Relativity - The Special and General Theory Part 4

eq. 15: file eq15.gif

but by the expression

eq. 16: file eq16.gif

This expression approaches infinity as the velocity v approaches the velocity of light c. The velocity must therefore always remain less than c, however great may be the energies used to produce the acceleration. If we develop the expression for the kinetic energy in the form of a series, we obtain

eq. 17: file eq17.gif

When eq. 18 is small compared with unity, the third of these terms is always small in comparison with the second,

which last is alone considered in cla.s.sical mechanics. The first term mc^2 does not contain the velocity, and requires no consideration if we are only dealing with the question as to how the energy of a point-ma.s.s; depends on the velocity. We shall speak of its essential significance later.

The most important result of a general character to which the special theory of relativity has led is concerned with the conception of ma.s.s.

Before the advent of relativity, physics recognised two conservation laws of fundamental importance, namely, the law of the canservation of energy and the law of the conservation of ma.s.s these two fundamental laws appeared to be quite independent of each other. By means of the theory of relativity they have been united into one law. We shall now briefly consider how this unification came about, and what meaning is to be attached to it.

The principle of relativity requires that the law of the concervation of energy should hold not only with reference to a co-ordinate system K, but also with respect to every co-ordinate system K1 which is in a state of uniform motion of translation relative to K, or, briefly, relative to every " Galileian " system of co-ordinates. In contrast to cla.s.sical mechanics; the Lorentz transformation is the deciding factor in the transition from one such system to another.

By means of comparatively simple considerations we are led to draw the following conclusion from these premises, in conjunction with the fundamental equations of the electrodynamics of Maxwell: A body moving with the velocity v, which absorbs * an amount of energy E[0] in the form of radiation without suffering an alteration in velocity in the process, has, as a consequence, its energy increased by an amount

eq. 19: file eq19.gif

In consideration of the expression given above for the kinetic energy of the body, the required energy of the body comes out to be

eq. 20: file eq20.gif

Thus the body has the same energy as a body of ma.s.s

eq.21: file eq21.gif

moving with the velocity v. Hence we can say: If a body takes up an amount of energy E[0], then its inertial ma.s.s increases by an amount

eq. 22: file eq22.gif

the inertial ma.s.s of a body is not a constant but varies according to the change in the energy of the body. The inertial ma.s.s of a system of bodies can even be regarded as a measure of its energy. The law of the conservation of the ma.s.s of a system becomes identical with the law of the conservation of energy, and is only valid provided that the system neither takes up nor sends out energy. Writing the expression for the energy in the form

eq. 23: file eq23.gif

we see that the term mc^2, which has. .h.i.therto attracted our attention, is nothing else than the energy possessed by the body ** before it absorbed the energy E[0].

A direct comparison of this relation with experiment is not possible at the present time (1920; see *** Note, p. 48), owing to the fact that the changes in energy E[0] to which we can Subject a system are not large enough to make themselves perceptible as a change in the inertial ma.s.s of the system.

eq. 22: file eq22.gif

is too small in comparison with the ma.s.s m, which was present before the alteration of the energy. It is owing to this circ.u.mstance that cla.s.sical mechanics was able to establish successfully the conservation of ma.s.s as a law of independent validity.

Let me add a final remark of a fundamental nature. The success of the Faraday-Maxwell interpretation of electromagnetic action at a distance resulted in physicists becoming convinced that there are no such things as instantaneous actions at a distance (not involving an intermediary medium) of the type of Newton's law of gravitation.

According to the theory of relativity, action at a distance with the velocity of light always takes the place of instantaneous action at a distance or of action at a distance with an infinite velocity of transmission. This is connected with the fact that the velocity c plays a fundamental role in this theory. In Part II we shall see in what way this result becomes modified in the general theory of relativity.

Notes

*) E[0] is the energy taken up, as judged from a co-ordinate system moving with the body.

**) As judged from a co-ordinate system moving with the body.

***[Note] The equation E = mc^2 has been thoroughly proved time and again since this time.

EXPERIENCE AND THE SPECIAL THEORY OF RELATIVITY

To what extent is the special theory of relativity supported by experience? This question is not easily answered for the reason already mentioned in connection with the fundamental experiment of Fizeau. The special theory of relativity has crystallised out from the Maxwell-Lorentz theory of electromagnetic phenomena. Thus all facts of experience which support the electromagnetic theory also support the theory of relativity. As being of particular importance, I mention here the fact that the theory of relativity enables us to predict the effects produced on the light reaching us from the fixed stars. These results are obtained in an exceedingly simple manner, and the effects indicated, which are due to the relative motion of the earth with reference to those fixed stars are found to be in accord with experience. We refer to the yearly movement of the apparent position of the fixed stars resulting from the motion of the earth round the sun (aberration), and to the influence of the radial components of the relative motions of the fixed stars with respect to the earth on the colour of the light reaching us from them. The latter effect manifests itself in a slight displacement of the spectral lines of the light transmitted to us from a fixed star, as compared with the position of the same spectral lines when they are produced by a terrestrial source of light (Doppler principle). The experimental arguments in favour of the Maxwell-Lorentz theory, which are at the same time arguments in favour of the theory of relativity, are too numerous to be set forth here. In reality they limit the theoretical possibilities to such an extent, that no other theory than that of Maxwell and Lorentz has been able to hold its own when tested by experience.

But there are two cla.s.ses of experimental facts. .h.i.therto obtained which can be represented in the Maxwell-Lorentz theory only by the introduction of an auxiliary hypothesis, which in itself -- i.e.

without making use of the theory of relativity -- appears extraneous.

It is known that cathode rays and the so-called b-rays emitted by radioactive substances consist of negatively electrified particles (electrons) of very small inertia and large velocity. By examining the deflection of these rays under the influence of electric and magnetic fields, we can study the law of motion of these particles very exactly.

In the theoretical treatment of these electrons, we are faced with the difficulty that electrodynamic theory of itself is unable to give an account of their nature. For since electrical ma.s.ses of one sign repel each other, the negative electrical ma.s.ses const.i.tuting the electron would necessarily be scattered under the influence of their mutual repulsions, unless there are forces of another kind operating between them, the nature of which has. .h.i.therto remained obscure to us.* If we now a.s.sume that the relative distances between the electrical ma.s.ses const.i.tuting the electron remain unchanged during the motion of the electron (rigid connection in the sense of cla.s.sical mechanics), we arrive at a law of motion of the electron which does not agree with experience. Guided by purely formal points of view, H. A. Lorentz was the first to introduce the hypothesis that the form of the electron experiences a contraction in the direction of motion in consequence of that motion. the contracted length being proportional to the expression

eq. 05: file eq05.gif

This, hypothesis, which is not justifiable by any electrodynamical facts, supplies us then with that particular law of motion which has been confirmed with great precision in recent years.

The theory of relativity leads to the same law of motion, without requiring any special hypothesis whatsoever as to the structure and the behaviour of the electron. We arrived at a similar conclusion in Section 13 in connection with the experiment of Fizeau, the result of which is foretold by the theory of relativity without the necessity of drawing on hypotheses as to the physical nature of the liquid.

The second cla.s.s of facts to which we have alluded has reference to the question whether or not the motion of the earth in s.p.a.ce can be made perceptible in terrestrial experiments. We have already remarked in Section 5 that all attempts of this nature led to a negative result. Before the theory of relativity was put forward, it was difficult to become reconciled to this negative result, for reasons now to be discussed. The inherited prejudices about time and s.p.a.ce did not allow any doubt to arise as to the prime importance of the Galileian transformation for changing over from one body of reference to another. Now a.s.suming that the Maxwell-Lorentz equations hold for a reference-body K, we then find that they do not hold for a reference-body K1 moving uniformly with respect to K, if we a.s.sume that the relations of the Galileian transformstion exist between the co-ordinates of K and K1. It thus appears that, of all Galileian co-ordinate systems, one (K) corresponding to a particular state of motion is physically unique. This result was interpreted physically by regarding K as at rest with respect to a hypothetical aether of s.p.a.ce.

On the other hand, all coordinate systems K1 moving relatively to K were to be regarded as in motion with respect to the aether. To this motion of K1 against the aether ("aether-drift " relative to K1) were attributed the more complicated laws which were supposed to hold relative to K1. Strictly speaking, such an aether-drift ought also to be a.s.sumed relative to the earth, and for a long time the efforts of physicists were devoted to attempts to detect the existence of an aether-drift at the earth's surface.

In one of the most notable of these attempts Michelson devised a method which appears as though it must be decisive. Imagine two mirrors so arranged on a rigid body that the reflecting surfaces face each other. A ray of light requires a perfectly definite time T to pa.s.s from one mirror to the other and back again, if the whole system be at rest with respect to the aether. It is found by calculation, however, that a slightly different time T1 is required for this process, if the body, together with the mirrors, be moving relatively to the aether. And yet another point: it is shown by calculation that for a given velocity v with reference to the aether, this time T1 is different when the body is moving perpendicularly to the planes of the mirrors from that resulting when the motion is parallel to these planes. Although the estimated difference between these two times is exceedingly small, Michelson and Morley performed an experiment involving interference in which this difference should have been clearly detectable. But the experiment gave a negative result -- a fact very perplexing to physicists. Lorentz and FitzGerald rescued the theory from this difficulty by a.s.suming that the motion of the body relative to the aether produces a contraction of the body in the direction of motion, the amount of contraction being just sufficient to compensate for the differeace in time mentioned above. Comparison with the discussion in Section 11 shows that also from the standpoint of the theory of relativity this solution of the difficulty was the right one. But on the basis of the theory of relativity the method of interpretation is incomparably more satisfactory. According to this theory there is no such thing as a " specially favoured "

(unique) co-ordinate system to occasion the introduction of the aether-idea, and hence there can be no aether-drift, nor any experiment with which to demonstrate it. Here the contraction of moving bodies follows from the two fundamental principles of the theory, without the introduction of particular hypotheses ; and as the prime factor involved in this contraction we find, not the motion in itself, to which we cannot attach any meaning, but the motion with respect to the body of reference chosen in the particular case in point. Thus for a co-ordinate system moving with the earth the mirror system of Michelson and Morley is not shortened, but it is shortened for a co-ordinate system which is at rest relatively to the sun.

Notes

*) The general theory of relativity renders it likely that the electrical ma.s.ses of an electron are held together by gravitational forces.

MINKOWSKI'S FOUR-DIMENSIONAL s.p.a.cE