Howard P. Robertson: Space

Howard P. Robertson was American mathematician and physicist. Explore interesting quotes on space.
Howard P. Robertson: 56   quotes 0   likes

“[T]he space constant K… "curvature" 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

Geometry as a Branch of Physics (1949)

“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

Geometry as a Branch of Physics (1949)

Nature

“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 curvature of the surface into the third dimension—whence it derives its name…”

Howard P. Robertson

Geometry as a Branch of Physics (1949)

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

Howard P. Robertson

Geometry as a Branch of Physics (1949)

“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

Footnote
Geometry as a Branch of Physics (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 cosmological constant.”

Howard P. Robertson

"On Relativistic Cosmology" (1928)

Time

“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

Geometry as a Branch of Physics (1949)

Appearance

“[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

Geometry as a Branch of Physics (1949)

“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

Geometry as a Branch of Physics (1949)

“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

Geometry as a Branch of Physics (1949)

Search , Light , Distant

“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

Geometry as a Branch of Physics (1949)

“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”

Howard P. Robertson

1 - \frac{Kr^2}{12} + …
Geometry as a Branch of Physics (1949)

“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.”

Howard P. Robertson

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...
Geometry as a Branch of Physics (1949)

Making love , Search