Notes on Relativity and Cosmology - Physics Department, UCSB

8.6. HOMEWORK PROBLEMS 221
8.6 Homework Problems
The following problems provides practice in working with curved spaces (i.e.,
not space-times) and the corresponding metrics.
8-1. In this problem, you will explore four effects associated with curvature on
each of 5 different spaces. The effects are subtle, so think about them
thoroughly, and be sure to read the instructions carefully at each stage.
The best way to work these problems is to actually build paper models
(say, for the cone and cylinder) and to try to hold a sphere and a funnel
shape in your hand while you work on those parts of this problem. I also
suggest that you review the discussion in section 8.2 before beginning this
problem.
The five spaces are:
i) Sphere ii) Cylinder iii) Cone iv) Double Funnel
and
v) This last one (see below) is a drawing by M.C. Escher. It is intended
to represent a mathematical space first constructed by Lobachevski. The
idea is that this is a ‘map’ of the space, and that all of the white fish
below (and, similarly, all of the black fish) are really the same size and
shape. They only look different because the geometry of the space being
described does not match the (flat) geometry of the paper. This is just
the phenomenon that we see all the time when we try to draw a flat map
of the round earth. Inevitably, the continents get distorted – Greenland
for example looks enormous on most maps of North America even though
it is actually rather small.