel trans-
ported an infinitesimal distance푐d휏. The quantities훤휎휈휇, calledaffine connections,are
readily derivable functions of the derivatives of the metric푔휇휈in curved space-time,
but they are not tensors. Their form is
훤휎휈휇=^1
2푔휇휌
(휕푔
휎휌
휕푥휈+
휕푔휈휌
휕푥휎
−
휕푔휎휈
휕푥휌
)
. (3.13)
With this definition Newton’s second law has been made generally covariant.
Figure 3.4Parallel transport of a vector around a closed contour on a curved surface.