General Relativity: An Introduction for Physicists · Problem 2.4

Manifolds and coordinates — Hobson, Efstathiou & Lasenby

Physics · stereographic-projection-induced-metric

Problem

Consider the surface of a 2-surface embedded in three-dimensional Euclidean space. In a stereographic projection, one assigns coordinates

(\rho, \phi)

to each point on the surface of the sphere. The

\phi

-coordinate is the standard azimuthal polar angle. The

\rho

-coordinate of each point is obtained by drawing a straight line in three dimensions from the south pole of the sphere through the point in question and extending the line until it intersects the tangent plane to the north pole of the sphere; the

\rho

-coordinate is then the distance in the tangent plane from the north pole to the intersection point. Show that the line element for the surface of the sphere in these coordinates is

ds^2 = \frac{d\rho^2}{(1+\rho^2/a^2)^2} + \frac{\rho^2}{1+\rho^2/a^2}\,d\phi^2.

At what point(s) on the sphere are these coordinates degenerate? If instead one works in terms of the Cartesian coordinates

x

and

y

in the tangent plane at the north pole, what is the corresponding form of the line element? At what point(s) on the sphere are these new coordinates degenerate?

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