Abstract

A ray of light is propagated in the XY plane and its angle with the X-axis is determined by two observers, S and S′, who are in relative motion to each other along the X-axis at a velocity q (expressed as a fraction of the velocity of light). If the angles measured by them are δ and δ′, then it is proved that $\text{tan}\frac{1}{2}\delta /\text{tan}\frac{1}{2}{\delta }^{\prime }=\text{cos}(45°+\frac{1}{2}\alpha )/\text{cos}(45°-\frac{1}{2}\alpha )$, where Sin α=q. This relationship is interpreted geometrically by means of a cone in oblique coordinates. The axes of coordinates are X, Y, and time T. The T and X axes are different for the two observers, and the four axes T, X, T′, X′, form a Lorentzian Plane to which Y is perpendicular. The properties of the cone are considered in detail and a mechanical model of the cone, built of wood and iron, is described. At a desired velocity q, one can compare directly the angles δ and δ′. It is also shown that the relationship between the angles δ, δ′, and α can be represented on a pyramid. References are made to other articles by the same writer, in which oblique axes are also used for the solution of problems in the theory of relativity.

© 1924 Optical Society of America

Straight-Line Relativity in Oblique Coordinates; also Illustrated by a Mechanical Model†

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