Howard P. Robertson Quotes

The value of the intrinsic approach is especially apparent in considering 3-dimensional congruence spaces... The intrinsic geometry of such a space of curvature <math>K</math> provides formulae for the surface area <math>S</math> and the volume <math>V</math> of a "small sphere" of radius <math>r</math>, whose leading terms are 3)<math>S = 4 \pi r^2 (1 - \frac{Kr^2}{3} + ...)</math>,
<math>V = \frac{4}{3} \pi r^3 (1 - \frac{Kr^2}{5} + ...)</math>.

[W]e propose... to deal exclusively with properties intrinsic to the space... measured within the space itself... in terms of... inner properties.

[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.

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.

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.

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.

This... is an outrageously over-simplified account of the assumptions and procedures...

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!

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>

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...

These formulae [in (1) and (2) above] may be shown to be valid for a circle or a triangle in the hyperbolic plane... for which <math>K < 0</math>. Accordingly here the perimeter and area of a circle are greater, and the sum of the three angles of a triangle are less, than the corresponding quantities in the Euclidean plane. It can also be shown that each full line is of infinite length, that through a given point outside a given line an infinity of full lines may be drawn which do not meet the given line (the two lines bounding the family are said to be "parallel" to the given line), and that two full lines which meet do so in but one point.

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 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.

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.

[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>.