The Differential Equations of the Gravitational Field

term of (21) gives only a vanishing contribution for infinitely great values of c. This follows from the fact that (1) vanishes everywhere where the influence of the exponential term of (24) has become unnoticeable. We compute the contribution of the second term in (21) by omitting the exponential term from the start and obtain, after a short calculation, as the final result, with u= 2roσo

… (25)

This equation when compared with the relation

allows an easy discussion of the essential properties of clusters of this type. First it is easy to see that we have extremely simple relations when we change M but keep fixed σo (0 < < 2√3) and thereby the tangential velocity of the particles as measured in light velocity units. When M is multiplied by z the gravitating mass will be zu and the diameter of the cluster will be z.2r. The mean density will be multiplied by 1/2".

In order to obtain a survey of all possibilities it is therefore sufficient to keep fixed the number of constituting particles and thereby M and to vary do together with the diameter 2r, and the gravitating mass μ. We obtain for M = 1

…

The following table gives μ and 2ro for M = 1 as functions of do (approximately):

When the cluster is contracted from an infinite diameter its mass decreases at the most about 5%. This minimal mass will be reached when the diameter 2ro is about 9.

The diameter can be further reduced down to about 5.6, but only by adding enormous amounts of energy. It is not possible to compress the cluster any more while preserving the chosen mass distribution. A further addition of energy enlarges the diameter again.

In this way the energy content, i.e. the gravitating mass of the cluster, can be increased arbitrarily without destroying the cluster. To each possible diameter there belong two clusters (when the number of particles is given) which differ with respect to the particle velocity. Of course, these paradoxical results are not represented by anything in physi- cal nature.

Only that branch belonging to smaller σo values contains the cases bearing some resemblance to real stars, and this branch only for diameter values between and 9M.

The case of the cluster of the shell type, discussed earlier in this paper, behaves quite similarly to this one, despite the different mass distribution. The shell type cluster, however, does not contain a case with infinite, given a finite M.

The essential result of this investigation is a clear understanding as to why the “Schwarzschild singularities” do not exist in physical reality.

Although the theory given here treats only clusters whose particles moye along circular paths it does not seem to be subject to reasonable doubt that more general cases will have analogous results. The “Schwarzschild singularity” does not appear for the reason that matter cannot be concentrated arbitrarily. And this is due to the fact that otherwise the constituting particles would reach the velocity of light.

This investigation arose out of discussions the author conducted with Professor H. P. Robertson and with Drs. V. Bargmann and P. Bergmann on the mathematical and physical significance of the Schwarzschild singularity. The problem quite naturally leads to the question, answered by this paper in the negative, as to whether physical models are capable of exhibiting such a singularity.