5 3 6 A P P E N D I X E
When the ’s vanish, this reduces to (43). This gives us our generalization. gives the
influence of the gravitational field. The tensor satisfies the equation:
(47)
where
(48)
This equation, like (43) is only valid when no gravitational field exists. In a gravitational
field, we must have:
(48)
which is the conservation equation, analogous to Poisson’s equation (42) which holds in the
presence of gravitation.
How does one obtain the paths of a freely moving particle in such a field? In the special
theory, or in a Galilean system, it describes a straight line. In general the geodesic is defined
by:
(49) giving (50)
In the special relativity theory, (51) and
means corresponding to the velocity of light.
III. T
HE
COSMOLOGICAL PROBLEM
In the classical mechanics, space is given a priori, but in the general theory of relativity
this is not the case. Space is itself determined by the matter in it. This was foreshadowed by
Mach. Inertia is troublesome. Consider two rotating heavenly bodies. It may happen that
one of them is flattened by centrifugal force. Why does this not act on the other, since
nothing distinguishes one from the other? Mach’s contention was that the resistance due to
inertia was not the absolute resistance in itself to acceleration (which is undefinable) but
resistance to acceleration relative to another body. What properties does the insertion of
matter cause in space?
A very large sphere has a certain mean density. Let us now imagine a large part of the
world, of spherical shape. The number of lines of force entering it is proportional to the
mass contained in it, or The surface is proportional to The strength of the field at
the surface is proportional to the number of lines of force per unit of area, or R, and hence
increases without limit when the radius of the sphere is increased. But this is impossible,
since the stars would have to possess infinite velocities to prevent them from falling into the
center of the field. This required the assumption that the density of matter decreased with
increasing distance from the center. This is an ugly conception.
Rim
Rim
1
2
--gimR -
–
0=
R gimRim =
Rim
1
2
--gimR - – kTim –=
ds 0=
d2x
ds2
----------
sd
dx
sd
dx
+ 0 =
ds2 dx2 dy2 dz2 dt2 – + + d 2 dt2 – = =
ds 0=
td
d
1=
[p. 9]
R3. R2.