5.5 Geodesics as Extremum Curves
73
Similarly for the x
2
= ϕ equation
¨
ϕ +
2
ρ
˙
ρ ˙
ϕ ⇔ 0 − ¨
ϕ +
2
i j ˙
x
i
˙
x
j
= 0.
(5.45)
From this there are only two equal nonzero connections with an upper index
2,
2
21 =
2
12 =
1
ρ
.
(5.46)
The ease of the technique is apparent, especially so since the metric is diagonal
and most of the connections are zero. It is often a large labor-saving technique.
In the rest of this book we will make frequent use of the technique in Example
5.3 for calculating the connections.
5.6 Affine Connections, Abstract View
Let us see how we may motivate and interpret the coefficients of affine connection
using the abstract view introduced in Sect. 4.3. Recall that a vector may be expanded
in a coordinate basis, that is vectors aligned along the coordinate axes, according to
V = V
j
e j ,
e j = coordinate basis.
(5.47)
If we think of moving the vector to a nearby point it will change due to a change in
its components and also a change in the basis vectors,
d
V = =
e i dV
i
+ V
j d e j .
(5.48)
As we discussed previously the vector spaces associated with different points in a
Riemann space are ab initio independent. As such it is necessary to postulate a way
to relate them. This leads to the idea of vector transplantation and the specific version
of transplantation called parallel displacement that we discussed in Sects. 5.1 and
5.3. We can think of this in the present abstract view as giving an effective change in
the coordinate basis, which we assume is a bilinear expression in the basis vectors
and the coordinate displacement; it is a rather compelling assumption. That is we
postulate
d e j =
i
k j
e i dx
k
.
(5.49)
73
Similarly for the x
2
= ϕ equation
¨
ϕ +
2
ρ
˙
ρ ˙
ϕ ⇔ 0 − ¨
ϕ +
2
i j ˙
x
i
˙
x
j
= 0.
(5.45)
From this there are only two equal nonzero connections with an upper index
2,
2
21 =
2
12 =
1
ρ
.
(5.46)
The ease of the technique is apparent, especially so since the metric is diagonal
and most of the connections are zero. It is often a large labor-saving technique.
In the rest of this book we will make frequent use of the technique in Example
5.3 for calculating the connections.
5.6 Affine Connections, Abstract View
Let us see how we may motivate and interpret the coefficients of affine connection
using the abstract view introduced in Sect. 4.3. Recall that a vector may be expanded
in a coordinate basis, that is vectors aligned along the coordinate axes, according to
V = V
j
e j ,
e j = coordinate basis.
(5.47)
If we think of moving the vector to a nearby point it will change due to a change in
its components and also a change in the basis vectors,
d
V = =
e i dV
i
+ V
j d e j .
(5.48)
As we discussed previously the vector spaces associated with different points in a
Riemann space are ab initio independent. As such it is necessary to postulate a way
to relate them. This leads to the idea of vector transplantation and the specific version
of transplantation called parallel displacement that we discussed in Sects. 5.1 and
5.3. We can think of this in the present abstract view as giving an effective change in
the coordinate basis, which we assume is a bilinear expression in the basis vectors
and the coordinate displacement; it is a rather compelling assumption. That is we
postulate
d e j =
i
k j
e i dx
k
.
(5.49)