The paths of the particles and their special distribution

(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.

u is the gravitating mass.

Can a field be built up containing such singularities with the help of actual gravitating masses?

Or whether such regions with vanishing g44 do not exist.

Schwarzschild himself investigated the gravitational field which is produced by an incompressible liquid.

He found that in this case, too, there appears a region with vanishing g44 if only, with given density of the liquid, the radius of the field-producing sphere is chosen large enough.

This argument, however, is not convincing.

The concept of an incompressible liquid is not compatible with relativity theory as elastic waves would have to travel with infinite velocity.

The compressible liquid would have an equation of state that excludes the possibility of sound signals that are faster than the speed of light.

But the equation of state would be arbitrary within wide limits.

One could not make assumptions of physical possibilities.

Can matter be introduced in such a way that questionable assumptions are excluded from the very beginning?

This can be done by choosing, as the field-producing mass, many small gravitating particles which move freely under the influence of the field produced by all of them together.

This is a system resembling a spherical star cluster.

Here, the field, in which the particles are moving, were produced by a continuous mass distribution of spherical symmetry corresponding to the circular paths of the particles.

We can assume that all particles move along circular paths around the center of symmetry of the cluster.

Even here, we can choose arbitrarily the radial distribution of mass density.

This makes:

• g44 not zero anywhere
• the total gravitating mass which may be produced by distributing particles within a given radius, always remains below a certain bound.

1. The paths of the particles and their special distribution

By a suitable choice of the radial coordinate, it is possible to obtain the gravitational field of the cluster of spherical symmetry in the form(2)

…

whereby a and b are functions of r = (x + x + x)'.

First we shall investigate the circular motion of one particle around the center of symmetry. Suppose, for instance, this motion takes place within the plane = 0. Through the introduction of polar coördinates

….

(2) assumes the form

…

The field is characterized by

…

where all the rest of the g, vanish.

The particle under consideration satisfies the equation

…

In addition its motion is determined by the conditions

…

It turns out that (3) is satisfied when

…

Because of (2a), we have

…

Thus, de/dt and ds/dt are determined when the field is given.

Because ds has to be positive for the world line of a particle in motion we have

…

or

…

By applying this condition to Schwarzschild’s field (1) we obtain

…(6a)

It follows that in the case of a Schwarzschild field, a particle is bound to follow a path with a radius greater than (2+3) times the radius of the Schwarzschild singularity.

This fact has the greatest significance for the following investigation.

In the outermost layer of our particle cluster (and beyond it) the gravitational field is given by (1).

It follows that the total gravitating mass of the cluster determines a lower limit for the radius of the cluster.

This radius is (in coördinate measure) more than (2 + √3) times greater than the radius of the Schwarzschild singularity as defined by the field in the empty space outside the cluster.

The normal to the plane in which the particle considered moves has the direction of x.

If it is assumed that the normals to an infinite number of such planes are distributed at random and also that the phase angles of the paths are subject to a random distribution, then we obtain a cluster of particles of spherical symmetry whose paths have the radius r.

The most general cluster to be considered by us consists of an infinite number of clusters of this special type which belong to all values of r.

More accurately, the whole cluster consists of a finite number of particles so that a field is created which only pproximates spherical symmetry.

In order to formulate the conditions of dynamical equilibrium of the cluster under the influence of its own gravitational field, we first have to compute the energy tensor belonging to such a cluster.

For this purpose we assume, for the sake of simplicity that all particles have the same mass m.