The Common Sense of Relativity

by Norman Robert Campbell

The Common Sense of Relativity.

By Norman Campbell^[1].

The Principle of Relativity has been discussed so often and in so many ways that it is perhaps presumptuous to attempt to add anything to the discussion except by offering original developments. But it appears to me that the needs of "the man in the laboratory" — to paraphrase a convenient modern expression — have been insufficiently considered by expositors. He has been offered profound mathematical investigations, which are intensely important and interesting, but tend to obscure the fundamental points at issue in the mind of one who thinks physically rather than mathematically. And on the other hand he has been offered collections of apparently paradoxical conclusions deduced from the Principle, which are sometimes elegant and entertaining, but more often fallacious. As a natural result he is inclined to think that this new development of science, the most important, in my opinion, since the days of Newton, is extremely abstruse and incomprehensible. In the following pages T desire to attempt to remove this misconception and to show that the view of the relations of moving systems adopted by the Principle of Relativity is very much simpler than that which it displaces, and that all its apparent difficulties are due to confusions of thought and misapprehensions. A few of the observations offered may be of interest to those who have studied the matter deeply, but it is to those whose knowledge is superficial that these remarks are primarily addressed.

As a basis of the discussion the admirable authoritative summary given by the original propounder of the Principle himself will be used (Einstein, Jahrbuch der Radioaktivität, Bd. iv. pp. 411 &c, 1907). The notation used there will be adopted without explanation.

I. The Nature of the Principle.

2. Let us first inquire exactly what the Principle of Relativity is and what it asserts.

The Principle is what is more often termed a "theory" — that is to say, it is a set of propositions from which experimental laws may be logically deduced. It can be proved to bo true or false in a manner convincing to everybody only by comparing the laws so deduced with those found experimentally ; but a theory which never conflicted with experiment might yet (as I hold) be judged objectionable on other grounds, and, conversely, a theory which was not in complete accord with experiment might yet be judged satisfactory.

The special laws which it is the business of the Principle of Relativity to explain (that is, those which it is specially important to be able to deduce from the theory) are those which are met with in the study of the optical and electrical^[2] properties of systems in relative motion, but in this case, as in most cases, it turns out that laws other than those contemplated originally are deducible from the theory. It is important to notice that there is another theory, that of Lorentz^[3], which explains completely all the electrical laws of relatively moving systems, that the deductions from the Principle of Relativity are identical with those from the Lorentzian theory, and that both sets of deductions agree completely with all experiments that have been performed^[4]. If, then, anyone prefers one theory to the other it must be either on the ground of differences in the laws not contemplated originally which are predicted respectively by the two theories, or because of some general grounds independent of experimental considerations.

3. The fundamental propositions of which the theory consists will now be enumerated and a few remarks made upon each. For the sake of brevity, I shall call a system the parts of which are all relatively at rest a "quiet" system, and one of which the parts are in relative motion a "disturbed" system. The terms are convenient to distinguish quiet and disturbed systems from those which are moving as wholes relatively to each other. Two quiet systems may be in relative motion as wholes.

(A.) The first assertion of the Principle of Relativity concerns quiet systems only. It asserts that any law and, consequently, any theory from which laws can be deduced, which has been found to hold for one quiet system which includes all the particles of which mention is made in the laws, will hold for all quiet systems which are not accelerated relatively to that quiet system^[5]. In particular the theory expressed by the fundamental equations of the electron theory has proved perfectly satisfactory so long as it is applied to a quiet system. So long, that is, as all distances are measured relatively to axes fixed in the earth, all times are measured on clocks fixed in the earth, and the phenomena considered are those of charged bodies fixed in the earth in magnetic fields produced by instruments fixed in the earth, or those of sources of light fixed in the earth observed by instruments fixed in the earth, no conclusions have ever been obtained which are not consistent with that theory. The Principle asserts that, if the whole system of axes, scales, clocks, charges, magnets, light-sources, telescopes, and observers were placed on a ship moving uniformly relative to the earth and the experiments repeated, exactly the same relation would be found between the quantities measured.

This proposition is known as the First Postulate of Relativity. The justification for it is that it is in accordance with all known experimental facts: it is, moreover, directly implicated in the usual formulation of the theory of dynamics.

Now it is the main object of the Principle of Relativity to establish a connexion between the laws of a quiet system and those of a disturbed system. In order to establish such a connexion Einstein shows that it is only necessary to introduce a small number of additional fundamental propositions into his theory, and he shows also — it is this which makes his work so brilliantly ingenious — that the additional propositions which must be introduced do not concern the laws of any quiet system. Consequently, if the theory is true, it will tell us how to change from the form suitable for a quiet system to that suitable for a disturbed system not only the laws which it is the special business of the theory to investigate, but any laws whatsoever. Instead of working out, as heretofore, the transformation necessary for each special law, we shall arrive at a transformation which is valid for all laws.

The three chief^[6] propositions necessary for this purpose are as follows:—

(B.) "Space is homogeneous and three-dimensional: time is homogeneous and one-dimensional." Mathematically this means that the transformation of the space and time coordinates is to be linear. It would take us too far afield to inquire what it means in terms of observations, and since no difficulty appears to have been felt in connexion with it, it is unimportant for our present purpose.

(C.) "If the velocity of a system S' relative to S is determined by an observer on S to be $v$, then the velocity of S relative to S' determined by an observer on S' is $v$"^[7] The necessity for introducing this proposition is generally overlooked. But it is a proposition which can be reasonably doubted and of which the truth can be tested by experiment only. Of course, all the evidence that there is is favourable: if it were not, we should not speak of "relative velocity."

(D.) "The velocity of light determined by all observers who are not accelerated relatively to each other is the same, whatever may be the relative velocities of the observers." This proposition is known as the Second Postulate of Relativity and more will be said about it hereafter.

4. The result which represents the attainment of the primary object of the Principle of Relativity is deduced from these fundamental propositions by purely mathematical argument. It may be stated as follows:—

Suppose that the disturbed system consists of two parts, A and B, each of which, regarded as a complete system, is quiet: let A be the part which contains the observer and his instruments for measuring $x,y,z,t$, and let the relative velocity of A and B be $v$. Then if A and B formed together a quiet system, the known laws for quiet systems would state a relation between the time and the coordinates of the various parts of the system on the one hand, and some quantities P, Q, R, representing the physical state of the system on the other (forces, for example). Let this relation be represented by

$f(x,y,z,t,\mathrm {P,Q,R} \dots )=0$

It is shown, as a consequence of the Principle of Relativity, that the analogous relation for the disturbed system is obtained by substituting for each set of coordinates $x,y,t$, belonging to a particle of B, the quantities $x',y',z',t'$, where

$(x',y',z',t')=\left(\beta (x-vt),\ y,\ z,\ \beta \left(t-{\frac {vx}{c^{2}}}\right)\right)$

It must be noted that the quantities P, Q, R, . . . will often involve implicitly the coordinates and the time, that is to say the values of P, Q, R, . . ., will be determined by certain measurements of $x,y,z,t$ for certain identified particles. In fact there will be a relation of the form

$\phi (P,Q,R\dots ,x,y,z,t)=0$

In this case the substitution of $x',y',z',t'$ for $x,y,z,t$ must be carried out consistently, and for P, Q, R, . . . must be substituted P', Q', R', . . ., where these latter quantities are given by

$\phi (P',Q',R'\dots ,x',y',z',t')=\phi (P,Q,R\dots ,x,y,z,t)$

In this manner the relation is obtained

$f(x',y',z',t',\mathrm {P',Q',R'} \dots )=0$

giving directly the relationship which holds for the disturbed system between $x,y,z,t$ on the one hand, and P, Q, R,... on the other, all measurements being still made by the observer on A with the measuring instruments which form part of his quiet system. This relation is that which we set out to seek, the law for the disturbed system as observed by an observer who forms part of it^[8].

So far surely nobody can find any difficulty: anything more beautifully straightforward it would be hard to conceive. Not only is the result magnificently simple, but it furnishes us with a mathematical instrument of extraordinary power. In place of the elaborate calculations which have hitherto been necessary in dealing with moving systems, all that we have to do now is to solve the problem under consideration for the limiting case of infinitesimal velocity, and then effect a mere algebraical transformation. The only objection that seems likely to be raised is that the Principle proves too much, that it appears impossible that such far-reaching conclusions can be drawn from such simple assumptions: the only difficulty, in fact, is that the thing is too easy.

5. That the arguments by which the conclusion is attained are valid can, of course, only be proved by examining them, but I think a few remarks of a general nature may remove one cause of uneasiness. It is felt that the universal importance attributed to the velocity of light is strange, when it is proposed to apply the principle to laws which have nothing to do with optics. Why. it may be questioned, do we drag in the velocity of light rather than that of sound or of the trains on the twopenny tube? Some part of this uneasiness may arise from the unfortunate way in which Einstein introduces the Second Postulate in his paper: he seems almost to try to deduce it from the First Postulate. In describing the First Postulate he says:— "In particular the same number must be found for the velocity of light in vacuo for both reference systems." It is very pertinent to ask here why, then, the velocity of light rather than that of sound.

Of course the Second Postulate cannot really be deduced from the First. What the First Postulate asserts is that all laws must be the same for all quiet systems having no relative acceleration: in particular, for all such systems, the velocity of light or the velocity of sound determined from a source which forms part of the system to a receiver which forms part of the system must be the same. But the proposition which is necessary for the argument is quite different from this. It is that the velocity of light from some source common to two systems will be found to be the same by observers on both systems, even if those systems are in relative unaccelerated motion. Since the source is common to the two systems in relative motion, it is clear that both systems, if they both include the source, cannot be quiet, and, therefore, that the First Postulate, which refers only to quiet systems, can have nothing to do with the matter.

The Second Postulate is really made up of three distinct propositions^[9]. The first is that there is some velocity which is found to be the same by all relatively unaccelerated observers; the second is that this velocity has been measured; the third is that it is the velocity of light. Only the first proposition is implicated in the result stated in the last paragraph: the quantity $c$ is this universal velocity, whatever it may turn out to be. The second and third propositions are not introduced until the result is applied to the deduction of the optical and electrical laws for a disturbed system from those for a quiet system.

The first proposition is that which is really characteristic of the Principle of Relativity, and is the feature which distinguishes it from all other theories. It seems at first sight rather startling, but perhaps it may be made to appear more plausible, if it is pointed out that it means some velocity must be "physically infinite" — that is to say, it must be such that the addition to it or subtraction from it of finite velocities do not change its magnitude. If its magnitude had turned out to be represented on the scale, of measurement of velocities ordinarily adopted by a mathematically infinite number, no difficulty would have been felt with regard to it: it is the fact that the second part of the Second Postulate proposes to represent the physically infinite velocity by a mathematically finite number which causes surprise. There is, however, nothing more difficult in such a representation than there is in the representation of the physically infinite low temperature by the mathematically infinitesimal number zero; both representations are merely consequences of the definitions of velocity and temperature adopted, and physical and mathematical infinity could be easily brought into agreement by a change of definition^[10].

But if the first and second parts of the Second Postulate be accepted, there can be no doubt about the third, for if we are going to identify the physically infinite velocity with any velocity which has ever been observed, there is, on general grounds, no doubt as to its identification with the velocity of light. For this velocity clearly cannot be less than any velocity which has been measured: to suppose that it could would be self-contradictory. Hence if we are to identify the physically infinite velocity with any velocity which has been measured, it must be with the greatest velocity which has been measured, the velocity of light in vacuo. The agreement of the propositions deduced from the Principle of Relativity with the aid of this identification with the experimental work of Bucherer is strong evidence for the second part of the postulate — that is, for the view that the physically infinite velocity has actually been measured. The first and third parts o£ the postulate are scarcely dubitable.

But perhaps such arguments are unconvincing to the physical instinct, so I proceed to considerations which should overcome an)difficulty which is felt in connexion with the great importance attributed to the velocity of light when dealing with phenomena which appear to have nothing to do with light. These considerations are based on the fact, obvious when it is pointed out, that such velocities as distinguish practically a disturbed from a quiet system, velocities, that is to say, which are not physically infinitesimal, can only be measured by optical or electrical means. When we hear of the "velocity of the earth relatively to the sun" or the "velocity of a $\beta$-particle relatively to its source," association leads us first to think of a quantity measured by the distance travelled relatively to a metre scale during the passage of the hand of a chronometer over a certain part of its face. But a little reflection will show that this is not what we mean by these velocities: we have never held a metre rod up against the sun or an electron and observed the change in relative position. It would lead us too far to inquire here what exactly we do mean by such an expression as the "velocity of a $\beta$-particle," but it could be shown quite easily that that expression has no meaning whatever, unless we assume the truth of the fundamental equations of the electron theory^[11]. And since those equations involve the velocity of light, it is not surprising that that quantity enters when we are considering the velocity of an electron. "The velocity of $\beta$-particle" is called a "velocity" because, within a certain range of values, the number representing it is the same as the number representing a velocity measured by a scale and clock (as is shown by the Rowland experiment), but, outside the range within which the scale-and-clock measurements of velocity are applicable, "the velocity of an electron is dependent for its meaning on certain theories. To inquire whether, outside this range, this velocity would agree with that determined by the scale and clock is as absurd as to inquire whether, if all triangles had four sides, all circles would be square.

II. The Consequences of the Principle.

6. But the chief objections which are raised against the Principle of Relativity are urged, not so much against the foundations of the Principle, as against its consequences. Two consequences seem to cause especial difficulty, and these will be considered.

The first difficulty concerns the "composition of velocities." The Principle of Relativity leads to the conclusion that, if an observer on a quiet system S measures the velocity of a quiet system S' relative to him and finds it $u$, and if an observer on S' finds the velocity relative to him of a third quiet system S" to be $v$, then the observer on S will find the velocity of S" relative to him to be

$w={\frac {u+v}{1+{\frac {uv}{c^{2}}}}}$

and not, as experience with small velocities might lead us to expect, $u+v$. ($u$ and $v$ are taken in the same direction.)

This conclusion seems absurd to many people. Let us inquire into the consequences of rejecting it and substituting the law $w=u+v$. We must then, of course, reject one of the fundamental propositions (B), (C), or (D); the rejection of (A) would not help us, because this proposition is not implied in the conclusion. Now, if an objector proposed to reject (B) or (C) I should have no argument to use against him, for the experimental evidence for these propositions is just as strong and just as weak as that for the proposition $w=u+v$. All these propositions, as well as (A), can be tested only by comparing the experiences of different observers, who have been moving relatively to each other with high velocities, when they meet again on a quiet system. Now since no two human beings have ever, within historic times, moved relatively to each other with a uniform velocity of $10^{4}$ cm./sec. and subsequently compared their experiences, and since, on the other hand, we do not expect to detect divergencies from the laws of a quiet system or from the laws predicted by the Principle of Relativity until the relative velocity reaches at least $10^{6}$ cm./sec, the evidence for all these propositions is extremely precarious. Nor does it seem in the least likely to become less precarious: so far as I know, nobody has made the faintest suggestion as to how a relative velocity of more than $10^{4}$ between two human beings might be attained in such a way that they could perform delicate measurements. The one proposition among those which are fundamental to the Principle of Relativity which there appears to be some hope of establishing definitely is (D): we have the source of light relative to which we are moving with a velocity of $3\times 10^{6}$ always available in the stars, and it is not too much to hope that some day experimental ingenuity will succeed in measuring the velocity of the light from it with an accuracy of one part in ten thousand. It seems to me incredible that anyone, who understands what he is doing, will really propose to reject definitely a proposition which he may hope to prove in the near future in favour of one for which there is never likely to be the smallest direct experimental evidence.

But I think these people do not understand what they are doing: they have been confused by the most fruitful cause of confusion, the habit of using one word to denote two quite different ideas. "Velocity" is commonly used to mean either "mathematical velocity or "physical velocity." Mathematical velocity is defined as the ratio of a certain variable $x$ to a certain variable $t$. From the definition of a variable and a ratio, it follows that

${\frac {x_{1}}{t}}+{\frac {x_{2}}{t}}={\frac {x_{1}+x_{2}}{t}}$

this is a perfectly purely logical conclusion, and to deny it would be absurd. On the other hand, "physical velocity" in its simplest meaning is a number equal to the ratio of two numbers — one representing the groups of metre roils that have to be placed together in order that their ends may coincide with certain points, and the other expressing the occurrence of certain events in an instrument called a clock. From the definition nothing whatsoever can be predicted as to the relations of $u,v$, and $w$, but experiment shows us that, for nil values of $u$ and $v$ which can be attained practically in this way, it $u={\tfrac {x_{1}}{t}}$, $v={\tfrac {x_{2}}{t}}$, then $w={\tfrac {x_{1}+x_{2}}{t}}$. This experimental proposition has become so familiar, and the association of the experimental $u$ and $v$ with the mathematical ${\tfrac {x_{1}}{t}}$ and ${\tfrac {x_{2}}{t}}$ so habitual, that people who do not think very deeply about these things have come to believe that $u$ means the same thing as ${\tfrac {x_{1}}{t}}$; and that, therefore, since it would be absurd to deny that ${\tfrac {x_{1}}{t}}+{\tfrac {x_{2}}{t}}={\tfrac {x_{1}+x_{2}}{t}}$, it is absurd to deny that $u+v=w$. There is no more absurdity in being forced to deny this assertion in the face of fresh evidence than there was in the necessity for Mill's Central African philosopher having to deny in the face of fresh evidence his previously undoubted proposition that "all men are black."

7. But the greatest difficulties in connexion with the Principle of Relativity appear to concern certain propositions about length and time. In what follows I shall, for brevity, discuss only time: everything I say will apply, mutatis mutandis, to length.

The Principle of Relativity leads to the following conclusion. Suppose I examine a number of clocks which, with me and my instruments, form a quiet system, and I find that they all go $n$ times as fast as my standard clock. That is to say, for the quiet system, the "law" of these clocks is that they return to some standard state when $t={\tfrac {\mathrm {P} }{n}}$, where P is any integer. Now one of these clocks is transferred to a system moving relatively to me with a velocity $v$. Let us suppose that, at the moment when $t=0$, this clock is just passing me, so that $x=0$. Then the Principle of Relativity states that the "law" for the disturbed system of myself and the clock is that the clock returns to a standard position when $t'={\tfrac {\mathrm {P} }{n}}$, or when

$\beta \left(t-{\frac {vx}{c^{2}}}\right)={\frac {\mathrm {P} }{n}}$

or, since $x=vt$, when

$P=\beta {\frac {\mathrm {P} }{n}}$

That is to say, the clock now agrees, not with the clocks with which it formerly agreed on the quiet system, but with one on the quiet which goes ${\tfrac {1}{\beta }}$ as fast as those clocks.

There is nothing new in the form of this conclusion. The crudest arguments based on the oldest theory of light lead to the conclusion that the rate of a clock as observed by a certain observer must change with the relative motion of clock and observer. For, it will be argued, the observer does not see the clock "as it really is at the moment," but "as it was a time T earlier, where T is the time taken for light to reach the observer." And on these lines it is easy to show that the apparent rate of a clock moving away from the observer with a velocity $v$ is $\left(1-{\tfrac {v}{c}}\right)$ times the rate of the same clocks observed at rest. It is only the magnitude of the change concerning which the two theories differ.

"Yes," says our objector, "that is all very well: of course the apparent rate of the clock changes with motion, but does the real rate change?" We immediately inquire what the "real rate" means. He is at first inclined to assert that it is the rate observed by an observer travelling with the clock, but when we inquire relative to what clock that observer is to measure the rate he becomes uneasy. He cannot compare another clock travelling with him, for if the "real rate" of one clock has changed, so has the "real rate" of the other ; and he cannot use a clock which is not travelling with him, because he admits that he does not see such a clock "as it really is."

Pressing our inquiries, I think we shall get an answer of this nature. "If I take a pendulum clock to some place where gravity is different, the rate of the clock will change. It is a change of this nature which I call a change in the 'real rate,' and I want to know whether there is any change of that kind, on the theory of Relativity, when the clock is set in motion." Now why does our objector call a change of the first kind a change in the "real rate"? The reply is to be found in the history of the word "real." The word is intimately associated with the philosophic doctrine of realism, which holds that the most important thing that we can know about any body is not what we observe about it, but its "real nature," which is something that is independent of observation. Now, of course, a quantity which is wholly independent of observation cannot play any part in an experimental science, but there are quantities which are independent of observation in the more limited sense that they are observed to be the same by whatever observer the observation is made. The term "real" has come to be transferred from the philosophical conception to such quantities. The "real rate" of the clock is said to change when it is transferred to a place where gravitation is different, because all observers agree that the rate of the clock which has been moved has undergone an alteration relatively to that which has not been moved.

Now in the conditions which we are considering the observers do not agree. If A and B, each carrying a clock with him, are moving relatively to each other, they will not agree as to the rate of either of their clocks relative to A s standard or to B's standard or to any other standard. The conditions which, in the case of the alteration of gravitation, gave rise to the conception of a "real rate" are not present: in this case there is no "real rate," and it is as absurd to ask whether it has changed as it would be to ask a question about the properties of round square. However, some people, who in their eagerness to escape the reproach of being metaphysicians have adopted without inquiry the oldest and least satisfactory metaphysical doctrines, are so enamoured of the conception of "reality" that they refuse to give it up. Finding that the observations of different observers do not agree, they define a new function of those observations, such that it is the same for all observers, and proceed to call this the "real rate." This function, according to the Principle of Relativity, is $\beta n'$, where $n'$ is the rate of the clock as seen by an observer relative to whom it is travelling with the velocity $v$: according to that Principle, if we substitute in that function the appropriate values for any one observer, the resulting number will always be the same.

So far no overwhelming objection can be raised. The function is important in the theory, and, if care is taken to note the precise meaning now attributed to the word "real," there is no harm in calling it by that name. But now certain writers commit an extraordinary series of blunders. They not only inquire whether the real rate changes with the velocity, a question which, as the real rate is defined as that function which does not change with the velocity, is utterly trivial, but they actually give a negative answer. They see that the expression for the real rate contains $v$ explicitly and rush to the absurd conclusion that the real rate changes with the velocity. No wonder that they soon involve themselves in a hopeless maze of paradox.

As a matter of fact the "crude argument" given above shows that the second definition of "real" had been introduced before the Principle of Relativity. It had been recognised already that observers would not agree as to the rate of a clock: the conception of the clock "as it really is," introduced in that argument, means (if it means anything) that function of the observed rate of the clock and its velocity relative to the observer which is the same for all observers. But the logical order of the argument is reversed. Instead of proving from the "real rate" of the clock, which we do not know, the observed rate, which we do know, we should say that the observed rate of the clock is $n$, and that our theory of light leads to the conclusion that $n/\left(1-{\tfrac {v}{c}}\right)$ will be the same for all observers. Whether that conclusion or the conclusion reached by the Principle of Relativity is correct can only be determined by experiment, and the experiment has not yet been tried.

It is the great merit of the Principle of Relativity that it forces on our attention the true nature of the concepts of "real time" and "real space" which have caused such endless confusion. If we mean by them quantities which are directly observed to be the same by all observers, there simply is no real space and real time. If we mean by them, as apparently we do mean nowadays, functions of the directly observed quantities which are the same for all observers, then they are derivative conceptions which depend for their meaning on the acceptance of some theory as to how the directly observed quantities will vary with the motion,position, etc. of the observers. "Real" quantities can never be the starting point of a scientific argument; by their very nature they are not quantities which can be determined by a single observation: the term "real" has always kept its original meaning of some property of a body which is not observed simply.

All the difficulties and apparent paradoxes of the Principle of Relativity will vanish it the attention is kept rigidly fixed upon the quantities which are actually observed. If anyone thinks he discovers that that Principle predicts some experimental result which is incomprehensible, let him dismiss utterly from his mind the conception of reality. Let him imagine himself in the laboratory actually performing the experiment: let him consider the numbers which he will record in his note-book and the subsequent calculation which he will make. He may then find that the result is somewhat unexpected — to meet with unexpected results is the usual end of performing experiments, — but he will not find any contradiction or any conclusion which is not quite as simple as that which he expected.

8. There is one further point sometimes raised in connexion with the Principle on which a few words may be said.

It is sometimes objected that the Principle "has no physical meaning," that it destroys utterly the old theory of light based on an elastic æther and puts nothing in its place, that, in fact, it sacrifices the needs of the physical to the needs of the mathematical instinct. That the statement is true there can be no doubt, but the absence of any substitute for the elastic æther theory of light may simply be due to the fact that the Principle has been developed so far chiefly by people who are primarily mathematicians. It is well to ask, can any physical theory of light be produced which is consistent with the Principle?

The answer depends on what is meant by a "physical theory." Hitherto the term has always meant a "mechanical theory," a theory of which the fundamental propositions are statements about particles moving according to the Newtonian dynamical formulas. In this sense a physical theory is impossible if the Principle of Relativity be accepted, for the same reason that a corpuscular theory of light is impossible, if the undulatory theory of light be accepted. Newtonian dynamics and the Principle of Relativity are two theories which deal in part with the same range of facts; they both pretend to be able to predict how the properties of observed systems will be altered by movement. If they are not logically equivalent they must be contradictory: in either case an "explanation" of one in terms of the other is impossible.

It can be easily shown that they are contradictory: if the Principle of Relativity is true, Newtonian dynamics must be abandoned^[12]. I shall deal with this point rather fully in a later paper; here it will suffice to point out that Einstein has been forced in his development of the subject to deny Newtonian dynamics at an early stage. He states that the fundamental equations of his electron theory are

$m{\ddot {x}}=\epsilon X$, etc.,

and then puts ${\dot {x}}=v$, where $v$ is the velocity of the electron relative to the instrument exerting the force $\epsilon X$. But, if Newtonian dynamics are true, ${\dot {x}}$ is not this relative velocity, but the velocity of the electron relative to the centre of mass of the electron and the instrument. Since the mass of the electron can conceivably become infinite, the distinction, negligible in practice, is of great importance theoretically.

On the other hand, if a "physical theory" of light means, as I think it means, a theory which draws an analogy between light propagation and the propagation of a disturbance through some mechanism, composed of rods and strings and fluids and such things, then there is no reason apparent why a physical theory of light should not be constructed which is consistent with the Principle of Relativity. But, of course, the laws according to which rods and strings and so on are supposed to act, must be changed from those predicted by Newtonian dynamics to some laws predicted by a mechanical theory consistent with the Principle. This development also is left for future discussion.

Summary.

1—5. The assumptions made by the Principle of Relativity are stated and an attempt made to render some of them more plausible at first sight.

6. A difficulty connected with the composition of velocities is examined and found to be due to verbal confusion.

7. The confusions introduced by the word "real" are discussed.

8. The relation between dynamics and relativity is considered briefly.

Leeds, November 1910.

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1. Communicated by the Author
2. "Optical" and "electrical" will be employed throughout as equivalent terms, one or the other being used according to the context.
3. As given in the Encyclo. d. Mathemat. Wissen.
4. This statement is only true if quantities of a higher order than the second in $v/c$ are left out of account. Since such terms cannot be detected experimentally the conclusion given is not affected, and the terms will be neglected throughout our argument. In saying that both theories agree completely with experiment, I do not wish to offer any opinion as to the value of the experiments of Bucherer and others which have been subjected to critisism: I merely assume that the results announced in them will he accepted ultimately, because, if they are not, the Principle of Relativity would seem to be unworthy of further discussion.
5. A difficulty arises from the fact that no system where anything "happens" can possibly be quiet, for almost all the changes which physics investigates involve relative motion of the parts of the system investigated. The logical questions rained by this difficulty will be discussed more fully in another place: for the present we may regard "laws for a quiet system" as propositions toward which the laws of actual systems tend as the relative velocity of the parts of those systems tends to zero. Thus the fundamental equations of the electron theory, involving the terms $\rho v$, cannot be applied to quiet systems, since they consider the motion of an electron (which is part of the system) relative to the rest of the system: they may be regarded as the limiting form of the equations valid for disturbed systems, as the velocity $v$ tends to zero. There is no practical difficulty about this extrapolation, for the form of laws is not found to change with the relative velocity of the parts of disturbed systems over a large range limited, on the one hand, by a relative velocity of about $10^{6}$ cm./sec, and, on the other, by the smallest velocity which can be detected experimentally. The laws for a quiet system are, then, the laws which hold within this range of velocities.
6. The other propositions are those which are implied in all physical measurement and in all theories. The propositions given are not those given by Einstein explicitly, but those which seem to be implied by his argument.
7. The sign attributed to $v$ is a matter of pure convention; so long as each observer adheres to bis own convention throughout, it does not signify whether the same or contrary signs are attributed to the relative velocity by the two observers.
8. The best examples of the process are, of course, those worked out by Einstein in the paper referred to.
9. The complexity of the Second Postulate appears very clearly in Minkowski's "Raum und Zeit.' Minkowski's treatment is somewhat different from that of Einstein and involves an entire rejection of the conceptions of space and time.
10. This line of thought will be developed in a later paper.
11. Some remarks on this subject are to be found in a paper on "The Principles of Dynamics," Phil. Mag. xix. p. 168.
12. This conclusion is reached by Sommerfeld in a recent paper, Ann. d. Phys. xxxiii. p. 684, &c. (1910).