Einstein's Theories of Relativity and Gravitation / A selection of material from the essays submitted in the competition for the Eugene Higgins prize of $5,000

by J. Malcolm Bird

Some Further Consequences

I need not trespass upon the subject matter of those essays which appear in full by going here into any details with regard to the manner in which time and space are finally found to depend upon one another and to form the parts of a single universal whole. But I may appropriately point out that if time and space are found to be relative, we may surely expect some of the less fundamental concepts that depend upon them to be relative also. In this expectation we are not disappointed. For one thing,]* [mass has always been assumed to be a constant, independent of any motion or energy which it might possess. Just as lengths and times depend upon relative motion, however, it is found that mass, which is the remaining factor in the expression for energy due to motion, also depends upon relative velocities. The dependence is such that if a body takes up an amount of energy E with respect to a certain system, the body behaves, to measurements made from that system, as though its mass had been increased by an amount

, where C is as usual the velocity of light.]^[194]

[This should not startle us. The key to the situation lies in the italicized words above, which indicate that the answer to the query whether a body has taken up energy or not depends upon the seat of observation. If I take up my location on the system S, and you on the system S′, and if we find that we are in relative motion, we must make some assumption about the energy which was necessary, initially, to get us into this condition. Suppose we are on two passing trains.]* [The chances are that either of us will assume that he is at rest and that it is the other train which moves, although if sufficiently sophisticated one of us may assume that he is moving and that the other train is at rest.]^[272] [Whatever our assumption, whatever the system, the localization of the energy that is carried in latent form by our systems depends upon this assumption. Indeed, if our systems are of differing mass, our assumptions will even govern our ideas of the amount of energy which is represented by our relative motion; if your system be the more massive, more energy would have to be localized in it than in mine to produce our relative motion. If we did not have the universal principle of relativity to forbid, we might make an arbitrary assumption about our motions and hence about our respective latent energies; in the presence of this veto, the only chance of adjustment lies in our masses, which must differ according to whether you or I observe them.]*

[For most of the velocities with which we are familiar

is, like the difference between K and unity, such an extremely small quantity that the most delicate measurements fail to detect it. But the electrons in a highly evacuated tube and the particles shot out from radioactive materials attain in some cases velocities as high as eight-tenths that of light. When we measure the mass of such particles at different velocities we find that it actually increases with the velocity, and in accordance with the foregoing law.]^[194] [This observation, in fact, antedates Einstein’s explanation, which is far more satisfactory than the earlier differentiation between “normal mass” and “electrical mass” which was called upon to account for the increase.]*

[But if the quantity

is to be considered as an actual increase in mass, may it not be possible that all mass is energy? This would lead to the conclusion that the energy stored up in any mass is

. The value is very great, since C is so large; but it is in good agreement with the internal energy of the atom as calculated from other considerations. It is obvious that conservation of mass and of momentum cannot both hold good under a theory that translates the one into the other. Mass is then not considered by Einstein as conservative in the ordinary sense, but it is the total quantity of mass plus energy in any closed system that remains constant. Small amounts of energy may be transformed into mass, and vice versa.]^[194]

[Other features of the theory which are often displayed as consequences are really more in the nature of assumptions. It will be recalled that when we had agreed upon the necessity of employing signals of some sort, we selected as the means of signalling the speediest messenger with which we happened to be acquainted. Our subsequent difficulties were largely due to the impossibility of making a proper allowance for this messenger’s speed, even though we knew its numerical value; and as a consequence, this speed enters into our formulae. Now we have not said in so many words that C is the greatest speed attainable, but we have tacitly assumed that it is. We need not, therefore, be surprised if our formulae give us absurd results for speeds higher than C, and indicate the impossibility of ever attaining these. Whatever we put into a problem the algebra is bound to give us back. If we look at our formula for K, we see that in the event of v equalling C, lengths become zero and times infinite. The light messenger itself, then, has no dimension; and for it time stands still.

If we suppose v to be greater than C, we get even more bizarre results, for then the factor K is the square root of a negative number, or as the mathematician calls it an “imaginary” quantity; and with it, lengths and times become imaginary too.

The fact that time stops for it, and the fact that it is the limiting velocity, give to C certain of the attributes of the mathematician’s infinity. Certainly if it can never be exceeded, we must have a new formula for the composition of velocities. Otherwise when my system passes yours at a speed of 100,000 miles per second, while yours passes a third in the same direction at the same velocity, I shall be passing this third framework at the forbidden velocity of 200,000 miles per second—greater than C. In fact Einstein is able to show that an old formula, which had already been found to connect the speed of light in a material medium with the speed of that medium, will now serve universally for the composition of velocities. When we combine the velocities v and u, instead of getting the resultant

as we would have supposed, we get the resultant

or

This need not surprise us either, if we will but reflect that the second velocity effects a second revision of length and time measurements between the systems involved. And now, if we let either v, or u, or even both of them, take the value C, the resultant still is C. In another way we have found C to behave like the mathematician’s infinity, to which, in the words of the blind poet, if we add untold thousands, we effect no real increment.