Howard P. Robertson Quotes

[O]nly in a homogeneous and isotropic space can the traditional concept of a rigid body be maintained.

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!

What is needed is a homely experiment which could be carried out in the basement with parts from an old sewing machine and an Ingersoll watch, with an old file of Popular Mechanics standing by for reference! This I am, alas, afraid we have not achieved, but I do believe that the following example... is adequate to expose the principles...

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.

The search for the curvature <math>K</math> indicates that, after making all known corrections, the number N seems to increase faster with <math>d</math> than the third power, which would be expected in a Euclidean space, hence <math>K</math> is positive. The space implied thereby is therefore bounded, of finite total volume, and of a present "radius of curvature" <math>R = \frac{1}{K^\frac{1}{2}}</math> which is found to be of the order of 500 million light years. Other observations, on the "red shift" of light from these distant objects, enable us to conclude with perhaps more assurance that this radius is increasing...

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.

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.

The solution of (1), which represents a homogeneous manifold, may be written in the form:<math>ds^2 = \frac{d\rho^2}{1 - \kappa^2\rho^2} - \rho^2 (d\theta^2 + sin^2 \theta \; d\phi^2) + (1 - \kappa^2 \rho^2)\; c^2 d\tau^2, \qquad (2)</math>where <math>\kappa = \sqrt \frac{\lambda}{3}</math>. If we consider <math>\rho</math> as determining distance from the origin... and <math>\tau</math> as measuring the proper-time of a clock at the origin, we are led to the de Sitter spherical world...

[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>.

We should, of course, expect that any universe which expands without limit will approach the empty de Sitter case, and that its ultimate fate is a state in which each physical unit—perhaps each nebula or intimate group of nebulae—is the only thing which exists within its own observable universe.

Now it is the practice of astronomers to assume that brightness falls off inversely with the square of the "distance" of an object—as it would do in Euclidean space, if there were no absorption... We must therefore examine the relation between this astronomer's "distance" <math>d</math>... and the distance <math>r</math> which appears as an element of the geometry.

The field equation may... be given a geometrical foundation, at least to a first approximation, by replacing it with the requirement that the mean curvature of the space vanish at any point at which no heat is being applied to the medium—in complete analogy with... the general theory of relativity by which classical field equations are replaced by the requirement that the Ricci contracted curvature tensor vanish.

[T]he space constant <math>K</math>... "" may in principle at least be determined by measurement on the surface, without recourse to its embodiment in a higher dimensional space.

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.