Top 15 Howard P. Robertson Quotes

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...

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

[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>.

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>

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.

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.

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.

Let a thin, flat metal plate be heated... so that the temperature T is not uniform... clamp or otherwise constrain the plate to keep it from buckling... [and] remain [reasonably] flat... Make simple geometric measurements... with a short metal rule, which has a certain coefficient of expansion c... What is the geometry of the plate as revealed by the results of those measurements? ...[T]he geometry will not turn out to be Euclidean, for the rule will expand more in the hotter regions... [T]he plate will seem to have a negative curvature <math>K</math>... the kind of structure exhibited... in the neighborhood of a "."

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 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>.

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.

[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.