The stereographic chart from the north pole of a sphere of radius assigns the plane…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Relativity & Gravitation · Differential geometry of curved spacetime

The Geometry of Gravity, Ch Manifolds, vectors, tensor fields and the metric

Problem

The stereographic chart from the north pole of a sphere of radius

a

assigns the plane radius

R_N=a\cot(\theta/2)

and azimuth

\phi

to the point

(\theta,\phi)

of the standard spherical chart. ((a)) Starting from

ds^2=a^2(d\theta^2+\sin^2\theta\,d\phi^2)

, use the chain rule to rewrite the metric in the coordinates

(R_N,\phi)

, i.e. translate its representation from the

(\theta,\phi)

chart to the stereographic chart. ((b)) Evaluate

g_{R_NR_N}

and

g_{\phi\phi}

\theta=60^\circ

(where

R_N=a\sqrt3

), and state the signature there. ((c)) Explain why the position dependence of the conformal factor in

(R_N,\phi)

never makes the metric degenerate for any finite

R_N>0

, even though the components change from point to point, and say what happens at

R_N=0

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