An "empty world," i.e., a homogeneous manifold at all points at which equations (1) are satisfied, has, according to the theory, a constant Riemann c… - Howard P. Robertson

The general theory of relativity considers physical space-time as a four-dimensional manifold whose line element coefficients <math>g_{\mu \nu}</math> satisfy the differential equations<math>G_{\mu \nu} = \lambda g_{\mu \nu} \qquad .\;.\;.\;.\;.\;.\; (1)</math>in all regions free from matter and electromagnetic field, where <math>G_{\mu \nu}</math> is the contracted Riemann-Christoffel tensor associated with the fundamental tensor <math>g_{\mu \nu}</math>, and <math>\lambda</math> is the .

Euclidean geometry is only one of several congruence geometries... Each of these geometries is characterized by a real number <math>K</math>, which for Euclidean geometry is 0, for the hyperbolic negative, and for the spherical and elliptic geometries, positive. In the case of 2-dimensional congruence spaces... <math>K</math> may be interpreted as the of the surface into the third dimension—whence it derives its name...

All the light which is radiated... will, after it has traveled a distance <math>r</math>, lie on the surface of a sphere whose area <math>S</math> is given by the first of the formulae (3). And since the practical procedure... in determining <math>d</math> is equivalent to assuming that all this light lies on the surface of a Euclidean sphere of radius <math>d</math>, it follows...<math>4 \pi d^2 = S = 4 \pi r^2 (1 - \frac{K r^2}{3} + ...);</math>whence, to our approximation 4)<math>d = r (1- \frac{K r^2}{6} + ...),</math> or
<math>r = d (1 + \frac{K d^2}{6} + ...).</math>