D O C . 1 G R A V I T A T I O N A L W A V E S 1 5
and where is the scalar of the density of matter. The T11, T12, . . . T33, therefore,
represent pressure components; T14, T24, T34 or T41, T42, T43 resp. is the vector of
momentum density multiplied by , or the density of the energy current, where-
as T44 is the negative value of the energy density. The interpretation of the re-
ferring to the gravitational field follows from this in analogy.
As an example we next treat the field of a mass point M at rest. From (7) and
(10) follows immediately
(11)
while all other vanish. According to (11), (3a) and (1), one gets for the
the values that were first given by De Sitter:
0 0 0
0 0 0
0 0
0
.
(11a)
0 0 0
The speed of light which is generally given by the equation
follows here from the relation
.
Therefore, under the choice of coordinates we favored, the velocity of light
(12)
ρ
–1
tμσ
[14]
γ44
′
κ
2π
------
M
r
---- -
=
γ′μν gμν
[15]
κ
4π
------ –
M
r
---- -
–1
κ
4π
------ –
M
r
---- - –1
κ
4π
------ –
M
r
---- -
–1
κ
4π
------ +
M
r
-----
–1
c
0
ds2
μν
∑gμνdxμdxν
= =
1
κ
4π
------
M⎞
r
---- -
+
⎝ ⎠
⎛
dx2 dy2 dz2)
+ + ( 1
κ
4π
------
M⎞
r
---- -
–
⎝ ⎠
⎛
–
dt2
0 =
[p. 159]
c
dx
2
dy
2
dz
2
+ +
dt2
------------------------------------- 1
κM
4πr
-------- - – = =