Consider a 2-sphere of radius R parametrized by the 2 spherical polar coordinates θ and φ. Write down the standard metric in these coordinates.
1. Show that lines of constant longitude are geodesics, and that the only line of constant latitude is the equator.
2. How does a tangent vector transform if it is parallel-transported around a line of latitude?
i don't know how to make equations in here. this is going to be difficult. i used the equation for the christoffel connection, to find the christoffel components in these coordinates. now i am trying to prove that the longitude and latitude lines satisfy the geodesic equation, which are basically (i'll try):
((d^2)x^mu)/(dlambda)^2 + (GAMMA^rho)_munu((dx^mu)/dlambda)((dx^nu)/dlambda) = 0.
that is probably incomprehensible.
The Attempt at a Solution
ok so i wrote down the std metric which is
ds^2 = (R^2)(dθ)^2 + (R^2)(sin^2(θ))(dφ)^2
then i calculated the christoffel components, using the equation, and they came out to be
(GAMMA^θ)_φφ = sinθcosθ
(GAMMA^φ)_θφ = (GAMMA^φ)_φθ = cot(θ)
and all the others are zero.
now i don't know much about lines of longitude and latitude except that for longitude are constant φ and latitude are constant θ. so i was working with that to try to show that the geodesic equations are satisfied by longitudes, but not latitudes unless θ = pi/2. now i got the geodesic equations to be:
(d^2)θ/(dlambda)^2 + sinθcosθ(dφ/dlambda)^2 = 0
(d^2)φ/(dlambda)^2 + 2cotθ(dθ/dlambda)(dφ/dlambda) = 0
so for longitude with phi constant, all the dφs are zero, so the second equation is automatically satisfied and the first we are left with
(d^2)θ/(dlambda)^2 = 0.
i said that this was okay because on longitudinal lines θ is changing at a constant rate so (d^2)(θ)/(dlambda)^2 is 0??? knowing the equations for latitude and longitude lines in terms of θ and φ would really help.
so then it was on to latitude which is where i ran in to the problem. the dθs vanish, so we're left with
sin(θ)cos(θ)(dφ/dlambda)^2 = 0
(d^2)φ/(dlambda)^2 = 0
well if i use the same argument as i did for longitude, φ changes at constant rate so the second derivative should vanish and i get geodesics for all latitude lines, which is just wrong. alternatively, i can't figure out why it's only θ = pi/2 that satisfies the geodesic - i mean i know that it's true but i can't figure it out mathematically. when first envisioning the solution it was great because i thought i'd have to make both sinθcosθ and cotθ vanish, which would mean θ would have to be pi/2, but cotθ vanishes already because of the dθ vanishing. so i'm wrong somewhere but i don't know where please helppppppppppp
i haven't even begun to think about the second part i just want to understand the first part first. please help please help.
oh it would also be nice to understand how to write equations in here. thanks!
wait do i really have to do it by attachment???