This fact should be plausible, and can be proved by a straightforward
calculation as follows.

Parametrize the 2-sphere
*S*^{2} of radius 1 by
(longitude,co-latitude). Then the
usual metric is

Parametrize *C*= the cone on *S*^{2} by
where *r* is the
distance from the cone point. Then the usual metric on *C* (with the cone
point at distance *k* from the sphere) is

A curve
will have coordinates
for *t* in [0,1]. If the curve is differentiable,
its length is

Suppose the curve joins
two points in the cone over the great circle ,
i.e. that
.
Then the curve
given
by ,
which lies entirely in the cone
over the great circle
,
will have
same endpoints as
and will be shorter, since all the terms in
the length integral were positive, and the term
will now be zero.

The path of a light ray is the curve of shortest length between its endpoints. So if is the path of a light ray between two points in the cone over a great circle, then must lie entirely in that cone.

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