Notes on Relativity and Cosmology - Physics Department, UCSB

226 CHAPTER 8. GENERAL RELATIVITY AND CURVED SPACETIME
more like a sphere, and for which is it more like the Lobachevskian
geometry?
b) By using a different set of coordinates (ρ, θ), instead of (r, θ), I can
write these geometries in a different way. The angle θ still goes from 0
to 2π when you go around the circle once.
The two metrics below represent the same two geometries as the two
metrics in part A. Which is which? Hint: Compute the radius R of a
circle with ρ = constant around the origin. Remember that the radius
is the actual distance of the circle from the center, not just the value of
the radial coordinate. This means that it is given by the actual length
of an appropriate line. If the circumference is C, what is the radius R?
i) ds 2 = a 2 dρ2
a 2 −ρ 2 + ρ 2 dθ 2 .
ii) ds 2 = a 2 dρ2
a 2 +ρ 2 + ρ 2 dθ 2 .
c) Show that the geometries in (2b) are the same as the geometries in
(2a), but just expressed in terms of different coordinates. To do so,
rewrite metrics (2(b)i) and (2(b)ii)) in terms of the distance R from
the origin to the circle at constant ρ and then compare them with
(2(a)i) and (2(a)ii). Remember that dρ =
(
dρ
dR
)
dR =
(
dR
dρ
) −1
dR.