D O C . 9 E N E R G Y C O N S E RVAT I O N 5 9
replaced, with any desired accuracy, by that of a mass point. Then one has at infin-
ity
,
(22)
where is a constant which we have to call the gravitational mass of the system.
We have to determine this constant.
The field equation
(23)
is rigorously valid in the entire space.
If we denote the quantity in parentheses on the left-hand side as and integrate
over the interior of the spatially infinitely distant surfaces that encloses the sys-
tem, we get
(24)
Since the first integral of the left-hand side as well as the right-hand side, which
expresses the energy of the entire system, do not change in time, it follows that this
is also true for the second term on the left-hand side; it must therefore vanish
because the integral cannot continually change in the same sense. The calculation
of the surface integral on the left-hand side presents no difficulty because in the
spatially infinite we can limit ourselves to the first approximation; and this yields,
considering (22), the value . Therefore,
. (25)
This result constitutes a support for our interpretation of the energy theorem pre-
cisely because the definition of given above is independent of our definition of
energy. The gravitational mass of a system is equal to the quantity which we called,
above, its energy.
g44 1
κ
4π
------
M
r
---- - – =
M
∂
∂xα⎝
--------⎜
∂G
*
∂g
α
μ4
-----------
gμ4⎟
⎠
⎛ ⎞
U
4
4
– =
Fα
S
{3} F1 cosnx1 F2 cosnx2 F3 cosnx3)dS + +
∫(
d
dx4
-------
-∫
F
4
dx1dx2dx3 U
4
4
∫
dx1dx2dx3 . = +
[17]
–M
M U
4
4
∫
dx1dx2dx3 J4 E
°
= = =
M