5 2 D O C . 9 E N E R G Y C O N S E RVAT I O N
The conservation theorem (1), (2) also holds in a quasi-spherical world. But
there is no coordinate system that is regular everywhere. The square of the invariant
line element in a rigorously spherical world has, in polar coordinates, the value
. (7)
The variables range as follows:
(8)
.
The coordinate system becomes singular at the boundaries of and ,
because more than 4 coordinate lines intersect in such points and the
determinant vanishes there. An analogous choice of coordinates will also be
possible for the case of a quasi-spherical world (under a correspondingly modified
expression for ); here, too, we have to watch for singular point sets of the
coordinate system. The equations (1) are valid for all nonsingular points of the
coordinate system. A transition to the integral equations (3) is also possible when
the integral over vanishes (“boundary condition”) . This
would be the case when, for example,
6
.
(9)
vanishes for and
and vanishes for and
For in this case, after integrations of (1) with respect to over the entire
and closed space, all terms on the left-hand side vanish except those which origi-
nate from the term .
6
Details on this follow in §4.
{1}
ds2
dt2
R2[
dϑ1
2
ϑ1dϑ2
2
sin
2
ϑ1
2
sin ϑ2dϑ3
2
sin
2]
+ + – =
[p. 453]
x1 ϑ1 =
between 0 and π
x2 ϑ2 = between 0 and π
x3 ϑ3 = between 0 and 2π
x4 t = between + –∞and ∞
ϑ1 ϑ2
∞ many) (
gμν
ds2
∂Uσ-
1
∂x1
-----------
∂Uσ-
2
∂x2
-----------
∂Uσ-
3
∂x3
----------- + +
U1
1
U2
1
U3
1
U4
1
, , ,
ϑ1 0 = ϑ1
π}
=
U1
2
U2
2
U3
2
U4
2
, , ,
ϑ2 0 = ϑ2 π =
x1, x2, x3
∂U
σ-
4
∂x4
-----------