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An enquiry concerning the principles of natural knowledge cover

An enquiry concerning the principles of natural knowledge

Chapter 9: APPENDIX TO CHAPTER III
By Alfred North Whitehead
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About This Book

A systematic examination of how the fundamental scientific concepts of space and time arise from the simplest generalizations of perceptual experience. The text traces the roles of mathematics and measurement in shaping physical geometry, distinguishing abstract axiomatic systems from geometry rooted in empirical observation. Developments in contemporary physics, including the theory of relativity, are assessed for their implications while avoiding commitment to specific laws. Philosophical questions about the object of perceptual knowledge are separated from metaphysical accounts of the knower, and methodological puzzles are identified to guide further inquiry.

APPENDIX TO CHAPTER III

Let and be two consentient sets of the Newtonian group. Let () be the rectangular axis system in the space of , and be 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.


Fig. 3.

At the time let the event-particle which instantaneously happens at the point in the space of a happen at in the space of , and let the event-particle which happens at in the space of happen at in the space of . Let the axis be in the direction of the motion of in the -space, and the axis be in the direction reversed of the motion of in the -space. Also let be so chosen that lies on . Then the event-particles at the instant which happen on are the event-particles which happen at the instant on . Also we choose and so that the event-particles which happen at time on and respectively happen on straight lines in the -space which are parallel to and . Let be the velocity of in -space and be 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 to and for reference to respectively are not identical. Let be the measure of the lapse of time in the -system, and be 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 time in -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 let lie in the direction of the motion of in , and in the reverse direction of the motion of in . Also let lie on , so that event-particles which happen on also 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 to also happen simultaneously at points in -space on a plane perpendicular to . Accordingly the quasi-parallelism of to , and of to , is defined and secured in the same way as for Newtonian relativity.

The same meaning as above will be given to and ; also is the fundamental velocity which is the velocity of light in vacuo. Then we define

The formulae for transformation are

These formulae are symmetrical as between and , so that

It is evident that when is small, and when and are 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 of Appendix II to Chapter II, the formulae of transformation for Maxwell's equations are where () is the velocity of the charge at () at the time .

Also it immediately follows from formulae (5) that

Hence vanish 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.