5 8 D O C . 9 E N E R G Y C O N S E RVAT I O N
vanish for the special case we considered here; just as we presupposed in the pre-
vious paragraph. It is probable this would be the case for any closed world of spher-
ical topology when using polar coordinates, but still would require a separate proof.
The total energy of the static world under consideration is
.
Here is
and
8
.
Since is the volume of the spherical world, one finds
(21)
Therefore, in this case gravitation does not contribute to the total energy.
§5. The Gravitational Mass of a Closed System
We return again to a consideration of the case of a system embedded in a “Galilean
space,” i.e., we again neglect the -term in the field equations. We have proven in
§3 that the integral of a system freely floating in a Galilean space transforms
like a four-vector. This means that the quantity we interpreted as energy also plays
the role of the inertial mass—in agreement with the special theory of relativity.
Now we also want to show that the gravitational mass of the total system under
consideration agrees with the quantity that we interpreted as the energy of the sys-
tem. Let there be an arbitrary physical system in the neighborhood of the origin of
the coordinates, and this system as a whole shall be at rest relative to the coordinate
system. This system generates a gravitational field which, in spatial infinity, can be
8
See A. Einstein, these Sitzungsber. 6 (1917), pp. 142–152, eq. (14).
[15]
J4
J4 ρ
°
–g
λ
κ
-- - –g
R
κ
---
cos2ϑ1
sinϑ2⎠ – +
⎝
⎛ ⎞
∫
dϑ1dϑ2dϑ3 =
–g
R3
ϑ1
2
sin sinϑ2 =
λ
κ
-- -
ρ
°
2
-----
1
R2κ
--------- = =
V
2π2R3
=
J4 ρ
°
V. =
λ
Jσ
[16]
[p. 458]