A two-sphere of radius carries two charts

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

A two-sphere of radius

a

carries two charts. Chart 1 is the usual spherical chart

(\theta,\phi)

with polar axis along

z

x=a\sin\theta\cos\phi

y=a\sin\theta\sin\phi

z=a\cos\theta

, undefined at

\theta=0,\pi

. Chart 2 uses a colatitude

\theta'

and azimuth

\phi'

measured from the

x

-axis instead:

x=a\cos\theta'

y=a\sin\theta'\cos\phi'

z=a\sin\theta'\sin\phi'

, undefined at the two points on the

x

-axis. ((a)) Write the transition map giving

\theta'

and

\phi'

in terms of

\theta,\phi

, valid on the overlap (points on neither the

z

-axis nor the

x

-axis). ((b)) Evaluate

(\theta',\phi')

at the point

\theta=90^\circ

\phi=90^\circ

. ((c)) Write the metric in chart 2 and state its signature at that point. Compare with the signature read off from the chart-1 metric at the same point, and say what this illustrates about signature and coordinate choice.

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