Howard P. Robertson

"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 ."

"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."

"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."

"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 Lemaîtres primeval-atom hypothesis."

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

"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."

"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 showed that the uncertainty relation follows from the commutation rule."

"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."

"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..."

"Many of the theoretical investigations of cosmology in the 1920s were examinations of De Sitters 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 Sitters 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)."

"[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)."