D O C . 9 E N E R G Y C O N S E RVAT I O N 5 1
where is the “energy at rest,” and is the velocity (four-vector) of the system
(as a whole). equals the component if the coordinates are chosen such that
.
Despite the free choice of coordinates inside , the rest energy and rest mass of
the system remain precisely defined quantities which do not depend upon the
choice of coordinates. This is the more remarkable as the components of energy
density cannot be given any invariant interpretation, because lacks tensorial
character.
If one imagines, for example, the inside of also to be empty, then the system
defined in this manner has vanishing total energy; but with our choice of coordi-
nates inside we can control and induce varied distributions of energy—which,
however, all yield the integral 0. We are thus led—contrary to our present habits of
thinking—to assign more weight of reality to an integral than to its differentials.
§3. The Integral Theorem for a Closed World
In order to talk at all about an isolated system, we had to assume that the metric
continuum is Galilean in a sufficient distance from the system, a precondition that
is satisfied with good approximation for an area in the order of magnitude of our
planetary system. But in a paper published last
year4
I could show that there are
considerable objections from the general theory of relativity to view the world at
large as approximately Galilean (or Euclidean, resp.); the universe would have to
be essentially empty, i.e., the larger the sphere considered, the lesser would the
mean density of ponderable matter in it deviating from zero. It appears probable
that the space of the universe at large is quasi-spherical (or quasi-elliptical, resp.).
This interpretation demands the addition of a term (the “ -term”) in the field equa-
tions of gravitation. With these amended equations, a part of the world which is free
of matter cannot behave as “Galilean.” Then it will not be possible to choose coor-
dinates such as demanded in §2; the less so, the more extended the considered sys-
tem
is.5
In case of this finite world there emerges the interesting question whether or not
the conservation theorems apply to the universe as a whole; and the latter has to be
viewed by all means as an “isolated system.” We can limit our analysis to a quasi-
spherical world since the quasi-elliptical case follows from it by adding another
condition of symmetry.
4
These Berichte 6 (1917), p. 142.
5
For the spaces considered in astronomy, the assumptions of §2 should be adequate, so
that the following is merely of speculative interest.
E0
dxσ--------
ds
E0 J4
J1 J2 J3 0 = = =
B
Uσ
ν
B
B
[p. 452]
[9]
[10]
λ
[11]