- #1
Whitehole
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Homework Statement
This is a problem from A. Zee's book EInstein Gravity in a Nutshell, problem I.5.5
Consider the metric ##ds^2 = dr^2 + (rh(r))^2dθ^2## with θ and θ + 2π identified. For h(r) = 1, this is flat space. Let h(0) = 1. Show that the curvature at the origin is positive or negative according to whether h(r) starts to turn downward or upward. Calculate the curvature for ##h(r) = \frac{sin(r)}{r}## and for ##h(r) = \frac{sinh(r)}{r}##
Homework Equations
The Attempt at a Solution
##ds^2 = dr^2 + (rh(r))^2dθ^2 = dr^2 + (r^2h(r)^2)dθ^2##
If I let r → 0 then h(r) → 1 but (r^2h(r)^2)dθ^2 → 0
How can I tell what would be the behavior of h(r) if it will already be gone if I let r → 0.
For ##h(r) = \frac{sin(r)}{r}## and ##h(r) = \frac{sinh(r)}{r}## I know that when I let r → 0 from calculus the answer would be 1 and -1 respectively, but I don't know what to do with the "show" part. Can anyone give me hints?