62
5 Affine Connections and Geodesics
of vector transplantation we may obtain the transformation law for the connections,
which we will find are not tensors. Moreover several theorems that result from the
transformation law are basic and important for both mathematics and physics.
To find the transformation law we make the natural demand that a vector remains
a vector as it is transplanted to a nearby point: that is it must obey the transformation
law (4.14) at both P and P
. The transplanted vector, at P
in the barred and unbarred
coordinate systems, is
V
∗i
= V
i
−
i
pq dx
p V
q
, V
∗ j = V
j −
j
mn dx
m V
n .
(5.6)
Here the vector and connections on the right side of (5.6) are evaluated at P. The
transformation matrix at P
may be gotten with a Taylor series expansion from that
at P,
∂ x
j
∂ x i
P
=
∂ x
j
∂ x i
P
+
∂
∂ x l
∂ x
j
∂ x i
P
dx
l
=
∂ x
j
∂ x i
P
+
∂
2 x
j
∂ x l ∂ x i
P
dx
l
. (5.7)
We use these expressions and impose the vector transformation law on the vector at
P
,
V
∗ j =
∂ x
j
∂ x l
P
V
∗l so
V
j −
j
mn dx
m V
n =
∂ x
j
∂ x i
P
+
∂
2 x
j
∂ x l ∂ x i
P
dx
l
V
i
−
i
pq dx
p V
q
=
∂ x
j
∂ x i
P
V
i
−
∂ x
j
∂ x i
P
i
pq dx
p V
q
+
∂
2 x
j
∂ x l ∂ x i
P
dx
l V
i
.
(5.8)
The first terms on each side of this equation cancel because V
i is a vector at P. We
relabel the dummy indices and the remaining terms tell us that
−
j
ml dx
m V
l =
−
∂ x
j
∂ x n
P
n
qi +
∂
2 x
j
∂ x q ∂ x i
P
dx
q V
i
.
(5.9)
Next, we express the vector and coordinate differential in the unbarred system in
terms of those in the barred system using the vector transformation equations, and
find
−
j
ml dx
m V
l
=
−
∂ x
j
∂ x n
∂ x
q
∂ x
m
∂ x
i
∂ x
l
P
n
qi +
∂
2 x
j
∂ x q ∂ x i
P
∂ x
q
∂ x
m
∂ x
i
∂ x
l
dx
m V
l .
(5.10)