As a matter of fact, this critical volume could never be reached; for even before it could be attained the pressure at the centre of the sphere would have become infinite, and calculation shows that this condition, too, would be accompanied by an arrest in the flow of time. In fine, we discover that the greatest possible volume our fluid sphere could ever occupy would be about
of the volume mentioned previously.
It follows that the fluid could never fill completely the spherical volume its density had created in space. Using a two-dimensional analogy, it would be as though on the surface of an ordinary sphere the entire surface could not be covered; an uncovered portion like a cap would always remain. On this account it is sometimes said that space would not close round on itself, since part of the spherical space created by the fluid could never be filled.
Just as a fluid of given density can never increase in volume beyond a certain point, so, conversely, a fluid of given mass could never be compressed into a volume which would cause its density to increase beyond a certain limit. The argument would be the same. The pressure would become infinite, time would stop, and nothing further could happen. A sphere of fluid possessing the same total mass as the earth could never be compressed into a volume less than that of a thimble.
The numerical values mentioned show that there is only a very slim chance of putting these anticipations of the theory to an experimental test. Even in the case of a giant star such as Betelgeuse, its density is so small that the critical volume is well beyond the actual size of the star. Nevertheless, from a theoretical point of view, it is interesting to note that there must exist a definite limit to the size of a star of given density. If these critical limits should ever be reached, time would be arrested, and it is difficult to foresee what the consequences might be. Einstein has called these critical conditions the “catastrophes.” When pressed to give his opinion as to what would happen were they ever to be reached, he is said to have replied that very possibly we should witness the disruption of matter into radiant or electromagnetic energy. However, he preferred not to venture an opinion, as anything he might say would be little better than a guess.
CHAPTER XXXII
THE PRINCIPLES OF CONSERVATION
CLASSICAL science had assumed that the mass of a body was an invariant, which would not change in value when the relative motion of the body was changed. It is true that highly refined experiments on electrons moving at enormous speeds (Bucherer’s experiment) had shown that the mass of an electron increased with its velocity; but it was assumed that this increase in mass was due to other causes, either to a modification in the electromagnetic field of the electron, or possibly to the mass of the ether which the electron in its rapid motion would drag along with it.
One of the first triumphs of the special theory of relativity was to prove that mass, just like time and space, must be a relative; and that a body in motion with respect to the observer would suffer an increase in mass in all ways identical with that disclosed by Bucherer’s experiment. This increase could no longer be attributed to the drag of the ether, since in the special theory of relativity exactly the same increase would have to be expected, regardless of whether it was the observer or the electron that was in motion through the ether. As in all previous examples, the essential factor was the relative motion between observer and observed. The ether played no part.
Calculation then showed that the mass of a body would become infinite when the velocity of the body reached that of light; for this reason no material body could ever move faster than light, since it would require an infinite force to increase its velocity any further.