1 2 D O C . 1 G R A V I T A T I O N A L W A V E S

It is seen from (6) that gravitational fields propagate at the speed of light. With

given

,

the can be calculated from them in the manner of retarded poten-

tials. If are the real-valued coordinates of the point , , , under

consideration, i.e., the point for which the are to be calculated, and , ,z

are the spatial coordinates of the volume element , the spatial distance be-

tween the latter and the point under consideration, then one gets

. (7)

§2. The Energy Components of the Gravitational Field

Some time

ago3

I gave the energy components of the gravitational field explicitly

in case the choice of coordinates satisfies the condition

,

a condition which, for the approximation considered here, is equivalent to

.

This, however, is in general not satisfied with our present choice of coordinates.

Therefore, the most simple method for obtaining the energy components follows a

separate consideration.

However, we have to consider the following difficulty. Our field equations (6)

are correct only up to the first order of magnitude, while the energy equations—as

is easily concluded—are small of the second order of magnitude. But we reach our

goal easily by the following consideration. The energy components (of matter)

3Ann.

d. Phys. 49 (1916), eq. (50).

Tμν γμν

x y z t , , , x1 x2 x3

x4-

i

----

γμν

′

x

°

y

° °

dV

°

r

γ′μν

κ

2π

–------

Tμν( x0, y0, z0, t r) –

r

-----------------------------------------------dV

°

∫

=

g gμν 1 = =

γ

α

∑γαα

0 = =

[7]

Tμ

σ

[8]