P R I N C E T O N L E C T U R E S A B S T R A C T S 5 3 7
How can we have a world with an average density of matter different from zero? Seeliger
assumed that empty space acted like negative matter. But the law of attraction would not
then be Newton’s. How will this be dealt with in the general relativity theory? We must con-
sider the curvature relations of space. Is inertia a mutual action between masses? If so it
must be increased near large masses. Motion over a spherical shell must accelerate the
masses inside of it. If the body rotates, then the equations of gravitation show that the mass-
es inside will rotate with it. Where does the gravitational field come from? It is one solution
of:
(52)
When the g’s are constant, the left, and consequently the right side vanishes. For empty
space we have (53)
Actually the distribution of matter is not uniform, but its [velocity is?] very small com-
pared to the velocity of light. Hence practically vanishes. We can also assume that the
density of matter is practically the same throughout the universe. We must arrive at a static
solution. We can obtain a simple conception of the physical meaning of the g’s, by consid-
ering a system in which the g’s are equal to those for the Galilean case plus small quantities,
. We have:
(54) where (55)
Since and are small, by considering the orders of the terms, we find that only
must be taken into consideration. A similar argument applies to the covariant tensor,
giving:
(56)
This is the same equation as that obtained from the Newtonian theory. is the Newtonian
potential function.
If the matter is at rest with regard to the co-ordinate system, as in the static problem, we
have:
(57)
determines the geometrical character of the three dimensional space, f determines
the gravitational field.
In (55) with index 4 4, the right side is the density of matter, consequently the equation
is not valid unless the matter has the density zero. This follows from the Newtonian theory.
Let us now consider in the general theory of relativity the fundamental gravitational
equations, ten in number.
Rik
1
2
--gikR - – kTik –=
T 0=
[p. 10] T44
g
gik
1
2
------
Tik
1
2
--gikT - –
r
-------------------------------dV
- –= Tik
sd
dxidxk
sd
=
sd
dxi
sd
dxk
T44
T44
d2x
dt2
---------i
1dg44
2dxi
-- - + 0 =
g44
ds2 g dx dx f2dt2 yikdxidxk – = =
yik