D O C . 1 G R A V I T A T I O N A L W A V E S 2 5
Let the acting wave be an energy-transporting one in which only the component
of the gravitational field is different form zero. Because of (22) one
then has
. (34)
For a given wave and given mechanical process, the energy absorbed from the wave
can, therefore, be found by integration.
§6. Answer to an Objection of Mr. Levi-Civita
In a series of interesting investigations, Mr. Levi-Civita has in recent times contrib-
uted to clarification of problems in the general theory of relativity. In one of these
papers6
he takes, with respect to the conservation theorems, a position which dif-
fers from my point of view and denies, based upon his interpretation, the claim of
my conclusions regarding the radiation of energy through gravitational waves.
Even though we have in the meantime, through an exchange of letters, clarified the
question in a manner satisfactory to both of us, I consider it advisable in the interest
of the subject matter to add a few general remarks about the conservation theorems.
It is generally conceded that, according to the foundations of the general theory
of relativity, there exists a four-equation, valid under an arbitrary choice of the sys-
tem of reference, and it has the form
(35)
where are the energy components of matter and the are functions of the
and their first derivatives. But there are differences of opinion on whether or
not the have to be interpreted as the energy components of the gravitational
field. I consider this difference of opinions as irrelevant, as a mere question of
words. However, I claim that the undisputed equation, given above, provides the
facilitations of perspective which constitutes the value of the conservation
6Accademia
dei Lincei 26 (April 1, 1917).
γ23(= γ23)
′
dE
dt
------ -
1
2
-- -
∂γ23
∂t
--------- -
d
2
ℑ23
dt2
------------- -
=
[34]
[p. 166]
[35]
[36]
[37]
∂( Tσ
ν
tσ
ν
+ )
∂xν
-
ν
∑-----------------------------
0 = σ 1 ,2,3,4), = (
Tσ
ν
tσ
ν
gμν
tσ
ν