Recent Researches on Space, Time, and Force.
Celestial mechanics, which has hitherto been based on the Newtonian laws of motion, is profoundly affected by the discoveries which have been made in recent years regarding measurements of space, time, and force.
One of the oldest and most perplexing questions regarding space is whether any definite meaning can be attached to the terms "absolute rest" and "absolute velocity." Newton in the laws of motion speaks of "rest" and "motion" relative to some presupposed frame of reference, but this frame of reference need not itself be absolutely at rest, for Newton’s laws are valid if the frame of reference is any one of an infinite number of frames which have uniform velocities of translation relative to each other. It is therefore hopeless to look to purely dynamical considerations for guidance in the recognition of absolute rest.
Throughout the nineteenth century it was thought that absolute motion in space, or at any rate absolute motion relative to the general body of stars, could be determined by the astronomical study of proper motions; and the Sun was supposed to have "an absolute velocity of about fifteen miles a second towards a point in the constellation Hercules." It is needless to relate here how this result has been overthrown by the labours of Kapteyn. But even when the failure of dynamics and astronomy to reveal absolute motion was admitted, it was still hoped that the solution might be found by aid of the theories of light and electricity. For suppose a luminous or electro-magnetic disturbance to be set up at a definite instant of time at some definite point of space, the disturbance will travel outwards from this point as a wave-front, which, if the space is unoccupied by matter, will be a sphere of gradually increasing size. During this process the centre of the sphere preserves its position unchanged relative to the æther; and thus it would seem as if a definite meaning could be attached to the term "absolute rest." Hopes might even be entertained of the measurement of absolute velocities relative to the æther by experimental methods.
The example which has just been given belongs to the theory of light. It is not difficult to imagine also electrical systems which might be expected to give information regarding absolute motion. For instance, two electric charges repel each other according to the law of electrostatics, but if the charges are in motion with equal and parallel velocities at right angles to the line joining them, the moving charges will be equivalent to elements of parallel currents, and the electrostatic repulsion will consequently be weakened by an electro-dynamic attraction superposed on it. Absolute rest could thus be characterised as that state in which the force between the charges has its maximum value.
Numerous optical and electrical experiments based on principles similar to these have been made at different times with the object of determining the absolute velocity of the Earth. But the expected effect has always failed to show itself; and at last physicists have been driven to the conclusion that a previously unrecognised compensatory influence must exist, which removes all effects of motion through the æther from the quantities which are measurable in the experiments.
The nature of this compensatory influence was first discovered by Fitzgerald. It is, that the dimensions of material bodies are slightly altered when they are in motion relative to the æther, the linear dimensions of a body in the direction of its motion being contracted in the ratio : 1, where w denotes the velocity of the body and c the velocity of light. For a body moving with the speed of the Earth the ratio w/c is only 1/10,000, so the contraction is only one two-hundred—millionth.
The full significance of Fitzgerald’s hypothesis was only gradually unfolded. The first development was the discovery that if a purely electrical system is in equilibrium when at rest, the members of the same system will be in equilibrium when they have a common translatory motion, provided the system experiences an alteration of dimensions precisely the same as that which Fitzgerald attributed to material bodies. The obvious deduction from this is, that the forces of cohesion which determine the size of material bodies are really electrical in their origin: if this be accepted, the Fitzgerald contraction follows as a necessary consequence.
It will be seen that under these circumstances we cannot hope to measure experimentally either the Fitzgerald contraction itself or any other effect of motion through the æther so long as we ourselves move with the moving system, for all these effects of motion form part of a general sympathetic change which takes place in everything with which we can operate, so that no alteration in the mutual relations of the members of the system is recognisable from within. The change, such as it is, is best described as a variation of the standards of length, time, and force.
It is this last-mentioned aspect of the matter which has chiefly occupied attention of late. On what principles are measurements of length, time, and force ultimately based?
It may be remarked that the measurement of space cannot be effected without introducing the idea of time, for of the events with which we deal in natural philosophy each is perceived to happen at some definite location at some definite instant of time, and we have no obvious means of preserving the identity of this location from one moment to another. Experience deals directly, not with a three-dimensional world of space, but with a four-dimensional world of time and space combined, and the choice of a system of measurement really means the choice of a particular projection of this four-dimensional world into a three-dimensional world of space and a one-dimensional world of time. In practice we perform this act of choice in such a way as to simplify the description of natural phenomena as much as possible; thus we lay down the condition that the increment of the time-variable in the interval between any two consecutive beats of a pendulum carried along with the observer shall be the same as its increment in the interval between any other two consecutive beats; by measuring time in this way we are enabled to formulate the laws of dynamics in simple terms.
We have already seen reason to suppose that the fundamental branch of physical science is the theory of the æther, and we are consequently led to measure space, time, and force in such a way as to give the simplest possible form to the laws of æthereal disturbance. Accordingly two philosophers, situated respectively on two stars which are in motion relative to each other, will not choose the same standards of length and time; each of them will in fact choose his standards so as to satisfy the condition that the velocity of propagation of æthereal disturbance, relative to a framework which moves with his own star, is to be reckoned equal in all directions.
The projection of the four-dimensional world of space and time into the three-dimensional world of space and the one-dimensional world of time is therefore really arbitrary, it may be done in an infinite number of ways, no one of which has any absolute primacy over the others; and each observer chooses the way which best suits the circumstances of his own motion. If we wish to describe natural phenomena in a way independent of the bias of the particular observer, we must have recourse to the language of four-dimensional analysis. We begin with a "substantial point," which represents the location of a definite particle, together with the instant at which the particle occupied this location, so that four scalar quantities are required to specify a substantial point. We then proceed to define various four-dimensional vectors, the "absolute velocity," "absolute acceleration," and "absolute force," and formulate the law of motion in the form
mass × absolute acceleration = absolute force.
This law may be expressed analytically in terms of the ordinary coordinates (x, y, z) of the particle m, in the form
where v denotes the velocity of the particle and (X, Y, Z) the force acting on it, in the ordinary sense of the term. These equations differ from those given by Newton’s laws owing to the presence of the factor , and it thus appears that Newton’s laws must henceforth be regarded as only approximately true.
E. T. W.