Annu. Rev. Astron. Astrophys. 1988. 26:
561-630
Copyright © 1988 by . All rights reserved |

**2.3. Line Lengths and Areas on a Sphere of Constant Curvature**

Experimental geometry can be illustrated by showing how the radius (of
curvature) of a sphere can be found by measurements of line lengths,
areas, and angles made entirely *on its surface*. The curvature
*K* = 1/*R*_{1}*R*_{2} is the product of
the reciprocals of the radii of the two
osculating circles to the geodesics drawn on the surface at any
particular point P, put in the directions of maximum and minimum
descent. Examples of surfaces of constant curvature are the sphere
(where *K* is positive) and the pseudo-sphere (where *K* is
negative).

Consider the experimental determination of the radius *R*
(i.e. *K*^{-1/2})
of a sphere found by measuring lengths, areas, or angles on its surface.
From any point P on the surface, proceed a distance *r* from P and
draw a
circle about P of radius *r* (*along the surface*). The length
of this circle is

(2) |

which for *r* small compared with *R* is, to second order,

(3) |

This differs from 2*r*
for a Euclidean plane (*K* = 0), permitting a
determination of *R* once *l* and *r* are measured *on
the surface itself!*

In a similar way, the areas of a spherical cap drawn about a point P
with radius *r* along the surface is

(4) |

For small *r*/*R*, Equation 4 can be expanded to

(5) |

which again differs from
*r*^{2} for a
space of zero curvature.

The deviation from Euclidean geometry is small. At the enormous
distance of *r* = *R*, along the surface, Equation 4 shows
that the area
is 0.92*r*^{2},
differing from the Euclidean case by only 8%. This
special case illustrates the general proposition that one must sample a very
large fraction (i.e. of the order of curvature radius *R*) of any
non-Euclidean space before deviations from the geometry of the
Euclidean tangent. space become measurable.

Besides lines and areas, the sum of the angles of triangles placed on the surface also measures the curvature. It can be shown that the difference of the angle sum from 180° for any triangle is the curvature times the area of the triangle, i.e.

(6) |

where ,
, and
are the interior angles
of the triangle, and *K* is
the curvature of the surface at the triangle. The special case of a
hemisphere illustrates the theorem. The area of the hemisphere is
2*R*^{2}. The sum
of the angles of the spherical triangle that forms the hemisphere is
2 +
/2 +
/2 =
3. The angular excess of
3 -
divided by
the area is *R*^{-2}, which is the curvature *K* as
stated by Equation 6. It can be shown that this equation holds for any
surface of constant Gaussian curvature.

Note that Equations 2, 4, and 6, which measure different aspects of
the geometry (lines, areas, and angles), all contain the common term
*R*^{-2} = *K*. A case for the reality of space
curvature would be strong if
the measurements of quite a different nature that are required to test
each of the three equations would give the same value of
*R*^{-2}. In a
similar way in cosmology, some confidence in the value of the
geometrical parameter *q*_{0}, which is related to the
curvature via
Equation 1, would obtain if multiple experiments of different kinds
gave the same result. This congruence of answers is the goal of the
observational quest.

Now to the details of the standard model.