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