D O C . 9 E N E R G Y C O N S E RVAT I O N 5 3
Here too one can prove, just as has been shown above, that the have the same
value for all those coordinate systems that can be obtained by continuous deforma-
tion from the first one used. The proof is analogous to the one given above, except
that the choice of coordinates outside of now has no analog. For a close world
of spherical topology, the are independent of the specific choice of coordinates,
provided the “boundary condition” is kept
intact.7
It then can be shown that the “components of momentum” for such a
closed world necessarily vanish. We first give the proof for and . Further
down we prove that one can go, by continuous change, from a coordinate system
to a new one when both are connected by the substitution
(10)
.
This is a linear transformation. Since the preserve tensorial character for linear
substitutions, it follows from (10) that everywhere
.
From this follows immediately that also
}.
(11)
7A
rigorous application of §2 gives the following result. If and are two coordinate
systems, and are two associated spatial cuts, and and are
the associated values of , then and are always equal if there is a continuous
transition between and that preserves the “boundary condition.”
Jσ
B
Jσ
K K′
x4 const. = x′4 const. = Jσ J′σ
Jσ Jσ J′σ
K K′
J1, J2, J3
J1 J2
[p. 454]
K K '
ϑ′1 π ϑ1 – =
ϑ′2 π ϑ2 – =
ϑ′3 ϑ3 =
t′ t =
Uν
σ
U1
4′ U14
– =
U2
4′
U2
4
– =
J1 ′ J1 – =
J′
2
J2 – =