D O C . 9 E N E R G Y C O N S E RVAT I O N 5 5
When we rotate the -system about the center of the sphere, the -system ro-
tates with it, and the relations (13) also hold for the system in its rotated position.
In a Euclidean space it is possible to rotate the Cartesian coordinate system such
that only two axes move and the others remain fixed. Among these rotations those
about the angle are distinguished, which correspond to substitutions of the type
.
(14)
Another one is
.
(15)
(14) and (15), resp., produce, with respect to (13) and the corresponding equations
for the primed system, directly the substitutions (10) and (10a), resp., which, con-
sequently, can be generated by continuous changes of the -system.
With this, the desired proof has been completed (except the proof that the
“boundary condition” is satisfied). For the closed universe as a whole, the momen-
tum vanishes; the amount of the total energy is independent of time and the choice
of coordinates.
§4. The Energy of the Spherical World
Now we want to calculate the for a spherical world with uniformly distributed,
incoherent matter. We do this primarily in order to check if condition (9) is at least
satisfied for this most simple case, because the results of the previous paragraph are
based upon this. We have to put
, (16)
uν ϑν
π
u′1 u1 – =
u′2 u2 – =
u′3 u3 =
u′4 u4 =
u′1 u1 – =
u′2 u2 =
u′3 u3 =
u′4 u4 – =
ϑν
Uσ
ν
Uσ
ν
Tσ
ν tν
σ
( )1
tν
σ
( )2 + + =