# A revision of the Fundamental Laws of Matter and Energy

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Philosophical Magazine, 1908, 6 16 (95): 705-717, Online
THE
LONDON, EDINBURGH, AND DUBLIN
PHILOSOPHICAL MAGAZINE
AND
JOURNAL OF SCIENCE.
[SIXTH SERIES.]
NOVEMBER 1908.
RECENT publications of Einstein and Comstock on the relation of mass to energy have emboldened me to publish certain views which I have entertained on this subject and which a few years ago appeared purely speculative, but which have been so far corroborated by recent advances in experimental and theoretical physics that it seems desirable to subject these views to a strict logical development, although in so doing it will be necessary to modify those fundamental principles of the mechanics of ponderable matter which have remained unaltered since the time of Newton.
The recent experiments which indicate a change in the mass of an electron with the speed, together with the phenomenon of radioactivity, have in some minds created a doubt as to the exact validity of some of the most general laws of nature. In the following pages I shall attempt to show that we may construct a simple system of mechanics which is consistent with all known experimental facts, and which rests upon the assumption of the truth of the three great conservation laws, ​namely, the law of conservation of energy, the law of conservation of mass, and the law of conservation of momentum. To these we may add, if we will, the law of conservation of electricity.
The Relation of Mass to Energy.
When a black body is placed in a beam of light it is subject to a pressure or force which tends to move it in the direction in which the light is moving. If ${\frac {d\mathrm {E} }{dt}}$ denotes the time-rate at which the body receives energy, f the force, and V the velocity of light, we have in rational units the formula
| $f={\frac {1}{\mathrm {V} }}{\frac {d\mathrm {E} }{dt}}$. | (1) |
This important equation, which was obtained by Maxwell as a consequence of his electromagnetic theory, and by Boltzmann through the direct application of the laws of thermodynamics, has recently been verified with remarkable precision in the beautiful experiments of Nichols and Hull.
A body subjected to the pressure of radiation will acquire momentum, and if we are to accept the law of conservation of momentum, we must conclude that some other system is losing in the same direction an equivalent momentum. We are thus led inevitably, as Poynting has shown, to the idea that the beam of radiation carries not only energy but momentum as well.
The body subject to the constant force of radiation f, will obviously acquire momentum at the rate
| ${\frac {d\mathrm {M} }{dt}}=f$. | (2) |
Combining equations (1) and (2) gives
| ${\frac {d\mathrm {E} }{d\mathrm {M} }}=\mathrm {V}$. | (3) |
The ratio of the acquired energy to the acquired momentum is equal to the velocity of light. The beam of radiation must, therefore, possess energy and momentum in the same ratio. Hence for the beam itself, or any part of it,
| ${\frac {\mathrm {E} }{\mathrm {M} }}=\mathrm {V}$. | (4) |
Postulating the validity of the fundamental conservation laws mentioned above, we shall need in the following development only this one cardinal assumption, that a beam of radiation possesses not only momentum and energy, but also mass, travelling with the velocity of light, and that a body absorbing radiation is acquiring this mass as it also acquires the momentum and the energy of the radiation. Therefore a body which absorbs radiant energy increases in mass.
The amount of this increase is readily found. If in general we write momentum as the product of mass and velocity, then the momentum of any part of a beam of radiation having the mass m will be given by the equation :
| $\mathrm {M} =m\mathrm {V}$. | (5) |
The increase dM in the momentum of the body absorbing the radiation will therefore equal the increase dm in its mass, multiplied by the velocity of light,
| $d\mathrm {M} =\mathrm {V} dm$. | (6) |
Combining this equation with (3) we find
| $d\mathrm {m} ={\frac {d\mathrm {E} }{\mathrm {V} ^{2}}}$, | (7) |
or if we write $\mathrm {V} =3\times 10^{10}$ centimetres per second,
$dm=1.111\times 10^{-21}d\mathrm {E}$
Thus a body receiving or emitting radiant energy gains or loses mass in proportion and by the amount $1.111\times 10^{-21}$ grams for every erg. This is a small quantity, indeed, but one which is not to be neglected.
Assuming the fundamental conservation law, we must regard mass as a real property of a body which depends upon its state, and not upon its history. Hence it is obvious that if in any other way than by radiation the body gains or loses energy it must still gain or lose mass in just the above proportion. In other words, any change in a body's content of energy is accompanied by a definite change in its mass, ​regardless of the nature of the process which the energy change accompanies.
Since therefore when a body loses a given quantity of energy it always loses a definite quantity of mass, we might assume that if it should lose all its energy it would lose all its mass, or, in other words, that the mass of a body is a direct measure of its total energy, according to the equation,
| $m={\frac {\mathrm {E} }{\mathrm {V} ^{2}}}$. | (8) |
We should then regard mass and energy as different names and different measures of the same quantity, and say that one gram equals $9\times 10^{20}$ ergs in the same sense that we say one metre equals 39.37... inches.
Plausible as this view seems, it rests upon an additional hypothesis besides the fundamental postulate which we have chosen. We shall use therefore, not equation (8) but equation (7) as the basis of the following work.
It is to be noted that equation (8) has also been obtained by Einstein (loc. cit.), who derived it from the general equations of the electromagnetic theory, with the aid of the so-called principle of relativity. That a different method of investigation thus leads to the same simple equation we have here deduced, speaks decidedly for the truth of our fundamental postulate.
Comstock (loc. cit.) from electromagnetic theory alone has also concluded that mass is proportional to energy, but his equation is
$m={\frac {4}{3}}{\frac {\mathrm {E} }{\mathrm {V} ^{2}}}$
To investigate for the cases studied by Comstock the cause or the justification for this appearance in the equation of the factor 4/3 would lead too far into electromagnetic theory, from which in the present paper I wish to hold entirely aloof.
Before proceeding to develop fully the consequences of equation (7) it may be well to point out an apparent ​inconsistency in the equations for the momentum and the energy of a beam of radiation. The momentum of the beam of mass m we have given in equation (5) as
$\mathrm {M} =m\mathrm {V}$.
From our assumption that the energy of the beam is simply the kinetic energy of the moving mass m, we might expect from our knowledge of elementary mechanics to find for the energy the equation
$\mathrm {E} =l/2m\mathrm {V} ^{2}$;
whereas in fact we find from equations (4) and (5) that
| $\mathrm {E} =m\mathrm {V} ^{2}$. | (9) |
We shall see, however, in the next section that this comparison of equations (5) and (9) instead of demolishing our theory actually furnishes a remarkably satisfactory argument in its favour.
Non-Newtonian Mechanics.
One of the interesting branches of modern mathematics has grown out of the study of those geometries which would result from the change of one or more of the axioms of Euclid. These non-Euclidian geometries present the properties of purely imaginary kinds of space and are therefore so far mere exercises in logic, without any physical significance. But their investigation was doubtless prompted in some cases by the belief that experiment itself may sometime show that there are deviations from the ordinary laws of space when these laws are subjected to tests of a different order from those of common mensuration. Indeed it is not unlikely that Euclidian geometry may prove inadequate when we are able to subject to an accurate metric investigation the vast stretches of interstellar space or the minute regions which we believe to be encompassed within an atom or an electron.
The science of mechanics, like geometry, has been built up from a set of simple axioms, which were laid down by Newton. But the conclusions of the previous section lead us to modify one of these axioms and thus lay the foundation of a system of non-Newtonian mechanics.
The axiom which we must surrender is the one which states that the mass of a body is independent of its velocity. We have concluded that mass is proportional to content of energy. When a body is set in motion it gains kinetic energy and therefore its mass must change with its velocity. In place of ​the axiom which we have abandoned we must substitute equation (7).
Before investigating the consequences of this step it is necessary to define exactly the principal mechanical quantities which we are to use.
Extension in space (l) and time (t) will be measured in the usual way and the centimetre and the second will be employed as units.
Force (f) will be given its usual significance and the unit, the dyne, will be that force which, acting upon the International standard kilogram, when the latter is at rest, imparts to it an initial acceleration of ⋅001${\frac {cm.}{sec.^{2}}}$.
The momentum (M) of a moving body will be measured by the time in which it is brought to rest under the influence of a constant opposing force of one dyne acting in the line of its motion.
The mass (m) of a moving body will be defined as the momentum divided by the velocity (v), that is,
| $m={\frac {\mathrm {M} }{v}}$. | (10) |
The limiting ratio of the momentum of a body to its velocity, when it is brought to rest, will be called its mass when at rest. The unit of mass is the gram.
The kinetic energy (E') of a body will be measured by the distance through which it will move before being brought to rest by a constant opposing force of one dyne, acting in the line or the body's motion. The unit of energy will be the erg.
These definitions, although somewhat unusual in form, are perfectly consistent with the ordinary definitions of Newtonian mechanics. But they have been so chosen as to be consistent also with equation (7) and the fundamental conservation laws. Obviously equation (7) itself is not inconsistent with these conservation laws, for although a body increases in mass as it gains kinetic energy, some other system is losing the same mass as it loses the same energy.
In accordance with the above definitions we may write
| $d\mathrm {M} =fdt$, | (11) |
| $d\mathrm {E} '=fdl$. | (12) |
Let us consider a body originally moving with a velocity v subjected for the time dt to a force f in the line of its motion. ​Its momentum and kinetic energy will change according to (11) and (12) by the amounts
* $d\mathrm {M} =fdt$,
* $d\mathrm {E} '=fdl=fvdt$.
Hence
| $d\mathrm {E} '=vd\mathrm {M}$. | (13) |
So far the equations are those of Newtonian mechanics, but now in substituting for M from equation (10) we must regard m as a variable and write
| $d\mathrm {E} '=mv\ dv+v^{2}dm$. | (14) |
This will be our fundamental equation connecting the kinetic energy of a body with its mass and velocity.
Introducing now the relation of mass to energy given in equation (7) we may write,
$d\mathrm {E} '=\mathrm {V} ^{2}dm$,
and combining this equation with (14) gives
$\mathrm {V} ^{2}dm=mv\ dv+v^{2}dm$.
This equation, containing only two variables, m and v and the constant V, may readily be integrated as follows. By a simple transformation
$\left(1-{\frac {v^{2}}{\mathrm {V} ^{2}}}\right)dm={\frac {mvdv}{\mathrm {V} ^{2}}}$.
Writing β=v/V, and noting that
${\frac {vdv}{\mathrm {V} ^{2}}}=-{\frac {1}{2}}d\left(1-\beta ^{2}\right)$,
we see that
${\frac {dm}{m}}=-{\frac {1}{2}}{\frac {d\left(1-\beta ^{2}\right)}{\left(1-\beta ^{2}\right)}}$.
Hence
$\log m=-{\frac {1}{2}}\log \left(1-\beta ^{2}\right)+\log m_{0}$,
where log m$_0$ is the integration constant. Therefore
$\log {\frac {m}{m_{0}}}=\log \left(1-\beta ^{2}\right)^{-{\frac {1}{2}}}$
or
| ${\frac {m}{m_{0}}}={\frac {1}{\left(1-\beta ^{2}\right)^{1/2}}}$. | (15) |
This is the general expression for the mass of a moving body in terms of β, the ratio of its velocity to the velocity of ​light. When β is zero, m=m$_0$. m$_0$ represents therefore the mass of the body at rest.
If we substitute in the equation numerical values of β we find that, while the quotient m/m$_0$ becomes infinite when the velocity equals the velocity of light, it remains almost equal to unity until the velocity of light is closely approached. Thus a ton weight given the velocity of the fastest cannon-ball would, according to this equation, gain in mass by less than one millionth of a gram. It is obvious that, except in those unusual cases in which we deal with velocities comparable with that of light, our non-Newtonian equations are identical with those of ordinary mechanics far within the limits of error of the most delicate experiments.
Recently, however, it has been possible to study, in the negative particles emitted by radioactive substances, bodies which sometimes move with a velocity only a little less than that of light. In a series of remarkably skilful experiments Kaufmann was able to measure the ratio of electric charge to mass (e/m) for negative particles moving at different speeds. Assuming that the charge is constant, the fact that e/m varies with the speed of the particle must be attributed to a variation of the mass with the speed. On this assumption it is possible to calculate from Kaufmann's experiments the values of m/m$_0$ at the different velocities.
The mass of a negative particle is usually spoken of as electromagnetic mass, but if we are to hold to our definitions we must recognize only one kind of mass. In general we have defined the mass of a moving body as the quotient of the time during which it will be brought to rest by unit force, divided by the initial velocity. It matters not what the supposed origin of this mass may be. Equation (15) should therefore be directly applicable to the experiments of Kaufmann. In the following table are given the values of
| m—⋅m$_0$ | β (observed) | β (calculated) |
| 1 1⋅34 1⋅37 1⋅42 1⋅47 1⋅54 1⋅65 1⋅73 2⋅05 2⋅14 2⋅42 | 0 ⋅73 ⋅75 ⋅78 ⋅80 ⋅83 ⋅86 ⋅88 ⋅93 ⋅95 ⋅96 | 0 ⋅67 ⋅69 ⋅71 ⋅73 ⋅76 ⋅80 ⋅82 ⋅88 ⋅89 ⋅91 |
It will be seen that the observed values of β follow to a remarkable degree the same trend as those which are calculated by equation (15), but are in every case six to eight per cent higher. I believe that these differences lie within the limits of experimental error of Kaufmann's measurements. It is true that he claims a higher degree of accuracy, but, notwithstanding the extreme care and delicacy with which the observations were made, it seems almost incredible that measurements of this character, which consisted in the determination of the minute displacement of a somewhat hazy spot on a photographic plate, could have been determined with the precision claimed. Moreover, Planck and Stark have pointed out certain corrections which probably should have been made by Kaufmann and which would produce a material change in his results.
That a charged particle must possess mass in virtue of its charge, and that this mass must vary with the velocity of the particle, was shown to be a consequence of the electromagnetic theory by J. J. Thomson and by Heaviside, and numerous attempts have been made to find the exact expression for the change of mass with the velocity. But before this can be done some assumption is necessary as to the shape of the particle and the distribution of its charge. The three theories of the simple negative particle or electron which are now most discussed are due to Abraham, Bucherer, and Lorentz. The first assumes that the electron is and remains a rigid sphere, the second assumes an electron which is spherical when at rest but which in motion contracts in the direction of its translation and expands laterally so as to keep a constant volume. The third assumes an electron similar to the second, which contracts in the direction of translation but which does not change its other dimensions. On the basis of these theories and from known electromagnetic principles, three equations have been obtained for the value ​of m/m$_0$ as a function of β, namely,
| (a) | ${\frac {m}{m_{0}}}={\frac {3}{4}}{\frac {1}{\beta ^{2}}}\left({\frac {1+\beta ^{2}}{2\beta }}\log {\frac {1+\beta }{1-\beta }}-1\right)$, |
| (b) | ${\frac {m}{m_{0}}}={\frac {1}{\left(1-\beta ^{2}\right)^{1/3}}}$, |
| (c) | ${\frac {m}{m_{0}}}={\frac {1}{\left(1-\beta ^{2}\right)^{1/2}}}$. |
The extraordinary significance of the similarity of the first two of these equations and the identity of the third with equation (15), which we have derived from strikingly different principles, needs no emphasis. Kaufmann shows that his results agree better with the first two of these equations than with the third, but to regard this as serious evidence as to the validity of equation (15) would, as Planck has pointed out, be laying too great a stress on the accuracy of the experimental observations.
The agreement of Kaufmann's results with the above equations has led him, and all others who have discussed his results, to the conclusion that all of the mass of an electron is electromagnetic.
Their argument is based on the assumption that ordinary mass, the mass of "ponderable matter," is independent of the velocity, while "electromagnetic mass" varies with the velocity according to one of the above equations. But in this paper we have assumed that all mass is one, and that any bodies, whether charged or not, moving at the velocities of Kaufmann's electrons would show the same values of m/m$_0$.
There is some hope that the correctness of this view may be decided by an experimental study of the mass of a positive or α particle at different speeds. According to the ordinary view, the mass of such a positive particle as issues from a radioactive source is chiefly that of its "ponderable" matter and only to a very small extent "electromagnetic mass." It would therefore be generally assumed that at the highest velocity of the particle, about one-tenth of the velocity of light, it would have substantially the same mass as at rest. According to our view, on the other hand, the mass of this or any other particle would change with the velocity in the same way as the mass of an electron. From equation (15) we should therefore expect the particle moving with one- tenth of the velocity of light to have a mass two per cent. greater than when at rest. The experimental difficulties in ​testing this point would be very great, of course, but perhaps not insurmountable.
The plausibility of our fundamental assumption which led directly to equation (15) has been greatly increased by the agreement, between this equation and Kaufmann's results, and also perhaps by the similarity between this equation and those deduced from electromagnetic theory. But the simplest as well as the strongest evidence of the correctness of our point of view comes from a consideration of the non-Newtonian equation for kinetic energy.
The integration of equation (14) obviously does not yield the simple Newtonian equation,
$\mathrm {E} '=1/2mv^{2}$
This equation must be replaced by one that is obtained as follows : —
Let a body, which at rest has the mass m$_0$, be given the velocity v. As its internal energy changes, its mass will change according to equation (7), and
$m-m_{0}={\frac {\mathrm {E} '}{\mathrm {V} ^{2}}}$
where E' is the acquired kinetic energy and m—m/m$_0$ is the increase in mass.
Eliminating m$_0$ between this equation and (15) gives
| $\mathrm {E} '=m\mathrm {V} ^{2}\left[1-\left(1-\beta ^{2}\right)^{1/2}\right]$. | (16) |
This is the general formula for the kinetic energy of a moving body. As usual β represents v/V, the ratio of this velocity to the velocity of light.
From equations (10), (15), and (16) may be constructed the whole science of non-Newtonian dynamics, of which Newtonian dynamics furnishes a limiting case, namely, for velocities which are negligible in comparison with the velocity of light.
For example, expanding (16) by the binomial theorem gives
| $\mathrm {E} '=m\mathrm {V} ^{2}\left({\frac {1}{2}}\beta ^{2}+{\frac {1}{8}}\beta ^{4}\dots \right)$. | (17) |
For low values of β the higher terms may be neglected and
$\mathrm {E} '={\frac {1}{2}}mv^{2}$
That is, the limit approached by the kinetic energy of a body as the velocity approaches zero is, as in ordinary mechanics, one half the product of the mass and the square of the velocity. At the other limit of velocity when β=1, it follows from (16) that
| $\mathrm {E} '=m\mathrm {V} ^{2}$. | (18) |
${\frac {1}{2}}mv^{2}<\mathrm {E} '<mv^{2}$.
The momentum and the kinetic energy of any mass moving with the velocity of light are, therefore,
* $\mathrm {M} =m\mathrm {V}$,
* $\mathrm {E} '=m\mathrm {V} ^{2}$,
but these equations are identical with (5) and (9) which we obtained for the momentum and the energy of a beam of light.
Further Consequences of the Theory.
The view here proposed, which appears at first sight a reversion to the old corpuscular theory of light, must seem to many incompatible with the electromagnetic theory. If it were really so I should not have ventured to advance it, for the ideas announced by Maxwell constitute what may no longer be regarded as a theory, but rather a body of experimental fact. The new theory is offered, not in any sense to replace, but to supplement the accepted theories of light. I hope in another paper to show that it is entirely consistent with those theories. Such a proof may constitute a step towards one of the obvious goals of present day science, the complete mechanical explanation of electromagnetic phenomena, or, what is very nearly the same thing, an electromagnetic explanation of the phenomena of ordinary mechanics. In the meantime a few of the more salient conclusions of our theory may be cursorily examined.
In the first place it should be noticed that, while the theory is consistent with a modified corpuscular theory of light, it does not necessarily imply that light is corpuscular. The stream of mass issuing from a radiating body may be made up of discrete particles or it may be continuous. Whatever it may be that is emitted it is not matter in the ordinary sense, as is to be seen from the following considerations : —
According to equation (15) any body of finite mass increases in mass as it increases in velocity and would possess infinite mass if it could be given the velocity of light. Therefore that which in a beam of light has mass, momentum, and energy, and is travelling with the velocity of light, would have no energy, momentum, or mass if it were at rest, or indeed if it were moving with a velocity even by the smallest fraction less than that of light. After this extraordinary conclusion it would be at present idle to discuss whether the same substance or thing which carries the radiation from the emitting body continues to carry it through space, or, indeed, whether there is any substance or thing connected with the process.
​If we assume an aether pervading space, and assume that this æther possesses no mass except when it moves with the velocity of light, it is obvious that an æther drift could in no way affect a beam of radiation nor could it be detected by any mechanical means. If we are to assume such an æther we may as well assume it to be at rest.
The question whether a method is conceivable by which absolute motion in space may be distinguished from relative motion must be answered definitely in the affirmative by one who accepts the above equations of non-Newtonian mechanics. A body is absolutely at rest when any motion imparted to it increases its mass, or when a certain force will give it the same acceleration in any direction. It is true that metaphysicians hold that in the strictest sense absolute motion is not only unknowable but unthinkable, but we may say at least that the above method permits theoretically the detection of absolute translational motion in the same sense that a study of centrifugal forces enables us to detect absolute rotational motion.
Summary.
It is postulated that the energy and momentum of a beam of radiation are due to a mass moving with the velocity of light.
From this postulate alone it is shown that the mass of a body depends upon its energy content. It is therefore necessary to replace that axiom of Newtonian mechanics according to which the mass of a body is independent of its velocity, by one which makes the mass increase with the kinetic energy.
Retaining all the other axioms of Newtonian mechanics and assuming the conservation laws of mass, energy, and momentum, a new system of mechanics is constructed.
In this system momentum is mv, kinetic energy varies between 1/2 mv² at low velocity and mv² at the velocity of light, while the mass of a body is a function of the velocity and becomes infinite at the velocity of light. The equation obtained agrees with the experiments of Kaufmann on the relation between the mass of an electron and its velocity. It is, moreover, strikingly similar to the equations that have been obtained for electromagnetic mass.
The new view leads to an unusual conception of the nature of light. It offers theoretically a method of distinguishing between absolute and relative motion.
    Research Laboratory of Physical Chemistry,
       Massachusetts Institute of Technology,
          May 14, 1908.
* ↑ Communicated by the author.
* ↑ Ann. Phys. xviii. p. 639 (1905).
* ↑ Phil. Mag xv. p. 1 (1908)
* ↑ In place of a black body we might consider a partially reflecting one. The equations thus obtained are more complicated but lead also to the simple equation (7).
* ↑ Phys. Review, xvii. pp. 26 and 91 (1903).
* ↑ I was first led to an investigation of the relation of mass to energy by the work of Landolt, on the change of weight in chemical reactions. But it is obvious from equation (7) that although there always will be a loss of mass in a reaction which is accompanied by the evolution of energy, this loss in the case of any ordinary reaction will be far too small to measure. In fact Landolt has very recently shown (Ber. Berlin. Akad. 1908, p. 354) that when all possible precautions are taken there is no measurable change of weight in the reactions which he studied.
* ↑ Einstein, however, obtains (8) as an approximate formula, while we obtain both (7) and (8) as perfectly exact equations.
* ↑ Phys. Zeit. iv. p. 54 (1902); Ann Phys. xix. p. 487 (1906).
* ↑ The constancy of the difference between the observed and calculated values of β is striking, and would alone indicate some constant error in Kaufmann's results.
* ↑ Verhandlung Deutsch. Phys. Ges. ix. p. 301 (1907).
* ↑ Ibid. x. p. 14 (1908).
* ↑ In reply to Planck see Kaufmann, ibid. ix. p. 667 (1907).
* ↑ For a discussion of these theories see Abraham, Theorie der Elektricität, vol. ii. Leipzig, 1905; and Bucherer, Mathematische Einführung in die Elektrontheorie, Leipzig, 1904.
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