74
5 Affine Connections and Geodesics
The coefficients in the expansion will of course be identified as the affine connections.
They can then be obtained explicitly by asking that inner products be unchanged when
parallel displaced just as we did in Sect. 5.3. Thereby the affine space plus a geometric
or physical demand becomes the Riemann space we use in general relativity theory.
Now we substitute the basis change (5.49) in the vector change (5.48) and obtain
d
V =
dV
i
+
i
k j V
j dx
k
e i .
(5.50)
The condition that the vector components change according to the law (5.5) of vector
transplantation thus corresponds to d
V = 0. We have obtained an interpretation for
the affine connections as being related to the change in the coordinate basis vectors
(5.49).
Note that the defining expression (5.50) for the connections does not imply that
they must be symmetric in the lower indices; the same is true in the component view
for (5.5).
In the case of flat space the derivatives of the basis vectors may be calculated
explicitly and we could rewrite (5.49) in terms of those derivatives. This leads to an
explicit expression for the connections in that special case,
d e j = =
e j,k dx
k
≡
i
k j
e i
dx
k
, so
n
k j =
e j,k · ·
e i
g
ni
.
(5.51)
Geodesics in terms of vector transplantation fit naturally into the present scheme.
Suppose as usual that we have a curve C with arc length s. Then any vector
V defined
along the curve will change according to (5.50) and have a derivative along the curve
given by
d
V
ds
=
dV
i
ds
+
i
k j V
j dx
k
ds
e i .
(5.52)
Apply this relation now to a tangent vector to the curve, which we may take to be
τ =
dx
k
ds
e k .
(5.53)
Then the derivative of the tangent vector along the curve is
d τ
ds
=
d
2 x
i
ds 2 +
i
k j
dx
j
ds
dx
k
ds
e i .
(5.54)
Thus if we define a geodesic as having a constant tangent vector we find that the
curve is the same as the geodesic curve we defined in Sect. 5.4,
˙
x
i
+
i
k j ˙
x
k
˙
x
j
= 0, ˙
x
k
=
dx
k
ds
, geodesic equation.
(5.55)
5 Affine Connections and Geodesics
The coefficients in the expansion will of course be identified as the affine connections.
They can then be obtained explicitly by asking that inner products be unchanged when
parallel displaced just as we did in Sect. 5.3. Thereby the affine space plus a geometric
or physical demand becomes the Riemann space we use in general relativity theory.
Now we substitute the basis change (5.49) in the vector change (5.48) and obtain
d
V =
dV
i
+
i
k j V
j dx
k
e i .
(5.50)
The condition that the vector components change according to the law (5.5) of vector
transplantation thus corresponds to d
V = 0. We have obtained an interpretation for
the affine connections as being related to the change in the coordinate basis vectors
(5.49).
Note that the defining expression (5.50) for the connections does not imply that
they must be symmetric in the lower indices; the same is true in the component view
for (5.5).
In the case of flat space the derivatives of the basis vectors may be calculated
explicitly and we could rewrite (5.49) in terms of those derivatives. This leads to an
explicit expression for the connections in that special case,
d e j = =
e j,k dx
k
≡
i
k j
e i
dx
k
, so
n
k j =
e j,k · ·
e i
g
ni
.
(5.51)
Geodesics in terms of vector transplantation fit naturally into the present scheme.
Suppose as usual that we have a curve C with arc length s. Then any vector
V defined
along the curve will change according to (5.50) and have a derivative along the curve
given by
d
V
ds
=
dV
i
ds
+
i
k j V
j dx
k
ds
e i .
(5.52)
Apply this relation now to a tangent vector to the curve, which we may take to be
τ =
dx
k
ds
e k .
(5.53)
Then the derivative of the tangent vector along the curve is
d τ
ds
=
d
2 x
i
ds 2 +
i
k j
dx
j
ds
dx
k
ds
e i .
(5.54)
Thus if we define a geodesic as having a constant tangent vector we find that the
curve is the same as the geodesic curve we defined in Sect. 5.4,
˙
x
i
+
i
k j ˙
x
k
˙
x
j
= 0, ˙
x
k
=
dx
k
ds
, geodesic equation.
(5.55)