5.2 Transformation of the Affine Connections
63
Finally we observe that we may impose this relation on any vector and any displacement; because of this the coefficients of the array dx
m V
i on the two sides of (5.10)
must be equal, so the affine connections must transform according to
j
ml =
∂ x
j
∂ x n
∂ x
q
∂ x
m
∂ x
i
∂ x
l
n
qi −
∂
2 x
j
∂ x q ∂ x i
∂ x
q
∂ x
m
∂ x
i
∂ x
l
.
(5.11)
We have dropped the subscript P which is no longer necessary since everything in
the equation is evaluated at P. Notice that the first term in this relation is that of a
tensor transformation as defined in (4.38), but the second term is inhomogeneous and
independent of the connections. It is important that in general the affine connections
do not transform as tensors.
Several interesting properties of the connections follow from the transformation
law (5.11), which we will state as theorems.
Theorem 1 Under the special case of linear transformations the affine connections
do transform as tensors. This follows since the second derivatives in (5.11) vanish
for linear transformations.
Theorem 2 If the affine connections are symmetric in their lower indices in one
coordinate system then they are symmetric in all coordinate systems. The proof is
evident from the transformation law (5.11) since the second term is symmetric.
Theorem 3 If the affine connections vanish in one coordinatesystem then they are
symmetric in any coordinate system. This is also evident from (5.11).
Theorem 4 (A beautiful and fundamental theorem of Weyl) If the affine connections
are symmetric then there exists a coordinatesystem in which they vanish. We may
prove this at the origin of the coordinate system without loss of generality. To prove
the theorem consider a transformation of the form
x
j
= x
j
+
1
2
A
j
ik x
i x
k
.
(5.12)
Here A
j
ik is an array of constants to be determined. Then at the origin the following
equations follow
∂ x
j
∂ x i = δ
j
i ,
∂ x
k
∂ x
n = δ
k
n ,
∂
2 x
j
∂ x q ∂ x i =
1
2
A
j
iq + A
j
qi
.
(5.13)
It then follows from the transformation law that at the origin the transformed
connections are
j
ml =
j
ml −
1
2
(A
j
ml + A
j
lm ).
(5.14)
63
Finally we observe that we may impose this relation on any vector and any displacement; because of this the coefficients of the array dx
m V
i on the two sides of (5.10)
must be equal, so the affine connections must transform according to
j
ml =
∂ x
j
∂ x n
∂ x
q
∂ x
m
∂ x
i
∂ x
l
n
qi −
∂
2 x
j
∂ x q ∂ x i
∂ x
q
∂ x
m
∂ x
i
∂ x
l
.
(5.11)
We have dropped the subscript P which is no longer necessary since everything in
the equation is evaluated at P. Notice that the first term in this relation is that of a
tensor transformation as defined in (4.38), but the second term is inhomogeneous and
independent of the connections. It is important that in general the affine connections
do not transform as tensors.
Several interesting properties of the connections follow from the transformation
law (5.11), which we will state as theorems.
Theorem 1 Under the special case of linear transformations the affine connections
do transform as tensors. This follows since the second derivatives in (5.11) vanish
for linear transformations.
Theorem 2 If the affine connections are symmetric in their lower indices in one
coordinate system then they are symmetric in all coordinate systems. The proof is
evident from the transformation law (5.11) since the second term is symmetric.
Theorem 3 If the affine connections vanish in one coordinatesystem then they are
symmetric in any coordinate system. This is also evident from (5.11).
Theorem 4 (A beautiful and fundamental theorem of Weyl) If the affine connections
are symmetric then there exists a coordinatesystem in which they vanish. We may
prove this at the origin of the coordinate system without loss of generality. To prove
the theorem consider a transformation of the form
x
j
= x
j
+
1
2
A
j
ik x
i x
k
.
(5.12)
Here A
j
ik is an array of constants to be determined. Then at the origin the following
equations follow
∂ x
j
∂ x i = δ
j
i ,
∂ x
k
∂ x
n = δ
k
n ,
∂
2 x
j
∂ x q ∂ x i =
1
2
A
j
iq + A
j
qi
.
(5.13)
It then follows from the transformation law that at the origin the transformed
connections are
j
ml =
j
ml −
1
2
(A
j
ml + A
j
lm ).
(5.14)