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It is shown that the fundamental space-time-velocity relations of Einstein’s original restricted theory of relativity, when the observed point, line, or ray of light moves in the direction of the observers themselves, can be represented quantitatively and visualized by using two sets of oblique time-space coordinates, forming the Lorentzian plane. For the observer moving in the positive direction at the relative velocity q, the angle between the coordinate axes is 90° − α, for the observer moving in the negative direction the angle is 90° + α, where sin α = q/c; α has been named the velocity angle.
The concept of intrinsic coordinates of an event is introduced, and certain simple phenomena are expressed in these coordinates. A spatium is defined as a vector which can be differently resolved into space and time components by different observers. The general method is illustrated by showing graphically the Fitzgerald contraction of a length, the slowing down of a clock, the fact that two velocities always add to one less than that of light, the Doppler effect, reflection from a moving mirror, etc.
A simple mechanical model is described which conveys some of the foregoing relations directly to the eye and which can be used for demonstrations before a large audience.
| Terms and Notation used. | |
|---|---|
| a, a′ | acceleration of a moving point |
| c | velocity of light |
| Eo | energy added to a system of moving particles, Eq. (66) |
| h | intrinsic spatium of a moving line* |
| i | used as a subscript means incidence; Fig. 10 |
| l, l′ | lengths |
| MN | in Fig. 6, a spatium, that is, a vector which may be resolved into a length and a time interval* |
| OB, OBn | universal bisectors, positive and negative, Figs. 3 and 7* |
| OH | Fig. 7, space-time line or curve |
| q | relative velocity between S and S′ |
| r | spatium vector of a point in the Lorentzian plane, Fig. 3* |
| r | used as a subscript means rebound or reflection |
| S, S′ | two systems or observers moving at a relative velocity q with respect to each other; Fig. 1 |
| T, T′ | time intervals |
| t, t′ | ordinates along the time axes |
| v, v′ | velocities of a point |
| XX′OTT′ | Lorentzian plane, Fig. 3* |
| x, x′ | lengths plotted as abscissae |
| α | velocity angle, Eq. (3) and Fig. 3* |
| γ, γ′ | coordinate angles, between the coordinate axes and the universal bisector OB, Eqs. (4) and (5); Fig. 3* |
| θ | orientation angle of a space-time line, Fig. 7* |
| λ, λ′ | wave lengths of light |
| μ, μ′ | values of the refraction coefficient |
| ν, ν′ | frequencies of light |
| ϕ | orientation angle of a spatium vector, Fig. 3* |
© 1926 Optical Society of America
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