A Stubbornly Persistent Illusion
Page 2
We might, of course, content ourselves with time values determined by an observer stationed together with the watch at the origin of the co-ordinates, and co-ordinating the corresponding positions of the hands with light signals, given out by every event to be timed, and reaching him through empty space. But this co-ordination has the disadvantage that it is not independent of the standpoint of the observer with the watch or clock, as we know from experience. We arrive at a much more practical determination along the following line of thought.
If at the point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous with these events. If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the immediate neighbourhood of B. But it is not possible without further assumption to compare, in respect of time, an event at A with an event at B. We have so far defined only an “A time” and a “B time.” We have not defined a common “time” for A and B, for the latter cannot be defined at all unless we establish by definition that the “time” required by light to travel from A to B equals the “time” it requires to travel from B to A. Let a ray of light start at the “A time” tA from A toward B, let it at the “B time” tB be reflected at B in the direction of A, and arrive again at A at the “A time” t′A.
In accordance with definition the two clocks synchronize if
We assume that this definition of synchronism is free from contradictions, and possible for any number of points; and that the following relations are universally valid:—
1. If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the clock at B.
2. If the clock at A synchronizes with the clock at B and also with the clock at C, the clocks at B and C also synchronize with each other.
Thus with the help of certain imaginary physical experiments we have settled what is to be understood by synchronous stationary clocks located at different places, and have evidently obtained a definition of “simultaneous,” or “synchronous,” and of “time.” The “time” of an event is that which is given simultaneously with the event by a stationary clock located at the place of the event, this clock being synchronous, and indeed synchronous for all time determinations, with a specified stationary clock.
In agreement with experience we further assume the quantity to be a universal constant—the velocity of light in empty space.
It is essential to have time defined by means of stationary clocks in the stationary system, and the time now defined being appropriate to the stationary system we call it “the time of the stationary system.”
§ 2. ON THE RELATIVITY OF LENGTHS AND TIMES
The following reflexions are based on the principle of relativity and on the principle of the constancy of the velocity of light. These two principles we define as follows:—
1. The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion.
2. Any ray of light moves in the “stationary” system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body. Hence
where time interval is to be taken in the sense of the definition in § 1.
Let there be given a stationary rigid rod; and let its length be l as measured by a measuring-rod which is also stationary. We now imagine the axis of the rod lying along the axis of x of the stationary system of co-ordinates, and that a uniform motion of parallel translation with velocity v along the axis of x in the direction of increasing x is then imparted to the rod. We now inquire as to the length of the moving rod, and imagine its length to be ascertained by the following two operations:—
(a) The observer moves together with the given measuring-rod and the rod to be measured, and measures the length of the rod directly by superposing the measuring-rod, in just the same way as if all three were at rest.
(b) By means of stationary clocks set up in the stationary system and synchronizing in accordance with § 1, the observer ascertains at what points of the stationary system the two ends of the rod to be measured are located at a definite time. The distance between these two points, measured by the measuring-rod already employed, which in this case is at rest, is also a length which may be designated “the length of the rod.”
In accordance with the principle of relativity the length to be discovered by the operation (a)—we will call it “the length of the rod in the moving system”—must be equal to the length l of the stationary rod.
The length to be discovered by the operation (b) we will call “the length of the (moving) rod in the stationary system.” This we shall determine on the basis of our two principles, and we shall find that it differs from l.
Current kinematics tacitly assumes that the lengths determined by these two operations are precisely equal, or in other words, that a moving rigid body at the epoch t may in geometrical respects be perfectly represented by the same body at rest in a definite position.
We imagine further that at the two ends A and B of the rod, clocks are placed which synchronize with the clocks of the stationary system, that is to say that their indications correspond at any instant to the “time of the stationary system” at the places where they happen to be. These clocks are therefore “synchronous in the stationary system.”
We imagine further that with each clock there is a moving observer, and that these observers apply to both clocks the criterion established in § 1 for the synchronization of two clocks. Let a ray of light depart from A at the time* tA, let it be reflected at B at the time tB, and reach A again at the time t′A. Taking into consideration the principle of the constancy of the velocity of light we find that
where rAB denotes the length of the moving rod—measured in the stationary system. Observers moving with the moving rod would thus find that the two clocks were not synchronous, while observers in the stationary system would declare the clocks to be synchronous.
So we see that we cannot attach any absolute signification to the concept of simultaneity, but that two events which, viewed from a system of co-ordinates, are simultaneous, can no longer be looked upon as simultaneous events when envisaged from a system which is in motion relatively to that system.
§ 3. THEORY OF THE TRANSFORMATION OF CO-ORDINATES AND TIMES FROM A STATIONARY SYSTEM TO ANOTHER SYSTEM IN UNIFORM MOTION OF TRANSLATION RELATIVELY TO THE FORMER
Let us in “stationary” space take two systems of coordinates, i.e. two systems, each of three rigid material lines, perpendicular to one another, and issuing from a point. Let the axes of X of the two systems coincide, and their axes of Y and Z respectively be parallel. Let each system be provided with a rigid measuring-rod and a number of clocks, and let the two measuring-rods, and likewise all the clocks of the two systems, be in all respects alike.
Now to the origin of one of the two systems (k) let a constant velocity v be imparted in the direction of the increasing x of the other stationary system (K), and let this velocity be communicated to the axes of the co-ordinates, the relevant measuring-rod, and the clocks. To any time of the stationary system K there then will correspond a definite position of the axes of the moving system, and from reasons of symmetry we are entitled to assume that the motion of k may be such that the axes of the moving system are at the time t (this “t” always denotes a time of the stationary system) parallel to the axes of the stationary system.
We now imagine space to be measured from the stationary system K by means of the stationary measuring-rod, and also from the moving system k by means of the measuring-rod moving with it; and that we thus obtain the co-ordinates x, y, z, and ξ, η, ζ respectively. Further, let the time t of the stationary system be determined for all points thereof at which there are clocks by means of light signals
in the manner indicated in § 1; similarly let the time τ of the moving system be determined for all points of the moving system at which there are clocks at rest relatively to that system by applying the method, given in § 1, of light signals between the points at which the latter clocks are located.
To any system of values x, y, z, t, which completely defines the place and time of an event in the stationary system, there belongs a system of values ξ, η, ζ, τ, determining that event relatively to the system k, and our task is now to find the system of equations connecting these quantities.
In the first place it is clear that the equations must be linear on account of the properties of homogeneity which we attribute to space and time.
If we place x′ = x – vt, it is clear that a point at rest in the system k must have a system of values x′, y, z, independent of time. We first define τ as a function of x′, y, z, and t. To do this we have to express in equations that τ is nothing else than the summary of the data of clocks at rest in system k, which have been synchronized according to the rule given in § 1.
From the origin of system k let a ray be emitted at the time τ0 along the X-axis to x′, and at the time τ1 be reflected thence to the origin of the co-ordinates, arriving there at the time τ2; we then must have (τ0 + τ2) = τ1, or, by inserting the arguments of the function τ and applying the principle of the constancy of the velocity of light in the stationary system:—
Hence, if x′ be chosen infinitesimally small,
or
It is to be noted that instead of the origin of the co-ordinates we might have chosen any other point for the point of origin of the ray, and the equation just obtained is therefore valid for all values of x′, y, z.
An analogous consideration—applied to the axes of Y and Z—it being borne in mind that light is always propagated along these axes, when viewed from the stationary system, with the velocity , gives us
Since τ is a linear function, it follows from these equations that
where a is a function ϕ(v) at present unknown, and where for brevity it is assumed that at the origin of k, τ = 0, when t = 0.
With the help of this result we easily determine the quantities ξ, η, ζ by expressing in equations that light (as required by the principle of the constancy of the velocity of light, in combination with the principle of relativity) is also propagated with velocity c when measured in the moving system. For a ray of light emitted at the time τ = 0 in the direction of the increasing ξ
But the ray moves relatively to the initial point of k, when measured in the stationary system, with the velocity c – v, so that
If we insert this value of t in the equation for ξ, we obtain
In an analogous manner we find, by considering rays moving along the two other axes, that
when
Thus
Substituting for x′ its value, we obtain
where
and ϕ is an as yet unknown function of v. If no assumption whatever be made as to the initial position of the moving system and as to the zero point of τ, an additive constant is to be placed on the right side of each of these equations.
We now have to prove that any ray of light, measured in the moving system, is propagated with the velocity c, if, as we have assumed, this is the case in the stationary system; for we have not as yet furnished the proof that the principle of the constancy of the velocity of light is compatible with the principle of relativity.
At the time t = τ = 0, when the origin of the co-ordinates is common to the two systems, let a spherical wave be emitted therefrom, and be propagated with the velocity c in system K. If (x, y, z) be a point just attained by this wave, then
Transforming this equation with the aid of our equations of transformation we obtain after a simple calculation
The wave under consideration is therefore no less a spherical wave with velocity of propagation c when viewed in the moving system. This shows that our two fundamental principles are compatible.*
In the equations of transformation which have been developed there enters an unknown function ϕ of v, which we will now determine.
For this purpose we introduce a third system of co-ordinates K′, which relatively to the system k is in a state of parallel translatory motion parallel to the axis of X, such that the origin of co-ordinates of system k moves with velocity–v on the axis of X. At the time t = 0 let all three origins coincide, and when t = x = y = z = 0 let the time t′ of the system K′ be zero. We call the co-ordinates, measured in the system K′, x′, y′, z′, and by a twofold application of our equations of transformation we obtain
Since the relations between x′, y′, z′ and x, y, z′ do not contain the time t, the systems K and K′ are at rest with respect to one another, and it is clear that the transformation from K to K′ must be the identical transformation. Thus
We now inquire into the signification of ϕ(v). We give our attention to that part of the axis of Y of system k which lies between ξ = 0, η = 0, ζ = 0 and ξ = 0, η = l, ζ = 0. This part of the axis of Y is a rod moving perpendicularly to its axis with velocity v relatively to system K. Its ends possess in K the co-ordinates
and
The length of the rod measured in K is therefore l/ϕ(v); and this gives us the meaning of the function ϕ(v). From reasons of symmetry it is now evident that the length of a given rod moving perpendicularly to its axis, measured in the stationary system, must depend only on the velocity and not on the direction and the sense of the motion. The length of the moving rod measured in the stationary system does not change, therefore, if v and – v are interchanged. Hence follows that l/ϕ(v) = l/ϕ(– v), or
It follows from this relation and the one previously found that ϕ(v) = 1, so that the transformation equations which have been found become
where
§ 4. PHYSICAL MEANING OF THE EQUATIONS OBTAINED IN RESPECT TO MOVING RIGID BODIES AND MOVING CLOCKS
We envisage a rigid sphere* of radius R, at rest relatively to the moving system k, and with its centre at the origin of co-ordinates of k.
The equation of the surface of this sphere moving relatively to the system K with velocity v is
The equation of this surface expressed in x, y, z at the time t = 0 is
A rigid body which, measured in a state of rest, has the form of a sphere, therefore has in a state of motion—viewed from the stationary system—the form of an ellipsoid of revolution with the axes
Thus, whereas the Y and Z dimensions of the sphere (and therefore of every rigid body of no matter what form) do not appear modified by the motion, the X dimension appears shortened in the ratio i.e. the greater the value of v, the greater the shortening. For v = c all moving objects—viewed from the “stationary” system—shrivel up into plain figures. For velocities greater than that of light our deliberations become meaningless; we shall, however, find in what follows, that the velocity of light in our theory plays the part, physically, of an infinitely great velocity.
It is clear that the same results hold good of bodies at rest in the “stationary” system, viewed from a system in uniform motion.
Further, we imagine one of the clocks which are qualified to mark the time t when at rest relatively to the stationary system, and the time τ when at rest relatively to the moving system, to be located at the origin of the co-ordinates of k, and so adjusted that it marks the time τ. What is the rate of this clock, when viewed from the stationary system?
Between the quantities x, t, and τ, which refer to the position of the clock, we have, evidently, x = vt and
Therefore,
whence it follows that the time marked by the clock (viewed in the stationary system) is slow by seconds per second, or—neglecting magnitudes of fourth and higher order—by .
From this there ensues the following peculiar consequence. If at the points A and B of K there are stationary clocks which, viewed in the stationary system, are synchronous; and if the clock at A is moved with the velocity v along the line AB to B,
then on its arrival at B the two clocks no longer synchronize, but the clock moved from A to B lags behind the other which has remained at B by (up to magnitudes of fourth and higher order), t being the time occupied in the journey from A to B.
It is at once apparent that this result still holds good if the clock moves from A to B in any polygonal line, and also when the points A and B coincide.
If we assume that the result proved for a polygonal line is also valid for a continuously curved line, we arrive at this result: If one of two synchronous clocks at A is moved in a closed curve with constant velocity until it returns to A, the journey lasting t seconds, then by the clock which has remained at rest the travelled clock on its arrival at A will be second slow. Thence we conclude that a balance-clock* at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions.
§ 5. THE COMPOSITION OF VELOCITIES
In the system k moving along the axis of X of the system K with velocity v, let a point move in accordance with the equations
where wξ and wη denote constants.
Required: the motion of the point relatively to the system K. If with the help of the equations of transformation developed in § 3 we introduce the quantities x, y, z, t into the equations of motion of the point, we obtain