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Measurements which may be made on the surface of the earth... is an example of a 2-dimensional congruence space of positive curvature <math>K = \frac… - Howard P. Robertson
" "Measurements which may be made on the surface of the earth... is an example of a 2-dimensional congruence space of positive curvature <math>K = \frac{1}{R^2}</math>... [C]onsider... a "small circle" of radius <math>r</math> (measured on the surface!)... its perimeter <math>L</math> and area <math>A</math>... are clearly less than the corresponding measures <math>2\pi r</math> and <math>\pi r^2</math>... in the Euclidean plane. ...for sufficiently small <math>r</math> (i.e., small compared with <math>R</math>) these quantities on the sphere are given by 1):<math>L = 2 \pi r (1 - \frac{Kr^2}{6} + ...)</math>,
<math>A = \pi r^2 (1 - \frac{Kr^2}{12} + ...)</math>
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About Howard P. Robertson
Howard Percy Robertson (January 27, 1903 – August 26, 1961) was an American mathematician and physicist known for contributions related to physical cosmology and the uncertainty principle. He was Professor of Mathematical Physics at the California Institute of Technology and Princeton University.
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Howard Percy Robertson
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H. P. Robertson
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[T]he astronomical data give the number N of nebulae counted out to a given inferred "distance" <math>d</math>, and in order to determine the curvature... we must express N, or equivalently <math>V</math>, to which it is assumed proportional, in terms of <math>d</math>. ...from the second of formulae (3) and... (4)... to the approximation here adopted, 5)<math>V = \frac{4}{3} \pi d^2 (1 + \frac{3}{10} K d^2 + ...);</math>...plotting N against... <math>d</math> and comparing... with the formula (5), it should be possible operationally to determine the "curvature" <math>K</math>.
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 .
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