5 3 8 A P P E N D I X E
(58)
It follows from this, using:
(59)
that we have:
(60)
This equation contains one less condition than (58).
Is the existence of a non-zero density of matter compatible with these equations? We can
show that it is. They are satisfied when
(61) hold.
If matter is in equilibrium and uniformly distributed
these equations are satisfied. In that case space must be
of constant curvature. This must be closed. The two di-
mensional analogue is the sphere. Geometrical proper-
ties result from the possibilities on the plane, we can
obtain a flat picture of the sphere. The arrangement and
metric for the zones must be obtained from the projec-
tion. Hence we have a spherical geometry of the pro-
jected circles in the plane. We can remove the sphere.
Analogously, in place of the plane we can take a three dimensional space, and replace
the circles to be projected by spheres. It is easily seen, that with the projected metric, if the
spheres are equally spaced, only a finite number can be inserted.
When all the matter is electrically charged, how can we then solve the equations?
Poincare has explained the equilibrium in the world by a pressure . We can introduce a
pressure term in the equations
Can we eliminate also the pressure term from these equations?
The radius of the world is: (62)
This is determined when we know the mean density of matter . Also we have another pos-
sibility:
(63)
(64)
These show that R, the four dimensional curvature scalar, is independent of space and
time.
We can also ask the size of a universe containing a given amount of mass, m? The space
shrinks to a point when the mass vanishes, showing that space is dependent on the mass.
Rik
1
2
--gikR - – kTik –=
R kT =
Rik
1
4
--gikR - – k Tik
1
4
--gikT - – –=
Pik yikk = R
2
k
----- -=
[p. 11]
p
–gikp.
R
2
k
----- -=
Rik
1
4
--gikR - – 0 =
xi
R
0=