, it would never return. We see, therefore, that the value of this critical speed depends on the difference in value of the Newtonian potential at infinity and at the earth’s surface.
Now consider the case of a nucleus of stars. The stars will be moving with various velocities under the influence of their mutual gravitational attractions. Statistical mechanics shows that every now and then the total energy of the stellar universe will be transferred to one single star; this star will then be moving at a tremendous speed. The total energy of the sidereal universe being finite, this speed will never be infinite, so that there will exist a maximum finite speed which no star of the nucleus can ever surpass. If, therefore, we consider the Newtonian potential due to the entire nucleus of stars, and if the difference in the value of this potential at infinity and near the centre of the nucleus is sufficiently great, no star will ever be able to tear away and be lost to infinity.
From this, it might appear that a nucleus of stars could endure permanently under Newton’s law. Theoretically, this would be possible, but astronomical observation proves that this possibility must be rejected. For if there existed very great variations in the value of the potential throughout the universe (as the preceding hypothesis would demand), enormous star velocities would ensue. Now the outstanding astronomical fact is that the star velocities are extremely low, proving that the potential varies but slightly from place to place throughout the universe even over vast astronomical distances. It appears, then, that the nucleus is held together very loosely, and a star would not have to be possessed of an inordinately high velocity to tear away and be lost forever.
In addition to this first difficulty, when Boltzmann’s law of gas equilibrium is applied to the stars, treating the latter as so many molecules, the nuclear form of the universe appears to be utterly impossible.
Now, thus far, we have been considering the problem of the universe from the standpoint of classical science and Newton’s law. No great change need be introduced into our arguments when we treat the same problem in terms of Einstein’s law of gravitation and of the infinite quasi-Euclidean universe. Once again, a continually impoverished nucleus would appear to be in order. There is, however, a novel point that may be mentioned. In classical science, we were concerned with the Newtonian potential
, and we explored its value throughout the universe by observing the velocities of the stars. With the space-time theory, the potential
(or, more precisely, a certain mathematical expression into which