D O C . 9 E N E R G Y C O N S E RVAT I O N 4 9
to assign different nonzero values to them. Therefore, there are general doubts as
to the meaning of equation (1).
In contrast, I will demonstrate in the following that equation (1) defines the con-
cepts of energy and momentum as strictly as we are used to demanding it in clas-
sical mechanics. Energy and momentum of a closed system are—independent of
the choice of coordinates—completely determined as long as the state of motion of
the system (seen as a whole) is given relative to the coordinate system; for example,
the “energy at rest” of an arbitrary and closed system is independent of the choice
of coordinates. The following proof rests essentially only upon the fact that equa-
tion (1) is valid for any choice of coordinates.
§2. In Which Respect Are Energy and Momentum Independent of the Choice of
Coordinates?
For the following we choose the coordinate system such that all line elements
are timelike, all line elements are spacelike. The
fourth coordinate can then be called, in a certain sense, “the time.”
In order to be able to talk meaningfully of energy or momentum of a system, it
is necessary that the density of the energy or momentum, resp., vanish outside of a
certain domain B. In general, this is the case only if the are constant outside of
B; i.e., when the system under consideration can be embedded in a “Galilean
space” and we use “Galilean coordinates” to describe its environment. The domain
B extends to infinity in the direction of time, i.e., B intersects every hyperplane
Its intersection with the hyperplane is bounded every-
where. There is no “Galilean coordinate system” inside of B; the choice of coordi-
nates inside of B is subject to only one constraint: they must continuously match
the coordinates outside of B. In the following we shall consider several such coor-
dinate systems, all of which coincide outside of B.
The integral theorems of the conservation of momentum and energy follow from
(1) by integration of this equation with respect to over the domain B.
Since the vanish at the boundaries of B, one gets
. (3)
In my opinion, these 4 equations represent the momentum theorem
and the energy theorem . We shall call the integral in (3) . I now claim
[6]
[p. 450]
0 0 0 dx4) , , , ( dx1, dx2, dx3, 0) (
gμν
[7]
x4= const. x4= const.
x1, x2, x3
Uσ
ν
d-
dx4
-------
Uσ
4
∫
dx1dx2dx3 0 =
σ 1 to 3) = (
σ 4) = ( Jσ