Volume 7: The Berlin Years: Writings, 1918-1921 (English translation supplement) Page 12 (28 of 399)

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]

Previous Page Next Page

Extracted Text (may have errors)

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]

Help

D O C . 1 G R A V I T A T I O N A L W A V E S 1 1
If this equation is multiplied by and added to equation (2a), then the second
term on the right-hand side of (2a) cancels out. The left-hand side can be written
more comprehensively if one introduces, instead of the , the functions
. (3)
The equation then takes the form:
. (4)
However, this equation can be simplified considerably if one demands the
to satisfy not only equation (4) but also the relations
. (5)
On first sight it seems strange that the 10 equations (4) for 10 functions
should allow for 4 additional and arbitrary conditions without becoming overdeter-
mined. But the justification of this procedure is seen from the following. Equations
(2) are covariant with respect to arbitrary substitutions, i.e., they are satisfied for an
arbitrarily chosen coordinate system. Upon the introduction of a new coordinate
system, the of the new system depend upon 4 arbitrary functions which define
the transformation of the coordinates. These 4 equations can now be chosen such
that the of the new system satisfy four arbitrarily prescribed relations. We
imagine them to be such that they transform into equations (5) for the approxima-
tions we are interested in. These latter equations, therefore, represent our prescrip-
tion according to which the coordinate system has to be chosen. One obtains in
place of (4), due to (5), the simple equations
. (6)
δμν
γμν
γμν
′
γμν
1
2
--δμν∑γαα -
α
– =
∂2γμν
′
∂xα 2
α
∑-------------
∂2γμα
′
-----------------
α
∑∂xν∂xα
–
∂2γνα-
′
-----------------
α
∑∂xμ∂xα δμν∑∂xα∂xβ
∂2γαβ
′
-----------------
αβ
+ – 2κTμν =
γ′μν
∂γμα
′
∂xα
α
∑-----------
0 =
γμν′
gμν
[p. 156]
gμν
[6]
∂2γμν ′
∂xα 2
α
∑-------------
2κTμν =

Previous Page Next Page

Extracted Text (may have errors)

D O C . 1 G R A V I T A T I O N A L W A V E S 1 1
If this equation is multiplied by and added to equation (2a), then the second
term on the right-hand side of (2a) cancels out. The left-hand side can be written
more comprehensively if one introduces, instead of the , the functions
. (3)
The equation then takes the form:
. (4)
However, this equation can be simplified considerably if one demands the
to satisfy not only equation (4) but also the relations
. (5)
On first sight it seems strange that the 10 equations (4) for 10 functions
should allow for 4 additional and arbitrary conditions without becoming overdeter-
mined. But the justification of this procedure is seen from the following. Equations
(2) are covariant with respect to arbitrary substitutions, i.e., they are satisfied for an
arbitrarily chosen coordinate system. Upon the introduction of a new coordinate
system, the of the new system depend upon 4 arbitrary functions which define
the transformation of the coordinates. These 4 equations can now be chosen such
that the of the new system satisfy four arbitrarily prescribed relations. We
imagine them to be such that they transform into equations (5) for the approxima-
tions we are interested in. These latter equations, therefore, represent our prescrip-
tion according to which the coordinate system has to be chosen. One obtains in
place of (4), due to (5), the simple equations
. (6)
δμν
γμν
γμν
′
γμν
1
2
--δμν∑γαα -
α
– =
∂2γμν
′
∂xα 2
α
∑-------------
∂2γμα
′
-----------------
α
∑∂xν∂xα
–
∂2γνα-
′
-----------------
α
∑∂xμ∂xα δμν∑∂xα∂xβ
∂2γαβ
′
-----------------
αβ
+ – 2κTμν =
γ′μν
∂γμα
′
∂xα
α
∑-----------
0 =
γμν′
gμν
[p. 156]
gμν
[6]
∂2γμν ′
∂xα 2
α
∑-------------
2κTμν =

D O C . 1 G R A V I T A T I O N A L W A V E S 1 3
and (of the gravitational field) satisfy, according to the general theory, the re-
lations
,
.
From these follows
.
We find the if we bring the right-hand side in the form of the left-hand side,
thereby using the from the field equations. In case of the approximation con-
sidered here, the two factors on the right-hand side of this equation are small quan-
tities of the first order. In order to get the accurately as quantities of the second
order, one only needs to substitute the two factors on the right accurately in terms
of quantities of the first order. One, therefore, can replace
by
and by .
For we introduce the which, under the desired approximation, deviate in
value from the only in sign. With respect to the character of their indices, the
are quantities analogous to the . We have to determine the from the
equation
. (8)
We transform the right-hand side by observing that, due to (3),
tμ
σ
[p. 157]
∂Tσ
μ
∂xσ
-
σ
∑-----------
1
2∑∂gρσTρσ
-- -
∂xμ
-----------
ρσ
+ 0 =
[9]
∂( Tμ
σ
tμ
σ
+ )
∂xσ
σ
∑------------------------------
0 =
∂tμ
σ
∂xσ
-
σ
∑---------
1
2∑∂gρσTρσ
-- -
∂xμ
-----------
ρσ
=
tμ
σ
[10]
Tρσ
[11]{1}
tμ
σ
∂gρσ
∂xμ
-----------
∂γ
∂xμ
-----------ρσ
–
Tρσ Tρσ
tμ
σ
tρσ
tρ
σ
tρσ Tμσ tμσ
∂tμσ
∂xσ
σ
∑----------
1
2∑---------- -- -
∂γρσTρσ
∂xμ
-
ρσ
=

Previous Page Next Page

Page 1SELECTED TEXTS click to expand contents

Extracted Text (may have errors)

Help

loading

Extracted Text (may have errors)