72
5 Affine Connections and Geodesics
(ρ, ϕ) the line element and the corresponding T function are
ds
2
= dρ
2
+ ρ
2 dϕ
2
, T = ˙
ρ
2
+ ρ
2
˙
ϕ
2
.
(5.40)
From this we may get the Euler-Lagrange equations. For the ρ equation
∂ T
∂ ˙
ρ
= 2 ˙
ρ,
d
ds
∂ T
∂ ˙
ρ
= 2 ¨
ρ,
∂ T
∂ρ
= 2ρ ˙
ϕ
2
,
¨
ρ − ρ ˙
ϕ
2
= 0.
(5.41)
For the ϕ equation
∂ T
∂ ˙
ϕ
= 2ρ
2
˙
ϕ,
d
ds
∂ T
∂ ˙
ϕ
= 2ρ
2
¨
ϕ + 4ρ ˙
ρ ˙
ϕ,
∂ T
∂ϕ
= 0,
(5.42)
ρ
2
¨
ϕ + 2ρ ˙
ρ ˙
ϕ = 0, so ρ
2
˙
ϕ = const.
You should check that a ray, constant ϕ, is a solution to the last two equations.
There is a beautiful practical use for the two approaches to geodesics we have
just worked through. It is apparent that the connections could be tedious to calculate
from their definition since there may be a large number of them, 40 in the four
dimensions of relativity. However the relation we have just worked out between
the Euler-Lagrange equations and the geodesic equation provides a simple useful
shortcut. The geodesic equations may be written in both the Euler-Lagrange form
(5.36), which is often easy, and also the standard canonical form (5.39) containing
the connections. We need only compare the two to pick out the nonzero connections.
Example 5.3 illustrates this.
Example 5.3 To show how this shortcut works we will apply it to the case of
polar coordinates in Euclidean 2-space that we worked with above. We obtained
the Euler-Lagrange equations in the previous example, so we compare them
with the canonical form. For the x
1
= ρ equation in (5.42) and the canonical
form in (5.39)
¨
ρ − ρ ˙
ϕ
2
⇔ 0 − ¨
ρ +
1
i j ˙
x
i
˙
x
j
= 0.
(5.43)
From this it is apparent that only one of the connections with an upper index 1
is nonzero,
1
22 = −ρ.
(5.44)