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Howard P. Robertson

27 quotes

“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.”
Howard P. Robertson (1903–1961) first seen 1949 wikiquote, CC BY-SA 4.0 source
Recorded in: As quoted by Gerald James Whitrow, The Structure of the Universe: An Introduction to Cosmology (1949)
“The general theory of relativity considers physical space-time as a four-dimensional manifold whose line element coefficients g_{\mu \nu} satisfy the differential equationsG_{\mu \nu} = \lambda g_{\mu \nu} \qquad .\;.\;.\;.\;.\;.\; (1)in all regions free from matter and electromagnetic field, where G_{\mu \nu} is the contracted Riemann-Christoffel tensor associated with the fundamental tensor g_{\mu \nu}, and \lambda is the .”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“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.”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“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.”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“The solution of (1), which represents a homogeneous manifold, may be written in the form: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)where \kappa = \sqrt \frac{\lambda}{3}. If we consider \rho as determining distance from the origin... and \tau as measuring the proper-time of a clock at the origin, we are led to the de Sitter spherical world...”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“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.”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“[O]nly in a homogeneous and isotropic space can the traditional concept of a rigid body be maintained.”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“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.”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“Euclidean geometry is only one of several congruence geometries... Each of these geometries is characterized by a real number K, which for Euclidean geometry is 0, for the hyperbolic negative, and for the spherical and elliptic geometries, positive. In the case of 2-dimensional congruence spaces... K may be interpreted as the of the surface into the third dimension—whence it derives its name...”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“[W]e propose... to deal exclusively with properties intrinsic to the space... measured within the space itself... in terms of... inner properties.”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“Measurements which may be made on the surface of the earth... is an example of a 2-dimensional congruence space of positive curvature K = \frac{1}{R^2}... [C]onsider... a "small circle" of radius r (measured on the surface!)... its perimeter L and area A... are clearly less than the corresponding measures 2\pi r and \pi r^2... in the Euclidean plane. ...for sufficiently small r (i.e., small compared with R) these quantities on the sphere are given by 1):L = 2 \pi r (1 - \frac{Kr^2}{6} + ...), A = \pi r^2 (1 - \frac{Kr^2}{12} + ...)”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“In the sum \sigma 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)\sigma - \pi = K \deltawhere \delta is the area of the spherical triangle and the angles are measured in s (in which 180° = \pi [radians]). Further, each full line (great circle) is of finite length 2 \pi R, and any two full lines meet in two points—there are no parallels!”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“[T]he space constant K... "" may in principle at least be determined by measurement on the surface, without recourse to its embodiment in a higher dimensional space.”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“The value of the intrinsic approach is especially apparent in considering 3-dimensional congruence spaces... The intrinsic geometry of such a space of curvature K provides formulae for the surface area S and the volume V of a "small sphere" of radius r, whose leading terms are 3)S = 4 \pi r^2 (1 - \frac{Kr^2}{3} + ...), V = \frac{4}{3} \pi r^3 (1 - \frac{Kr^2}{5} + ...).”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“In all these congruence geometries, except the Euclidean, there is at hand a natural unit of length R = \frac{1}{K^\frac{1}{2}}; this length we shall, without prejudice, call the "radius of curvature" of the space.”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“We have merely (!) to measure the volume V of a sphere of radius r or the sum \sigma of the angles of a triangle of measured are \delta, and from the results to compute the value of K.”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“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...”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“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.”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“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.”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“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.”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“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" d... and the distance r which appears as an element of the geometry.”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“All the light which is radiated... will, after it has traveled a distance r, lie on the surface of a sphere whose area S is given by the first of the formulae (3). And since the practical procedure... in determining d is equivalent to assuming that all this light lies on the surface of a Euclidean sphere of radius d, it follows...4 \pi d^2 = S = 4 \pi r^2 (1 - \frac{K r^2}{3} + ...);whence, to our approximation 4)d = r (1- \frac{K r^2}{6} + ...), or r = d (1 + \frac{K d^2}{6} + ...).”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“[T]he astronomical data give the number N of nebulae counted out to a given inferred "distance" d, and in order to determine the curvature... we must express N, or equivalently V, to which it is assumed proportional, in terms of d. ...from the second of formulae (3) and... (4)... to the approximation here adopted, 5)V = \frac{4}{3} \pi d^2 (1 + \frac{3}{10} K d^2 + ...);...plotting N against... d and comparing... with the formula (5), it should be possible operationally to determine the "curvature" K.”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“This... is an outrageously over-simplified account of the assumptions and procedures...”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“The search for the curvature K indicates that, after making all known corrections, the number N seems to increase faster with d than the third power, which would be expected in a Euclidean space, hence K is positive. The space implied thereby is therefore bounded, of finite total volume, and of a present "radius of curvature" R = \frac{1}{K^\frac{1}{2}} 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...”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“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 K . 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.”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source
“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 K... the kind of structure exhibited... in the neighborhood of a ".”
Howard P. Robertson (1903–1961) said by 1961 at the latest wikiquote, CC BY-SA 4.0 source