Howard P. Robertson: Measure

Howard P. Robertson was American mathematician and physicist. Explore interesting quotes on measure.
Howard P. Robertson: 56 quotes0 likes

“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

"On Relativistic Cosmology" (1928)

“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

Geometry as a Branch of Physics (1949)

“[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)

“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 "saddle point."”

Howard P. Robertson

Geometry as a Branch of Physics (1949)

“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

Geometry as a Branch of Physics (1949)

“[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)

“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)