APPENDIX TO CHAPTER IIILetandbe two consentient sets of the Newtonian group. Let () be the rectangular axis system in the space of, andbe the rectangular axis system in the space of.First consider the traditional theory of relativity. Then the time-system is independent of the consentient set of reference.fig03Fig. 3.At the timelet the event-particle which instantaneously happens at the pointin the space of a happen atin the space of, and let the event-particle which happens atin the space ofhappen atin the space of. Let the axisbe in the direction of the motion ofin the-space, and the axisbe in the direction reversed of the motion ofin the-space. Also letbe so chosen thatlies on. Then the event-particles at the instantwhich happen onare the event-particles which happen at the instanton. Also we chooseandso that the event-particles which happen at timeonandrespectively happen on straight lines in the-space which are parallel toand. Letbe the velocity ofin-space andbe the velocity of a in-space. Then (with a suitable origin of time)These are the 'Newtonian' formulae for relative motion.Secondly consider the Lorentzian [or 'electromagnetic'] theory of relativity. The two time-systems for reference toand for reference torespectively are not identical. Letbe the measure of the lapse of time in the-system, andbe the measure of the lapse of time in the-system. The distinction between the two time-systems is embodied in the fact that event-particles which happen simultaneously at timein-space do not happen simultaneously throughout space. Thus supposing that an event-particle happens at () in-space and-time and at () in-space and-time, we seek for the formulae which are to replace equations (1) of the Newtonian theory.As before letlie in the direction of the motion ofin, andin the reverse direction of the motion ofin. Also letlie on, so that event-particles which happen onalso happen, on. One connection between the two time-systems is secured by the rule that event-particles which happen simultaneously at points in-space on a plane perpendicular toalso happen simultaneously at points in-space on a plane perpendicular to. Accordingly the quasi-parallelism ofto, and ofto, is defined and secured in the same way as for Newtonian relativity.The same meaning as above will be given toand; alsois the fundamental velocity which is the velocity of lightin vacuo. Then we defineThe formulae for transformation areThese formulae are symmetrical as betweenand, so thatIt is evident that whenis small,and whenandare not too largeThus the formulae reduce to the Newtonian type.Let,,stand for, etc., and,,, for, etc. Then it follows immediately from the preceding formulae thatWith the notation ofAppendix IItoChapter II, the formulae of transformation for Maxwell's equations arewhere () is the velocity of the charge at () at the time.Also it immediately follows from formulae (5) thatHencevanish together. This proves Einstein's theorem on the invariance of the velocity, so far as concerns the sufficiency of the Lorentzian formulae to produce that result.
Letandbe two consentient sets of the Newtonian group. Let () be the rectangular axis system in the space of, andbe the rectangular axis system in the space of.
First consider the traditional theory of relativity. Then the time-system is independent of the consentient set of reference.
fig03Fig. 3.
Fig. 3.
Fig. 3.
At the timelet the event-particle which instantaneously happens at the pointin the space of a happen atin the space of, and let the event-particle which happens atin the space ofhappen atin the space of. Let the axisbe in the direction of the motion ofin the-space, and the axisbe in the direction reversed of the motion ofin the-space. Also letbe so chosen thatlies on. Then the event-particles at the instantwhich happen onare the event-particles which happen at the instanton. Also we chooseandso that the event-particles which happen at timeonandrespectively happen on straight lines in the-space which are parallel toand. Letbe the velocity ofin-space andbe the velocity of a in-space. Then (with a suitable origin of time)These are the 'Newtonian' formulae for relative motion.
Secondly consider the Lorentzian [or 'electromagnetic'] theory of relativity. The two time-systems for reference toand for reference torespectively are not identical. Letbe the measure of the lapse of time in the-system, andbe the measure of the lapse of time in the-system. The distinction between the two time-systems is embodied in the fact that event-particles which happen simultaneously at timein-space do not happen simultaneously throughout space. Thus supposing that an event-particle happens at () in-space and-time and at () in-space and-time, we seek for the formulae which are to replace equations (1) of the Newtonian theory.
As before letlie in the direction of the motion ofin, andin the reverse direction of the motion ofin. Also letlie on, so that event-particles which happen onalso happen, on. One connection between the two time-systems is secured by the rule that event-particles which happen simultaneously at points in-space on a plane perpendicular toalso happen simultaneously at points in-space on a plane perpendicular to. Accordingly the quasi-parallelism ofto, and ofto, is defined and secured in the same way as for Newtonian relativity.
The same meaning as above will be given toand; alsois the fundamental velocity which is the velocity of lightin vacuo. Then we define
The formulae for transformation are
These formulae are symmetrical as betweenand, so that
It is evident that whenis small,and whenandare not too large
Thus the formulae reduce to the Newtonian type.
Let,,stand for, etc., and,,, for, etc. Then it follows immediately from the preceding formulae that
With the notation ofAppendix IItoChapter II, the formulae of transformation for Maxwell's equations arewhere () is the velocity of the charge at () at the time.
Also it immediately follows from formulae (5) that
Hencevanish together. This proves Einstein's theorem on the invariance of the velocity, so far as concerns the sufficiency of the Lorentzian formulae to produce that result.