66
5 Affine Connections and Geodesics
(g ik,l + g kl,i − g li,k ) − 2
r
il g kr = 0.
(5.18)
In obtaining (5.18) we have made use of the symmetry of the metric and also assumed
the connections are symmetric in the lower indices. To solve for the connections we
multiply (5.18) by g
kt and contract on k to find
n
il =
1
2
g
kn
(g ik,l + g kl,i − g li,k ).
(5.19)
Thus the connections are explicitly solved in terms of the metric and its derivatives.
Notice their explicit symmetry in the lower indices, which we will use often. The
connections defined in (5.19) are called the Christoffel connections or often the
Christoffel symbols. They apply specifically to parallel displacement rather than the
more general vector transplantation.
A historical note: there are also “Christoffel symbols of the first kind” used by
some authors, which are defined as
[il, k] =
1
2
(g ik,l + g kl,i − g li,k ), ,
t
il = g
kt [il, k],
(5.20)
We do not use these in this book. Also, Christoffel originally used a curly bracket
notation for the affine connections, but this is now seldom used (Pauli 1958; Adler
1975). In the rest of this book we will use only the connections for parallel displacement (5.19), denoted with a capital gamma, and refer to them as either Christoffel
connections or simply connections.
Let us summarize properties of parallel displacement: vector transplantation using
the connections defined in (5.19) gives a vector at the nearby point that is parallel to
the original one in a generalized sense of parallel. Explicitly, the change in a parallel
displaced vector is expressed by
dξ
n
+
n
li dx
l
ξ
i
= 0,
n
li =
1
2
g
kn
(g ik,l + g kl,i − g li,k ), parallel displacement.
(5.21)
One consequence of the definition is that under parallel displacement the inner
product of a vector with itself is unchanged, which means that its length remains
unchanged.
Although it may appear somewhat formal at this point the idea of parallel displacement turns out to have beautiful physical and geometric meaning. It leads to definitions for the generalized straight lines called geodesics and curvature in a Riemann
space. It is the central idea when we discuss covariant derivatives in tensor analysis.
It might look as if the Christoffel connections require a lot of algebra to calculate,
since there are 40 of them in 4-dimensional space and n
2
(n + 1)/2 in n-dimensional
space (see Exercise 5.1). Fortunately there is a shortcut method to obtain the nonzero
connections using the algebra of geodesics, which we will study in Sect. 5.5 and
Example 5.3.
5 Affine Connections and Geodesics
(g ik,l + g kl,i − g li,k ) − 2
r
il g kr = 0.
(5.18)
In obtaining (5.18) we have made use of the symmetry of the metric and also assumed
the connections are symmetric in the lower indices. To solve for the connections we
multiply (5.18) by g
kt and contract on k to find
n
il =
1
2
g
kn
(g ik,l + g kl,i − g li,k ).
(5.19)
Thus the connections are explicitly solved in terms of the metric and its derivatives.
Notice their explicit symmetry in the lower indices, which we will use often. The
connections defined in (5.19) are called the Christoffel connections or often the
Christoffel symbols. They apply specifically to parallel displacement rather than the
more general vector transplantation.
A historical note: there are also “Christoffel symbols of the first kind” used by
some authors, which are defined as
[il, k] =
1
2
(g ik,l + g kl,i − g li,k ), ,
t
il = g
kt [il, k],
(5.20)
We do not use these in this book. Also, Christoffel originally used a curly bracket
notation for the affine connections, but this is now seldom used (Pauli 1958; Adler
1975). In the rest of this book we will use only the connections for parallel displacement (5.19), denoted with a capital gamma, and refer to them as either Christoffel
connections or simply connections.
Let us summarize properties of parallel displacement: vector transplantation using
the connections defined in (5.19) gives a vector at the nearby point that is parallel to
the original one in a generalized sense of parallel. Explicitly, the change in a parallel
displaced vector is expressed by
dξ
n
+
n
li dx
l
ξ
i
= 0,
n
li =
1
2
g
kn
(g ik,l + g kl,i − g li,k ), parallel displacement.
(5.21)
One consequence of the definition is that under parallel displacement the inner
product of a vector with itself is unchanged, which means that its length remains
unchanged.
Although it may appear somewhat formal at this point the idea of parallel displacement turns out to have beautiful physical and geometric meaning. It leads to definitions for the generalized straight lines called geodesics and curvature in a Riemann
space. It is the central idea when we discuss covariant derivatives in tensor analysis.
It might look as if the Christoffel connections require a lot of algebra to calculate,
since there are 40 of them in 4-dimensional space and n
2
(n + 1)/2 in n-dimensional
space (see Exercise 5.1). Fortunately there is a shortcut method to obtain the nonzero
connections using the algebra of geodesics, which we will study in Sect. 5.5 and
Example 5.3.