A cone of half-angle has the metric with

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch The metric, the connection and geodesics

Problem

A cone of half-angle

\beta

has the metric

\mathrm{d}s^2=\mathrm{d}r^2+k^2r^2\,\mathrm{d}\phi^2

with

k=\sin\beta

, where

r

is the slant distance from the apex and

\phi

the angle round the axis. Its non-zero Christoffel symbols are

\Gamma^r{}_{\phi\phi}=-k^2r

and

\Gamma^\phi{}_{r\phi}=\Gamma^\phi{}_{\phi r}=1/r

. A vector is carried by parallel transport once round the circle

r=r_0

. ((a)) Write the transport equations for

V^r(\phi)

and

V^\phi(\phi)

along the circle. Show that the components

A=V^r

and

B=kr_0V^\phi

along unit vectors rotate at the constant rate

k

per unit of

\phi

, and find the turn of the vector against the local directions after one circuit. ((b)) The cone that touches a sphere of radius

a

along the latitude circle at colatitude

\theta_0

has

\beta=90^\circ-\theta_0

and

r_0=a\tan\theta_0

. Show that on that circle its two Christoffel symbols equal the sphere's, written with the meridian arc length

a\theta

in place of

\theta

, and compare the two turns. ((c)) Evaluate the turn, in degrees, for the cone that touches the Earth along the parallel of Paris, latitude

48.85^\circ\mathrm{N}

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