On the Theory of Relativity: Mass, Force and Energy

by Robert Daniel Carmichael

ON THE THEORY OF RELATIVITY: MASS, FORCE AND ENERGY.

By R. D. Carmichael.

Introduction.

In a previous paper^[1] I have given an analysis of the characteristic postulates on which the theory of relativity depends and have developed in a general way some of the fundamental conclusions of this theory. The plan of treatment adopted makes it possible to arrive at these results by arguments which are not complicated in form; and the notions which enter into them are of simpler character than those usually to be found in memoirs on this subject.

It seems desirable that the general theory should be further worked out along the same lines, even though this should require in part a repetition of some things already in the memoirs. Accordingly, in the present paper I develop the fundamental properties^[2] of mass, force and energy, using as a basis the theorems of my previous paper. In the derivation of these results the theory of electricity is not employed. Consequently the conclusions are of wider applicability than when they are proved by means of electrical theory; in particular, they may be applied to the derivation of results in the theory of electricity itself.

In § 1, after giving some definitions, I state the laws of conservation of momentum and energy and electricity and the principle of least action in the form in which I shall have occasion to use them. The law of conservation of electricity is employed in this paper only in the applications of the results concerning mass and energy.

In § 2 the question of the dependence of mass on velocity is treated. First the transverse mass of a moving body is determined by the elegant method of Lewis and Tolman. The relation between transverse mass and longitudinal mass is found by the method of Bumstead, and thus the longitudinal mass of a moving body is obtained.

In § 3 dimensional equations are employed to derive the relations of acceleration and force in two systems of reference.

In § 4 from considerations concerning the mass of a moving body two essential equivalents of postulate R are determined, each of which furnishes a possible means for the experimental proof or disproof of the theory of relativity. The researches of Bucherer have been thought to afford the requisite experimental confirmation in the first case; this matter is treated in § 5. In § 6 suggestions are given for a new crucial experiment for testing the theory of relativity, this being associated with the second essential equivalent of R in § 4. The writer desires to call especial attention to this proposed experiment.

In § 7 the intimate relation of the mass and the total energy of a body is pointed out and two theoretical means are suggested for determining the velocity of light indirectly, that is, without direct measurement of this velocity. These experiments, if they could be performed with sufficient accuracy, would afford an interesting and striking confirmation of the theory of relativity, provided of course that they turn out according to the predictions of this theory.

A few remarks on the principle of least action are found in § 8, and § 9 is given to some speculative considerations which are intended as brief suggestions of means by which one may represent to himself the conclusions of relativity as natural parts of a consistent view of physical phenomena.

§ 1. Fundamental Definitions and Postulates.

If m, M and v are respectively the mass, momentum and velocity of a body we shall assume (as in the classical mechanics) that they are connected by a relation of the form

$M=mv$;

and hence any one of these quantities is determined in terms of the other two (except that mass is not thus determined when velocity and momentum are zero). We shall take mass and velocity to be the two fundamental quantities, so that momentum is defined in terms of them.

Likewise we shall define the kinetic energy E of a moving body by means of the usual relation

$E=\int _{0}^{v}Mdv=\int _{0}^{v}mvdv$

Later we shall see that "mass" is variable and is not in general independent of the direction in which it is measured; consequently, we must take for m in the above formulae the mass of the body in the direction of its motion.

We shall take for granted the following laws of conservation of momentum and energy and electricity:

Postulate $C_{1}$. The sum total of momentum in any isolated system remains unaltered, whatever changes may take place in the system, provided that it is not affected by any forces from without.

Postulate $C_{2}$. The sum total of energy in any isolated system remains unaltered, whatever changes may take place in the system, provided that it is not affected by any forces from without.

Postulate $C_{3}$. The sum total of electricity in any isolated system remains unaltered, whatever changes may take place in the system, provided that the system as a whole neither receives electricity from nor gives out electricity to bodies not belonging to the system.

The "action" of a moving body in passing from one position to another may be defined as the space integral of the momentum taken over the path of motion. If we denote this action by A we have therefore

$A=\int Mds=\int mvds$

Now ds = vdt, so that we have also

$A=\int mv^{2}dt$

If several bodies are involved we have

$A=\sum \int mvds=\sum \int mv^{2}dt$

where the summation is for the various bodies in the system.

We may state the fundamental principle of least action in the following form:

Principle of Least Action. The free motion of a conservative system between any two given configurations has the property that the action A is a minimum, the admissible values ${\overline {A}}$ of the action with which A is compared being obtained from varied motions in which the total energy has the same constant value as in the actual free motion.

§ 2. Dependence of Mass on Velocity.

Suppose that we have two systems of reference $S_{1}$ and $S_{2}$ moving with a relative velocity v. We inquire as to whether, and in what way, the mass of a body as measured on the two systems depends on v. Will a given body have the same measure of mass when that mass is estimated in units of $S_{1}$ and in units of $S_{2}$? And will the mass of a body depend on the direction of its motion by means of which that mass is measured? Our purpose in this section is to answer these two questions.

The two most important directions in which to measure the mass of a body are, first, that perpendicular to the line of relative motion of $S_{1}$ and $S_{2}$ and, secondly, that parallel to this line of motion. For convenience in distinguishing these we shall speak of the "transverse mass" of a body as that with which we have to deal when we are concerned with the motion of the body in a direction perpendicular to the line of relative motion of $S_{1}$ and $S_{2}$; when the motion is parallel to this line we shall speak of the "longitudinal mass" of the body.

Lewis and Tolman^[3] determine what they call the "mass of a body in motion," employing for this purpose a very simple and elegant method. This "mass" is what we have just defined as the transverse mass of the body. We employ the excellent method of these authors in deriving the formula for transverse mass.

Suppose that an experimenter A on the system $S_{1}$ constructs a ball $B_{1}$ of some rigid elastic material, with unit volume, and puts it in motion with unit velocity in a direction perpendicular to the line of relative motion of $S_{1}$ and $S_{2}$, the units of measurement employed being those belonging to $S_{1}$. Likewise suppose that an experimenter C on $S_{2}$ constructs a ball $B_{2}$ of the same material, also of unit volume, and puts it in motion with unit velocity in a direction perpendicular to the line of relative motion of $S_{1}$ and $S_{2}$; we suppose that the measurements made by C are with units belonging to $S_{2}$. Assume that the experiment has been so planned that the balls will collide and rebound, over their original paths, the path of each ball being thought of as relative to the system to which it belongs.

Now the relation of the ball $B_{2}$ to the system $S_{1}$ is the same as that of the ball $B_{1}$ to the system $S_{2}$, on account of the perfect symmetry which exists between the two systems of reference in accordance with the results of the previous paper (already referred to). Therefore the change of velocity of $B_{2}$ relative to its starting point on $S_{2}$ as measured by A is equal to the change of velocity of $B_{1}$ relative to its starting point on $S_{1}$ as measured by C. Now velocity is equal to the ratio of distance to time: and in the direction perpendicular to the line of relative motion of the two systems the units of length are equal; but the units of time are unequal. Hence to either of the observers the change of velocity in the two balls, each with respect to its starting point on its own system, will appear to be unequal.

To A the time unit on $S_{2}$ appears to be longer than his own in the ratio ${\sqrt {1-\beta ^{2}}}:1$ (see previous paper, theorem IV., p. 167). Hence to A it must appear that the change in velocity of $B_{2}$ relative to its starting point is smaller than that of $B_{1}$ relative to its starting point in the ratio ${\sqrt {1-\beta ^{2}}}:1$. But the change in velocity of each ball multiplied by its mass gives its change in momentum. From postulate $C_{1}$ it follows that these two changes of momentum are equal. Hence to A it appears th at the mass of the ball $B_{1}$ is smaller than that of the ball $B_{2}$ in the ratio ${\sqrt {1-\beta ^{2}}}:1$.

Similarly, it may be shown that to C it appears that the mass of the ball $B_{2}$ is smaller than that of $B_{1}$ in the ratio ${\sqrt {1-\beta ^{2}}}:1$.

From the general theorems concerning the measurement of length (in the theory of relativity) it follows that if the ball which has been constructed by A were transferred to $C$'s system it would be impossible for C to distinguish $A$'s ball from his own by any considerations of shape and size. Likewise, as A looks at them from his own system he is similarly unable to distinguish them. It is therefore natural to take the mass of $C$'s ball as that which $A$'s would have if it had the velocity v with respect to $S_{1}$ of the system $S_{2}$. Thus we obtain a relation existing between the mass of a body in motion and at rest.

Now, "mass" as we have measured it above is the transverse mass of our definition. From the argument just carried out we are forced to conclude that the transverse mass of a body in motion depends (in a certain definite way) on the velocity of that motion. The result may be formulated as follows:

Theorem I. Let $m_{0}$ denote the mass of a body when at rest relative to a system of reference S. When it is moving with a velocity v relative to S denote by $t\left(m_{v}\right)$ its transverse mass, that is, its mass in a direction perpendicular to its line of motion. Then we have

$t\left(m_{v}\right)={\frac {m_{0}}{\sqrt {1-\beta ^{2}}}}$,

where $\beta =v/c$ and c is the velocity of light $\scriptstyle (MVLRC_{1})$.^[4]

In the statement of this theorem we have tacitly assumed that the mass of a body at rest relative to S, when measured by means of units belonging to S, is independent of the direction in which it is measured. If this assumption were not true we should have a means of detecting the motion of S, a conclusion which is in contradiction to postulate M.

In order to find the longitudinal mass of a moving body we first find the relation which exists between longitudinal mass and transverse mass. We employ for this purpose the elegant method of Bumstead.^[5]

Let us as usual consider two systems of reference $S_{1}$ and $S_{2}$ moving with a relative velocity v, observers A and B being stationed on $S_{1}$ and $S_{2}$ respectively. Suppose that B performs the following experiment: He takes a rod of two units length, whose mass is so small as to be negligible, and attaches to its ends two balls of equal mass. Then he suspends this rod by a wire so as to form a torsion pendulum. We assume that the line of relative motion of the two systems is perpendicular to the line of this wire.

Let us consider the period of this torsion pendulum in the two cases when the rod is clamped to the wire so as to be in equilibrium in each of the following two positions: (1) With its length perpendicular to the line of relative motion of $S_{1}$ and $S_{2}$; (2) with its length parallel to this line of motion.

As B observes it the period must be the same in the two cases; for, otherwise, he would have a means of detecting his motion by observations made on his system alone, contrary to postulate M. Then from the relation of time units on $S_{1}$ and $S_{2}$ it follows that the two periods will also appear the same to A. As observed by B the apparent mass of the balls is the same in both cases. We inquire as to how they appear to A. Let $m_{1}$ and $m_{2}$ be the apparent masses, as observed by A, in the first and second cases respectively. It is obvious that $m_{1}$ is the longitudinal mass and $m_{2}$ the transverse mass of the balls in question.

When the pendulum is in motion it appears to B that each ball traces a circular arc. From the relations between units of length in the two systems it follows that to A it appears that the balls trace arcs of an ellipse whose semi-axes are 1 and ${\sqrt {1-\beta ^{2}}}$ and lie perpendicular and parallel, respectively, to the line of relative motion of the two systems.

Let us now determine the period of each of these two pendulums as they are observed by A. By equating the expressions for these periods we shall find the relation which exists between $m_{1}$ and $m_{2}$.

Let x and y be the cartesian coordinates of a point as determined by A, the axes of reference being the major and minor axes of the ellipse in which the balls move. Let $x'$ and $y'$ be the coordinates of the same point as determined by B. Then the circular path of motion, as determined by B, has the equations

$x'=\cos \theta ,\ y'=\sin \theta$

the angle θ being measured from the major axis of the ellipse. The equations of the ellipse, as determined by A, are

$x=\cos \theta ,\ y={\sqrt {1-\beta ^{2}}}\sin \theta$

In the first case — when the rod is perpendicular to the line of relative motion of $S_{1}$ and $S_{2}$ — the amount of twisting in the wire when the ball is in a given position is the absolute value of the corresponding angle θ; and therefore the potential energy^[6] is proportional to θ², say that it is ${\tfrac {1}{2}}k\theta ^{2}$. Now from the values of y and x above we have

$y=x{\sqrt {1-\beta ^{2}}}\tan \theta$

For small oscillations we have x = 1 and tan θ = θ; and therefore

$y={\sqrt {1-\beta ^{2}}}\cdot \theta$

Hence the potential energy is

${\frac {1}{2}}{\frac {k}{1-\beta ^{2}}}y^{2}$;

and the equation of motion of the particle becomes

$m_{1}{\frac {d^{2}y}{dt^{2}}}=-{\frac {k}{1-\beta ^{2}}}y$

Hence the period $T_{1}$ of oscillation is

$T_{1}=2\pi {\sqrt {\frac {m_{1}\left(1-\beta ^{2}\right)}{k}}}$

In the second case — when the rod is parallel to the line of relative motion of <marh>S_1</math> and $S_{2}$ — the amount of twisting in the wire for a given position of the balls is the absolute value of $(\pi /2)-\theta$. The potential energy is ${\tfrac {1}{2}}k\left[(\pi /2)-\theta \right]^{2}$. We have

$x={\frac {y}{\sqrt {1-\beta ^{2}}}}\cot \theta$

For small oscillations we have

$y={\sqrt {1-\beta ^{2}}},\ \cot \theta =\tan \left({\frac {\pi }{2}}-\theta \right)={\frac {\pi }{2}}-\theta .$

Hence the potential energy is ${\tfrac {1}{2}}kx^{2}$ the period $T_{2}$ of oscillation is therefore

$T_{2}=2\pi {\sqrt {\frac {m_{2}}{k}}}.$

Equating the two periods of oscillation found above we have

$m_{2}=\left(1-\beta ^{2}\right)m_{1}.$

Remembering that $m_{1}$ and $m_{2}$ are the longitudinal mass and transverse mass, respectively, and making use of theorem I., we have the following result:

Theorem II. Let $m_{0}$ denote the mass of a body when at rest relative to a system of reference S. When it is moving with a velocity v relative to S denote by $l\left(m_{v}\right)$ its longitudinal mass, that is, its mass in a direction parallel to its line of motion. Then we have

$l\left(m_{v}\right)={\frac {m_{0}}{\left(1-\beta ^{2}\right)^{\frac {3}{2}}}}$

where $\beta =v/c$ and c is the velocity of light $\scriptstyle \left(MVLRC_{1}C_{2}\right)$.

§ 3. On the Dimensions of Units.

Denote the fundamental measurable physical entities mass, length and time by M, L and T respectively. Then the definition of derived entities gives rise to the so-called dimensional equations. Thus if B denote velocity, then from the definition of velocity we have the dimensional equation

$V={\frac {L}{T}}$.

That such equations must be useful in obtaining the relations of a derived unit in two systems of reference is obvious. Thus from the above dimensional equation for V we may at once derive the fundamental result (see previous paper, theorem VI., p. 170) concerning the relation of units of length in the line of relative motion of two systems not at rest relatively to each other. For this purpose it is sufficient to employ postulate V. and theorem IV. of the previous paper. The reader can easily supply the argument. Or, conversely, if one knows the relations which exist between units of length and units of time in two systems one concludes readily to the truth of postulate V.

Likewise, from the dimensional equation

acceleration = ${\frac {L}{T^{2}}}$

one may readily determine the relations which exist between units of acceleration on two systems, it being assumed that the relations of time units and length units are known. Making this assumption, then, the two dimensional equations above give us the following theorem:

Theorem III. Let two systems $S_{1}$ and $S_{2}$ move with a relative velocity v in the direction of a line l, and let $\beta =v/c$ where c is the velocity of light. Then to an observer on $S_{1}$ it appears that the unit of velocity [acceleration] on $S_{1}$ bears to the unit of velocity [acceleration] on $S_{2}$ the ratio $1:1\left[1:{\sqrt {1-\beta ^{2}}}\right]$ or $1:{\sqrt {1-\beta ^{2}}}\left[1:1-\beta ^{2}\right]$ according as the motion is parallel to I or perpendicular to I (MVLR).

Let us use F to denote force. Then from the dimensional equation

$F={\frac {ML}{T^{2}}}$,

we shall be able to draw an interesting conclusion concerning the measurement of force.

Suppose that an observer B on a system $S_{2}$ carries out some observations concerning a certain rectilinear motion, measuring the quantities M', L', T', so that he has the equation

$F'={\frac {M'L'}{T'^{2}}}$.

Another observer A on a system $S_{1}$ (having with respect to $S_{2}$ the velocity v in the line l) measures the same force, calling it F. Required the value of F in terms of $F'$, when the motion is parallel to l and when it is perpendicular to l, the estimate being made by A.

When the motion is perpendicular to l — that is, when the force acts in a line perpendicular to l — we have

$F_{1}={\frac {ML}{T^{2}}}={\frac {M'{\sqrt {1-\beta ^{2}}}\cdot L'}{T'^{2}\left(1-\beta ^{2}\right)}}={\frac {F'}{\sqrt {1-\beta ^{2}}}}$

When the motion is parallel to l we have

$F_{2}={\frac {ML}{T^{2}}}={\frac {M'\left(1-\beta ^{2}\right)^{\frac {3}{2}}\cdot L{\sqrt {1-\beta ^{2}}}}{T'^{2}\left(1-\beta ^{2}\right)}}=\left(1-\beta ^{2}\right)F'.$

These results may be stated in the following theorem:

Theorem IV. In the same systems of reference as in theorem III., let an observer on $S_{2}$ measure a given force F' in a direction perpendicular to l and in a direction parallel to l, and let $F_{1}$ and $F_{2}$ be the values of this force as measured in the first and second cases respectively by an observer on $S_{1}$. Then we have

$F_{1}={\frac {F'}{\sqrt {1-\beta ^{2}}}},\ F_{2}=\left(1-\beta ^{2}\right)F'$

$\scriptstyle \left(MVLRC_{1}C_{2}\right).$

It is obvious that a similar use may be made of the dimensional equation of any derived unit in determining the relation which exists between this unit in two relatively moving systems of reference.

§ 4. Equivalents of the Postulates.

The problem of determining sets of postulates logically equivalent to those which have been used as a basis of the theory of relativity is obviously important. So far as a justification of the theory is concerned it is, however, unnecessary to have complete logical equivalents; all that is essential is to find a set of postulates, experimentally demonstrable, by means of which it is possible to demonstrate the characteristic conclusions of relativity concerning the relations of units of time and units of length in two systems of reference.^[7] Such a set of postulates we shall call essential equivalents of the postulates of relativity. The object of this section is to determine essential equivalents of postulate R, that is, such postulates as may be taken in connection with postulates M, V, L, so that the new set shall be essentially equivalent to M, V, L, R.

For this purpose let us first consider the relation between the transverse mass of a moving body and its mass at rest as given in theorem I. Let us suppose that this theorem is true^[8] (whether proved experimentally or otherwise); and let us seek its consequences. Suppose that the experiment by means of which we proved theorem I. is now repeated. If we again assume the law of conservation of momentum and equate the two observed changes in momenta, it is clear that we shall have a relation between measurements of time as carried out in the two systems of reference, and that this relation will be precisely the same as in the usual theory of relativity. Having this relation concerning time units we can then proceed as in the first paragraph in § 3 to derive the usual relations between units of length. Hence we have the following result:

Theorem V. If $m_{0}$ and $t\left(m_{v}\right)$ have the same meaning as in theorem I. and if for any particular kind of matter whatever we have the relation

$t\left(m_{v}\right)={\frac {m_{0}}{\sqrt {1-\beta ^{2}}}}$

then this fact and postulates $\scriptstyle \left(MVLC_{1}\right)$ form an essential equivalent of postulates $\scriptstyle \left(MVLRC_{1}\right)$.

Next, let us suppose that for some particular kind of matter we have the relation

$t\left(m_{v}\right)=\left(1-\beta ^{2}\right)l\left(m_{v}\right)$

where $t\left(m_{v}\right)$ and $l\left(m_{v}\right)$ denote the transverse mass and the longitudinal mass, respectively, as above. Then repeat the experiments by means of which we proved theorem II. As before the balls will appear to B to move on arcs of the circle

$x'=\cos \theta ,\ y'=\sin \theta .$

Suppose that to A they appear to move along arcs of the ellipse^[9]

$x=\cos \theta ,\ y=\rho \sin \theta$

where ρ is a constant to be determined. As before, without the use of postulate R, it may be shown that to A the periods will be the same in the two cases. Then determine the periods as in the preceding discussion. The expressions for the period will contain ρ; in fact on equating them we shall find

$m_{2}=\rho ^{2}m_{1}.$

But $m_{1}=l\left(m_{v}\right)$ and $m_{2}=t\left(m_{v}\right)$ whence on account of the relations between $l\left(m_{v}\right)$ and $t\left(m_{v}\right)$, we have at once

$\rho ={\sqrt {1-\beta ^{2}}}$

This, in connection with postulate L, leads readily to the usual relations concerning the units of length in two systems of reference. Having these relations of length units, the dimensional equation

$V={\frac {L}{T}}$

taken in connection with postulate V leads at once to the usual relation of time units, provided we take the motion along the line of relative motion of the two systems. Hence we have the following theorem:

Theorem VI. If $l\left(m_{v}\right)$ and $t\left(m_{v}\right)$ have the same meaning as in theorems L and II. and if for any particular kind of matter whatever we have the relation

$t\left(m_{v}\right)=\left(1-\beta ^{2}\right)\cdot l\left(m_{v}\right)$

then this fact and postulates $\scriptstyle \left(MVLC_{1}C_{2}\right)$ are essential equivalents of postulates $\scriptstyle \left(MVLRC_{1}C_{2}\right)$.

§ 5. The Bucherer Experiment.

^[10] Our postulates V, L, $C_{1}$ have been universally accepted as part of the basis of the classical mechanics. Many persons have found no difficulty in accepting postulate M; certain it is at least that we have absolutely no evidence to contradict it. We have seen in theorem V, that these four postulates, taken in connection with the formula for transverse mass, form an essential equivalent of $\scriptstyle \left(MVLRC_{1}\right)$; in other words, the experimental demonstration of the formula for transverse mass carries with it the experimental proof of the theory of relativity, provided that postulates $\scriptstyle \left(MVLC_{1}\right)$ are accepted as experimentally proved.

Bucherer^[11] has carried out some investigations which have been supposed to furnish this experimental verification for the formula of transverse mass, and hence for the whole theory of relativity. In order to draw this conclusion from Bucherer's direct results it is necessary to make use of a law which we have not yet employed, namely, the law of conservation of electricity which we have stated as postulate $C_{2}$. Since this law has customarily been accepted and has not yet led to contradictions it should certainly still be supposed to hold. Accepting it, then, we have in Bucherer's results a partial experimental confirmation of the theory of relativity, as we now show.

Bucherer's investigations have to do with the mass of a moving electron. There seems to be no means at hand for a direct measurement of this mass, and Bucherer resorted to the expedient of determining the ratio of charge to mass. Let us denote the charge by e, which we suppose to be constant, in accordance with postulate $C_{3}$. As before let $m_{0}$ and $t\left(m_{v}\right)$ denote the mass at rest and the mass when moving with velocity v, of the electron in consideration. Bucherer's experiments were carried out to determine the relation which exists between $e/m_{0}$ and $e/t\left(m_{v}\right)$. The measurements agreed in a remarkable way, not only as to general characteristics but also as to exact numerical results, with the formula^[12]

${\frac {e}{t\left(m_{v}\right)}}={\frac {e{\sqrt {1-\beta ^{2}}}}{m_{0}}}$

Taking this formula as thus experimentally demonstrated we have at once our fundamental relation for transverse mass: ${\sqrt {1-\beta ^{2}}}t\left(m_{v}\right)=m_{0}$.

From this it follows that the experimental demonstration of the theory of relativity is complete when we have proved M, V, L, $C_{1}$ and $C_{3}$, provided that one accepts Bucherer's proof of the above relation between $e/m_{0}$ and $e/t\left(m_{v}\right)$. That is, the essentials of the theory of relativity flow from principles for each of which there is strong experimental confirmation. This important conclusion has often been pointed out.

To the present writer, however, it seems that one point especially should be subjected to further examination. Is it in fact true that the charge of a moving electron is independent of the velocity with which it moves? Let $e_{0}$ be the charge of the electron when at rest and denote by $t\left(e_{v}\right)$ its apparent charge when in motion with velocity v, the charge being measured by means of tests in which the line of action is perpendicular to the line of motion of the charge. In the above work we have assumed, in accordance with the usual practice, that $e_{0}=t\left(e_{v}\right)$. Suppose however that the true relation were different from this, that, in fact, we have

$t\left(e_{v}\right)=e_{0}{\sqrt {1-\beta ^{2}}}$

then Bucherer's experiment would lead to the conclusion that $t\left(m_{v}\right)=m_{0}$, and thus the whole theory of relativity would be overturned. Furthermore, if any relation other than $e_{0}=t\left(e_{v}\right)$ is the true one, some modification at least of the theory of relativity would have to be made or else one would have to give up postulate $C_{1}$ which asserts the law of conservation of momentum. This result brings to notice the great importance of the question of the constancy of electric charge on the electron. We shall treat this matter further in the next section.

§ 6. Another Means for the Experimental Verification of the Theory of Relativity.

Just as theorem V. was used for the theoretical basis of Bucherer's (partial) experimental demonstration of the theory of relativity so theorem VI. may be employed as the theoretical basis of a new experimental investigation which has not yet been carried out, one which bears the same essential relation as that of Bucherer to the confirmation or disproof of the entire theory of relativity. The object of this section is to indicate the nature of this experiment.

Let $e_{0}$ denote the charge of an electron when at rest with respect to a given system of reference. When it is in motion with a velocity v let $t\left(e_{v}\right)$ and $l\left(e_{v}\right)$ be the apparent charge when measured by means of tests whose lines of action are perpendicular and parallel, respectively, to the line of motion of the electron.

If we employ, postulate $C_{3}$ we conclude that $e_{0}=t\left(e_{v}\right)=l\left(e_{v}\right)$. We shall first assume the truth of one of these relations, namely, $t\left(e_{v}\right)=l\left(e_{v}\right)$, and we shall denote the common value of these two quantities by e. Now let us suppose that some means are found for measuring both the quantities $e/t\left(m_{v}\right)$ and $e/l\left(m_{v}\right)$, where $t\left(m_{v}\right)$ and $l\left(m_{v}\right)$ denote as usual the transverse mass and the longitudinal mass respectively of the moving electron, whose velocity is v. Bucherer's methods furnish the means of measuring the first of these ratios; it will be necessary to devise a way to determine the value of the second ratio.

Or, instead of finding a means of measuring the two quantities $e/t\left(m_{v}\right)$ and $e/l\left(m_{v}\right)$ it will be sufficient if one determines only their ratio, as will be obvious from the discussion following.

Suppose now that we find the relation predicted by the theory of relativity:

${\frac {e}{t\left(m_{v}\right)}}={\frac {e}{\left(1-\beta ^{2}\right)l\left(m_{v}\right)}}$

This equation leads to the relation $t\left(m_{v}\right)=\left(1-\beta ^{2}\right)\cdot l\left(m_{v}\right)$. According to theorem VI. this would give a new experimental confirmation of the theory of relativity. The importance of such a result is apparent.

But we should also have more than this. Having now concluded that the theory of relativity is confirmed and this result having been reached without the use of a relation between $e_{0}$ and $t\left(e_{v}\right)$ we may now use the experiment of Bucherer to draw further conclusions concerning electric charges in motion. In particular, it is obvious that we should have a proof of the fundamental relation

$e_{0}=t\left(e_{v}\right)$

That is to say, having assumed that $t\left(e_{v}\right)$ and $l\left(e_{v}\right)$ are equal we conclude further on experimental evidence that each of these is equal to $e_{0}$. Now it is difficult to conceive how $t\left(e_{v}\right)$ and $l\left(e_{v}\right)$ could be different, for this would imply that the notion of electric charge is in need of essential modification. In fact, if the charged body is moving, the notion of charge would be indefinite in meaning until we had assigned the direction along which such charge is to be measured. Thus, if the experiment should turn out as surmised above, we should not only have the strongest sort of experimental confirmation of the theory of relativity but we should also have a valuable verification of the fact that an electric charge does not vary in amount with the velocity of the body which carries it.

Suppose, on the other hand, that we make no assumption concerning the relation of $t\left(e_{v}\right)$ and $l\left(e_{v}\right)$ or of $t\left(m_{v}\right)$ and $l\left(m_{v}\right)$. On carrying out the experiments a relation of the form

${\frac {t\left(e_{v}\right)}{t\left(m_{v}\right)}}=k{\frac {l\left(e_{v}\right)}{l\left(m_{v}\right)}}$

will be obtained where k is a constant or a variable depending on v. If it is found that k is different from unity we shall be forced to the conclusion that either our conception of mass in the classical mechanics or our conception of charge in the classical electrical theory is in need of essential modification. Again, if k = I and if we assume, as is natural, that $t\left(e_{v}\right)=l\left(e_{v}\right)$ then we have an experimental disproof of the theory of relativity. In fact we have such a disproof unless $k=1/\left(1-\beta ^{2}\right)$, provided of course that we assume $t\left(e_{v}\right)=l\left(e_{v}\right)$.

From these remarks it is obvious that, whatever may be the result of the experiments, they will certainly lead to important conclusions of a fundamental nature; that is, we have here a crucial experiment, one that cannot fail to lead somewhither. It is to be hoped that some laboratory worker will soon perform the requisite experiments; the writer, who is a mathematician, can only regret that he cannot conveniently carry out the work himself.

A variation of the experiment of Bucherer would seem to be sufficient for the purpose here. Bucherer's results were obtained by subjecting the moving electron to a magnetic field and also to an electric field each at right angles to the line of motion. A variation of the direction of these fields relative to the line of motion of the electron would probably afford a means of making the necessary measurements for the experimental proof of the relations requisite for use in the preceding discussion.

§ 7. Mass and Energy.

If, as is frequently done, we employ for the definition of the kinetic energy E the relation (compare §1)

$E=\int _{0}^{v}Mdv=\int _{0}^{v}mvdv$

it is clear that for the mass m we should take the longitudinal mass $l\left(m_{v}\right)$. Then let $m_{0}$ denote the mass of the body at rest, $E_{0}$ its energy when at rest (that is, the energy due to its internal activity), and $E_{v}$ its energy when moving at the velocity v. Then clearly $E=E_{v}-E_{0}$, so that in view of theorem II. we have

$E=E_{v}-E_{0}=\int _{0}^{v}{\frac {m_{0}vdv}{\left(1-\beta ^{2}\right)^{\frac {3}{2}}}}$

whence, on integration, we have

$E=E_{v}-E_{0}=m_{0}c^{2}\left({\frac {1}{\sqrt {1-\beta ^{2}}}}-1\right)$

(1)

Hence for the kinetic energy of a moving body we have

$E=m_{0}c^{2}\left({\frac {1}{2}}\beta ^{2}+{\frac {3}{8}}\beta ^{4}+\dots \right)$

or, to a first approximation only,

$E={\frac {1}{2}}m_{0}v^{2}$

Therefore the usual formula for kinetic energy in the classical mechanics is only a first approximation.

Since relation (1) is to be true for all velocities v it is obvious that we have

$E_{v}={\frac {m_{0}c^{2}}{\sqrt {1-\beta ^{2}}}}+k,\ E_{0}=m_{0}c^{2}+k,$

where k is a constant, that is, a quantity independent of v. From the first of these equations we conclude further that

$E_{v}=c^{2}\cdot t\left(m_{v}\right)+k$

so that the total energy of a body, decreased by the constant k, is directly proportional to its transverse mass. In case the body is at rest its mass in one direction is the same as in another; hence $m_{0}=t\left(m_{0}\right)$. Bearing this in mind we have the following theorem:

Theorem VII. Let $m_{0}$ be the mass of a body when at rest with respect to a given system of reference and let $t\left(m_{v}\right)$ denote its transverse mass when it is moving with the velocity v (the case v = 0 is not excluded). Then the total energy $E_{v}$ which it possesses is $c^{2}t\left(m_{v}\right)+k$, where k is a constant.

The following relations are immediate consequences of equations written out above:

${\frac {E_{v}-E_{0}}{t\left(m_{v}\right)-m_{0}}}=c^{2},\ E_{v}-k={\frac {E_{0}-k}{\sqrt {1-\beta ^{2}}}}$

Now, suppose that an experimenter contributes to a body which is at rest a known amount of energy and determines the velocity which this causes the body to acquire. If the two measurements are made with sufficient accuracy one will be able, by substituting the results in the first of the above equations, to determine in this way the velocity of light. Actually to carry out this remarkable method for measuring c would doubtless be very difficult; but the obvious great importance of the result is certainly such as to justify a careful consideration of the problem. If the value of c determined in this way should agree well with its value as otherwise found, this would give us an interesting confirmation of the theory of relativity.

Let us consider the mass of a rotating top, the mass being measured along the axis of rotation. According to our results this mass should be different from that of the same top when at rest, and the difference should bear a definite relation to the amount of energy which is involved in the rotation. If the measurements here involved could be made with sufficient accuracy we would have another means, independent of light itself, for the measurement of the light-velocity c. Again, this experiment would afford us a measure of transverse mass and in that way could lead to a confirmation of the theory of relativity, provided that we assume c as known from independent measurements; and this confirmation, it is to be noticed, would be independent of electrical considerations.

It seems to be impossible to determine the constant k which enters into the above discussion. But in the absence of any evidence to the contrary it would appear natural tentatively to assume that k is zero. On the basis of this assumption we should have the following remarkable conclusions: The mass of a body at rest is simply the measure of its internal energy. The transverse mass of a body in motion is the measure of its internal energy and its kinetic energy taken together.^[13] Its longitudinal mass is its total energy multiplied by a simple factor. (The longitudinal mass, therefore, bears a simple ratio to the total energy.) One can hardly resist the conclusion that the transverse mass of a body depends entirely on its energy, and therefore that matter is merely one manifestation of energy.

§ 8. Remarks on the Principle of Least Action.

In the preceding section we have seen that in the theory of relativity the classical formula $E={\tfrac {1}{2}}mv^{2}$ for the measure of kinetic energy is true only as a first approximation. This is due to the fact that mass is a variable quantity. But the conclusion does not appear to necessitate our surrender of the law of conservation of energy.

The same causes which lead to a modification in the formula for E will also require a corresponding modification in the value of the action A as defined in § I. The question arises as to whether the principle of least action is left intact. I cannot enter upon the investigation here; but the problem seems to me to be of importance, and consequently I am stating it in the hope that some one will be led to consider its solution.

Undoubtedly the principle of least action is one which should be given up only when there are strong reasons for it. It is a mathematical formulation of the law that nature accomplishes her ends with the least possible expenditure of labor, so to speak. Certainly this law is one which appeals to our minds with strong force. There is something about it which is aesthetically satisfying in a high degree. It seems to me, however, that a fresh study of it should be made in the light of the theory of relativity.

§ 9. Speculative Considerations.

From some results in the preceding discussion it has appeared that the transverse mass of a body is merely a manifestation of its total energy, that it is in fact a measure of that energy. It is then natural to suppose, on the other hand, that anything which possesses energy has mass; and we thus conceive of mass and energy as coextensive.

Now a beam of light possesses energy; whence we conclude naturally that it also has mass. But we have seen that no "material body" can have a velocity as great as that of light. How are these two facts to be reconciled? If we define "matter" as that which possesses mass (and this is probably the best definition) we shall perhaps best be able to represent to ourselves the nature of matter if we think of it as a strain in the ether. Then the two facts which we have to reconcile would be entirely consistent if we suppose that the beam of light sets up a strain in the ether (whence its mass) but that this strain as a whole is not propagated with the velocity of light. In fact, if it moves at all it is probably with a velocity relatively much smaller than that of light.

Again, if mass is merely a manifestation of energy in the form of a strain in the ether it would follow that gravitation is simply an interaction among these several strains. A strain principally localized in one place would have lines of strain going out from it in all directions, and the action of these lines of strain upon one another would afford the effective means by which gravitation acts.

Whether these strains should be thought of as static in the ether or as due to the relative motion of the parts of the ether would probably be determined differently by different minds. If one inclines to the latter form of representation, the necessary movement might be conceived of as vortex whorls in the ether, setting up strains and lines of strain by aid of which we are able to interpret observed phenomena.

Indiana University,

November, 1912.

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1. Physical Review, Vol. 35 (1912), pp. 153-176.
2. Several applications of the principal conclusions are also given.
3. Phil. Mag., 18 (1909), 510-523. See especially pp. 517-518.
4. These letters attached to the theorem indicate that in its proof we have employed postulates M, V, L, R, $C_{1}$. Compare a similar usage in my previous paper.
5. American Journal of Science (4) 26 (1908), pp. 498-500.
6. That the potential energy is proportional to $\theta ^{2}$ when measured by B is obvious. Since A observes a different apparent angle $\theta '$ (say) corresponding to $B$'s observed angle $\theta$ it might at first sight appear that the potential energy as observed by A is proportional to $\theta '^{2}$. That this is not the case is seen from the fact that for a given twist in the wire $\theta '$ depends on the direction of equilibrium of the bar, that is, it depends on the way in which the bar is attached to the wire; hence, if the potential energy as observed by A were proportional to $\theta '^{2}$ it would depend on the way in which the bar is attached. Since this is obviously not the case we conclude that the potential energy is proportional to $\theta ^{2}$.
7. It is obvious that we should then be able to demonstrate theorems I. and II. concerning the mass of a moving body.
8. All that is essential to the argument is the truth of theorem I. for a particle of matter of some one kind; it need not be assumed to be true universally.
9. Since we are assuming postulate L it is clear that the path must be of this form.
10. Compare Tolman, Physical Review, 31 (191 o), p. 36.
11. Annalen der Physik (4) 28 (1909), 513-536.
12. As a matter of fact Bucherer did not measure the ratio $e/m_{0}$. Instead of this he considered the ratio $e/t\left(m_{v}\right)$ for a considerable range of values of v and noticed that its value always agreed with the formula $e/t\left(m_{v}\right)=k{\sqrt {1-\beta ^{2}}}$, where k is a constant. It appears natural, then, to assume that $m_{0}=e/k$, whence one has the formula in the text. It should be emphasized that this assumption is necessary in order that the Bucherer results may be associated with our theorem as in the text, and consequently the conclusions there reached can be accepted with no stronger confidence than that which one has in the accuracy of the above assumption. See the next section where a means of experimental verification of the theory of relativity is suggested which does not depend on this assumption for its validity.
13. Compare Lewis and Tolman, Phil. Mag. (6) 18 (1909), p. 521.