On A Stationary System With Spherical Symmetry Consisting Of Many Gravitating Masses on Superphysics

The paths of the particles and their special distribution

Mon, 01 Jan 0001 00:00:00 +0000

(Received May 10, 1939)

If one considers Schwarzschild’s solution of the static gravitational field of spherical symmetry

…

Then

…

vanishes for r = u/2.

This means that:

a clock kept at this place would go at the rate zero.
both light rays and material particles take an infinite long time (measured in x-coordinates time t) in order to reach the point r = u/2 when originating from a point r > u/2.

In this sense, the sphere r = u/2 is a place where the field is singular.

The Matter-Energy Tensor of the Cluster

Mon, 01 Jan 0001 00:00:00 +0000

We consider the motion of particles within a volume element on the x-axis.

The velocity vectors all have the same amount, they are perpendicular on the z-direction, and they are evenly distributed with respect to the directions within the 1, 2 - plane.

The Differential Equations of the Gravitational Field

Mon, 01 Jan 0001 00:00:00 +0000

The differential equation of a gravitational field which is due to a matter- energy tensor are

These equations have to be specialized for a static field of the type (2). By a straight forward calculation the following equations are obtained for a point on the z-axis:

Localization of the Particles within a Thin Spherical Shell

Mon, 01 Jan 0001 00:00:00 +0000

Outside the cluster, the gravitational field is represented by Schwarzschild’s solution which, with our choice of the coördinate system, is given by (1)

…

Inside the cluster, the field is determined by (14). Thereby, the function n is to be considered as given. However, n is not completely arbitrary, as the total radius of the cluster is restricted by the lower limit given by (6a).

Qualitative Discussion of the Case of Arbitrary Radial Mass Distribution

Mon, 01 Jan 0001 00:00:00 +0000

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.

The Case of Continuous Particle Density

Mon, 01 Jan 0001 00:00:00 +0000

The consideration given in part 5. leads toward the solution for continuous distributions of the particle density. We divide the interval 0 ≤rro into an infinite number of equal parts dr. We imagine that there is constructed in the center of each partition dr a shell of a two dimensional character of the type discussed in part 4.

A Specie Cued Continuous Mass Distribution

Mon, 01 Jan 0001 00:00:00 +0000

It is of some interest to investigate the case where … inside the cluster is constant 4u StAcrly specking eau outaide of our condiaons es ought to duce tow.d the point • • 0 et leset es lut r. in orders hat the density in the neighborhood of the center should Am finite. We een swirly Will condition by choosing n for instance (29) where c is to be an ANA, eon…

The Differential Equations of the Gravitational Field

Mon, 01 Jan 0001 00:00:00 +0000

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