Qualitative Discussion of the Case of Arbitrary Radial Mass Distribution

We consider the case of a given mass μ and a shell radius ro satisfying the inequality (6a). When a number N of particles is brought into this shell zone, as determined by (15), then the exterior gravitational field is just completely screened off from the interior I so that there the field will be Euclidean. This means that the line element in I is characterized by constant values of a and b, where b cannot reach its lower limit 1/√3.

If, however, the number of particles in S is chosen smaller than according to (15) then the field will not be screened off entirely (u is hereby regarded as being kept fixed). We can then satisfy the theory formally by replacing the Euclidean line element in I by a Schwarzschild line element of the form

where A, B, and μ are constants. μ will be smaller than which characterizes the field outside the shell. This interior field has a singularity of the Schwarz- schild type (b = 0) at r = μ1/2.

This singularity, however, can be removed by introducing a second shell S inside S, which has to be constructed so that the gravitational field in its interior will be Euclidean. The whole cluster will then consist of two shells S and Si and will have no Schwarzschild singularity.

Again this system can be modified by reducing the number of particles in S1 so that it will not screen off its exterior field (between S and S1) entirely; then a third shell S., of still smaller radius, may be constructed so that its exterior field is just screened off entirely from its interior.

This method can be reiterated up to the center of the cluster. Thus one obtains clusters with the most varied radial mass distributions. There will be also various steady distributions. It is impossible, however, that b should vanish anywhere. The radius of the cluster will always be greater than the limiting radius (2 + √3), and it will not be possible to concentrate the matter of the cluster arbitrarily densely near the center of the cluster.