This preview shows page 1. Sign up to view the full content.
Unformatted text preview: the metric for the surface of a sphere parameterized by a distance from a
given “north pole”, r, and an angle from a given “prime meridian”, θ, by:
ds2 = dr2 + R2 sin2 (r/R)dθ2 (1) where ds is the distance between two points (r, θ) and (r + dr, θ + dθ). If we choose
the coordinates such that the origin is the north pole and the object spans from (r, 0)
to (r, dθ). Therefore the width of this object (distance between its endpoints) can be
expressed with equation (1) where dr = 0 and we can solve for dθ.
dθ = ds
Rsin(r/R) (2) As r → π R, the denominator in (2) goes to zero so our angle dθ blows up. This is
because all geodesics from the north pole will meet at the south pole, no matter their 2 initial angular separation, meaning that the angular size of anything at the opposite end
of the sphere will have inﬁnite angular size; you will see it no matter which direction you
look. Problem 5 (Ryden 3.5)
(see attached scan) 3...
View Full Document