1. Spherical Geodesic
May 2, 2011
By the orthogonality of spherical polar coordinates; ηµν = (∂µ )2 = ηµ .
2
= ηµ 2
µ
µ
Take φ as the inclination angle, and θ as the azimuthal angle.

 = sin φ cos θ

= sin φ sin θ

= cos φ

So we have,

η = sin2 φ cos2 θ + sin2 φ sin2 θ + cos2 φ = 1

η = 2 cos2 φ cos2 θ + 2 cos2 φ sin2 θ + 2 sin2 φ = 2
 φ
ηθ = 2 sin2 θ sin2 φ + 2 sin2 φ cos2 θ = 2 sin2 φ

And therefor,
φ
2
2 2 φ2 + 2 sin2 φ θ 2 = θ + sin2 φ
θ
= +
Note that vanishes, as it is invariant with respect to θ. And now, for any
path from 1 → 2 on the surface of a sphere;
2 θ2
φ
2
{λ} = θ + sin2 φ
θ
=
1 θ1
which gives the arclength of a path φ(θ). The geodesic is said to be an ex-
φ φ
2
tremum of the action defined above. The function λ θ φ(θ) = θ + sin2 φ
will be a critical path if and only if it satisfies the Euler-Lagrange equation;
∂λ ∂λ
−
∂φ θ ∂( φ )
=0
θ
1

2. And now a slew of computation that I have checked 4 or 5 times now.
( φ ) +sin2 φ sin φ cos φ ( φ ) sin φ cos φ+sin3 φ cos φ
2 2
φ
∂λ
+ sin2 φ = sin φ2cos φ2 =
2
θ
∂φ = ∂φ θ φ
= θ
( φ ) +sin2 φ ( φ ) +sin2 φ
3/2 3/2
( θ ) +sin φ
2 2
θ θ
φ
∂λ ∂ φ
2
∂( φ )
= ∂( φ θ + sin2 φ = φ 2
θ
θ θ) ( ) θ +sin2 φ
φ φ 2φ
+sin φ cos φ φ
2φ
2φ
( φ) +sin2 φ
2
φ
∂λ θ θ θ2 θ θ2 θ
θ ∂( φ ) = θ φ 2
θ
= θ2
φ 2
− φ 2
= −
( φ)
3/2 3/2
φ φ +sin2 φ +sin2 φ
2
θ +sin2 +sin2
θ
( ) ( ) θ ( ) θ θ
φ φ 2φ
θ θ +sin φ cos φ φθ
θ2
φ 2
3/2
( )θ +sin2 φ
2φ
sin2 φ−sin φ cos φ φ
2
∂λ θ2 θ ( )
θ ∂( φ ) = φ
3/2
+sin2 φ
2
θ ( ) θ
The Euler-Lagrange equation becomes;
φ φ φ
2 2
sin φ cos φ + sin3 φ cos φ − sin2 φ + sin φ cos φ
2
θ θ2 θ
3/2
=0
φ
2
θ + sin φ 2
φ φ
2 2
sin φ cos φ + sin3 φ cos φ − sin2 φ = 0
θ θ2
=⇒ 2
φ φ
2 2
cos φ − sin φ = − sin2 φ cos φ
θ θ2
2
φ φ
2 2
−2 cot φ = sin φ cos φ
θ2 θ
How in the fuck do I solve this? It is clear that a constant solution of the form
π
φ = π ( N) or φ = 2 ( odd) is valid, and they do correspond to circles on
the surface of the sphere, but they do not necessarilly need to share the origin of
the surface.
Maybe constraining paths only to the paramterization φ → φ(θ) is not general
enough. Maybe both phi and theta need to be parameterized appropriately.
2