The Matter-Energy Tensor of the Cluster

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.

We know further that the matter-energy tensor depends also on the particle density and on the gravitational potentials, but not on the derivatives of the latter. It is, therefore, possible to determine this tensor by a straightforward calculation.

First we consider particles, with the mass m and the particle density no per unit volume, at rest with respect to a coördinate system of the theory of re- stricted relativity. In such a case of the energy tensor only the (44)-com- ponent exists,

…

With respect to coördinate systems in relative motion in the x-direction we have the components

Till

…

The particle density n with respect to such a system is determined by the equations:

…

where Vo and V denote the rest volume and the coördinate volume respectively. Therefore we have

…

We now consider the case when the velocity vector of the particle makes an angle a with respect to the x-axis, and is perpendicular to the x-axis.

By using the relations derived above and by introducing dl = dzi + dx, we obtain

…

all the other components of the energy tensor being zero. In the case that the velocity vectors are evenly distributed over all values of a the result is

…

We now proceed to the case that the components of the metric tensor are gu ….

The components of the energy tensor are obtained by applying the transformation law for tensors and by transforming the co- ordinates according to:

…

We obtain

…

dl and dx4, contained in T and Tu, are to be replaced dl by a’dl and dx4 by b’dt4.

We have to introduce the particle density with respect to the new coordinates, ñ, according to:

…

or

…

After having made all these transformations and substitutions, and omitting the bars denoting the new coördinate system, we obtain

…(7)

In these equations ds/dz, and dl/ds have to be replaced by the expressions given by (4) and (5) which were derived from the equations of the geodesic lines.

Further we write dt instead of dr, and rdp instead of dl. The final result is

…

where a and 8 denote the expressions

…