Howard P. Robertson

"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."

"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 ."

"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."

"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."

"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..."

"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."

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

"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."

"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..."

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

"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} + ...)"

"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!"

"[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."

"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 < 0. 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."

"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} + ...)."

"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."

"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."

"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..."

"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 ".""

"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."

"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 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."

"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."

"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} + ...)."

"[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."

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

"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..."

"It was the work of... Friedmann, Robertson and Walker, which resulted in the general mathematical framework that is still used today when discussing relativistic cosmological models of a homogeneous and isotropic universe."

"[I]nvestigations of cosmological issues, whose only observational basis was the observations. ...involved ...mostly of mathematicians like Howard Robertson and and astronomers with strong mathematical training such as Eddington, Lemaître, George McVittie, and William McCrea. They were mainly interested in how to apply general relativity to cosmological problems, which involved not only understanding cosmic dynamics but also solving the intricate problem of interpreting cosmological solutions to the Einstein equations, in particular separating time (which determined the evolution of the universe) from space (to which simplified assumptions concerning the structure of the universe, such as homogeneity and , were to be applied)."

"Distinguished scientist, selfless servant of the national interest, courageous champion of the good and the right, warm human being, he gave richly to us and to all from his own great gifts. We are grateful for the years with him. We mourn the loss of his presence but rejoice in the legacy of his wisdom and strength."

"Weyl published a third appendix to his Raum, Zeit, Materie, and an accompanying paper], where he calculated the redshift for the ‘de Sitter cosmology’,ds^2 = -dt^2 + e^{2{\sqrt{\frac{\Lambda}{3}t}}} (dx^2+dy^2+dz^2),the explicit form of which would only be found later, independently by Lemaître and Robertson."

"The basic cause and nature of cosmic expansion, along with its recently-observed acceleration, are significant problems of the standard model; so, condisering the evidence that the acceleration is best described by pure \Lambda, there is strong motivation to search for an alternative big bang model that would respect the pioneering concept of expansion, as a direct consequence of the ‘de Sitter effect’ in the modified . It is therefore worth investigating the axiomatic basis of the Robertson-Walker (RW) line-element."

"[I]n deriving the general line-element for the background geometry of FLRW [Friedmann–Lemaître–Robertson–Walker, sometimes called the Standard Model] cosmology, Robertson required four basic assumptions: i. a congruence of s, ii. , iii. homogeneity, and iv. . i. and ii. are required to satisfy of a causal coherence amongst s in the entire Universe, by which every single event in the bundle of fundamental world lines is associated with a well-defined three-dimensional set of others with which it ‘really’ occurs simultaneously. However, it seems that ii. is therefore mostly required to satisfy the concept that synchronous events in a given inertial frame should have occurred simultaneously, against which I’ve argued..."

"Note.—The second part of this paper was considerably altered by me after the departure of Mr. Rosen for Russia since we had originally interpreted our results erroneously. I wish to thank my colleague Professor Robertson for his friendly in his assistance in the clarification of the original error."

"Important contributions to what would later appear as mainstream big bang cosmology were made by Americans Howard P. Robertson and Richard Tolman... Robertson (and independently, A. G. Walker in England) deduced in 1935 the most general form of the metric for a space-time satisfying the : the postulate that the universe is spatially homogeneous in its large scale appearance. This metric became generally known as the Robertson-Walker metric. Together with Tolman, Robertson pioneered the study of thermodynamics in the theory of the expanding universe."

"Many of the theoretical investigations of cosmology in the 1920s were examinations of De Sitter's model, which is particular because it can be understood both as a static model (as De Sitter did) and as an expanding model (as became the view after 1930). It is clearly problematic to read the pre-1930 literature in light of later knowledge. 'Expanding' versions of the De Sitter universe were found by in 1922, Hermann Weyl in 1923, and Lemaître in 1925, but were not conceived as expanding in any real sense. These works, as well as later works by Howard Percy Robertson and Richard Chase Tolman in the United States, consisted in transforming De Sitter's line element in such a way that it formally became static, that is, included a term F(t) referring to the time parameter. The metric, giving the distance in space-time between two neighboring points, would then be in the form ds^2 = c^2 dt^2 - F(t)(dx^2 = dy^2 +dz^2)."

"In 1928 Robertson found a non-static line element similar to the one Lemaître had found three years earlier. Also like Lemaître, he derived a linear relationship between apparent recessional velocities and distances, and he discussed it in relation to observation data. Within the same tradition was Tolman's 1929 derivation of a 'Hubble law', that is, a relationship of the form v = kr, with v = \frac{d\lambda}{\lambda}. Robertson and Tolman generalized the De Sitter model to an arbitrary scale factor F(t), but they remained within the static paradigm and did not realize the significance of F(t). In a paper of 1929, Robertson wrote the general line element of what would later be known as the Robertson-Walker models and he even referred to Friedmann's work. And yet, although he had evidently studied Friedmann, he 'misread' him and failed to realize the significance of the expanding metric."

"Robertson and Tolman developed much of the mathematics of the expanding universe, but without concluding or predicting prior to 1930 that the universe actually expands. They were not discoverers or codiscoverers of the expanding universe (and never claimed that they were)."

"Howard Percy Robertson was a postdoctorate student at Göttingen and Munich from 1925 to 1927. While in Göttingen he completed an important paper on relativistic cosmology in which he derived a relation between the velocity of nebulae and their distances. For the radius of the observable world he calculated R = 2 \times 10^{25} m. Although he had a velocity-distance relation and referred to Slipher's s, he did not conclude that the universe was in a state of expansion."

"Robertson wrote an influential [1933] review of relativistic world models in which he specifically excluded those which have "arisen in finite time from the singular state R = 0." Although he included the Einstein-de Sitter paper in his bibliography, he did not mention it in the review. He also did not mention Lemaître's primeval-atom hypothesis."

"Howard P. Robertson showed that the uncertainty relation follows from the commutation rule."

"As Daniel Dennefink explains in his article on the story behind the writings of the Einstein-Rosen gravitational paper, the singularity that Einstein and Rosen had encountered was an apparent singularity introduced by their choice of coordinate system, similar to the singularity one encounters when attempting to find the longitude of the North Pole. In fact, in his referee report Robertson had indicated that the singularity was removed by a change to a cylindrical coordinate system."

"In June 1936, Einstein and Rosen sent the paper... "Do Gravitational Waves Exist?" to The Physical Review... [which] rejected the paper, provoking Einstein's furious reaction. Einstein told the editor he... saw no reason to address the erroneous comments of his anonymous expert [Howard Percy Robertson] and... preferred to publish the paper elsewhere. ...Leopold Infeld arrived in Princeton to replace Rosen as... assistant. Einstein explained to him his proof of the non-existence of gravity waves. ...Infeld told Robertson [then professor of theoretical physics at Princeton] about Einstein's... paper[.] Robertson... found a trivial mistake [by] Infeld [and] clarified... the mistake in Einstein's explanation... The linearized approximation [led] to plane transverse gravitational waves... introduc[ing]... coordinate singularities... not real singularities. ...Robertson ...suggested ...the so-called Einstein-Rosen metric... be transformed... to cylindrical coordinates. ...the singularity can be regarded as describing a material source. The solution describe[s]...cylindrical... rather than plane gravitational waves. ...with Robertson's help (still not knowing it was Robertson who had [refereed] The Physical Review) ...Einstein ...revis[ed the] ...paper and added a section: "Rigorous Solution for Cylindrical Waves"... The new version of the paper was re-titled "On Gravitational Waves"..."