Howard P. Robertson Quotes

We have merely (!) to measure the volume <math>V</math> of a sphere of radius <math>r</math> or the sum <math>\sigma</math> of the angles of a triangle of measured are <math>\delta</math>, and from the results to compute the value of <math>K</math>.

In all these congruence geometries, except the Euclidean, there is at hand a natural unit of length <math>R = \frac{1}{K^\frac{1}{2}}</math>; this length we shall, without prejudice, call the "radius of curvature" of the space.

In considerations involving the nature of the world as a whole the irregularities caused by the aggregation of matter into stars and stellar systems may be ignored; and if we further assume that the total matter in the world has but little effect on its macroscopic properties, we may consider them as being determined by the solution of an empty world.

In what respect... does the general theory of relativity differ...? The answer is: in its universality; the force of gravitation in the geometrical structure acts equally on all matter. There is here a close analogy between the gravitational mass M...(Sun) and the inertial mass m... (Earth) on the one hand, and the heat conduction k of the field (plate)... and the coefficient of expansion c... on the other. ...The success of the general relativity theory... is attributable to the fact that the gravitational and inertial masses of any body are... rigorously proportional for all matter.

In the sum <math>\sigma</math> of the three angles of a triangle (whose sides are arcs of s) is greater than two right angles [180°]; it can... be shown that this "spherical excess" is given by 2)<math>\sigma - \pi = K \delta</math>where <math>\delta</math> is the area of the spherical triangle and the angles are measured in s (in which 180° = <math>\pi</math> [radians]). Further, each full line (great circle) is of finite length <math>2 \pi R</math>, and any two full lines meet in two points—there are no parallels!

is a congruence geometry, or equivalently the space comprising its elements is homogeneous and isotropic; the intrinsic relations between... elements of a configuration are unaffected by the position or orientation of the configuration. ...[M]otions of are the familiar translations and rotations... made in proving the theorems of Euclid.

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 .

That the existence of these motions (the "axiom of free mobility") is a desideratum, if not... a necessity, for a geometry applicable to physical space, has been forcefully argued on a priori grounds by von Helmholtz, Whitehead, Russell and others; for only in a homogeneous and isotropic space can the traditional concept of a rigid body be maintained.

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>

What is the true geometry of the plate? ...Anyone examining the situation will prefer Poincaré's common-sense solution... to attribute it Euclidean geometry, and to consider the measured deviations... as due to the actions of a force (thermal stresses in the rule). ...On employing a brass rule in place of one of steel we would find that the local curvature is trebled—and an ideal rule (c = 0) would... lead to Euclidean geometry.

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>

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 curvature, and any deviation from this fundamental solution is to be directly attributed to the influence of matter or energy.