6 0 D O C . 9 E N E R G Y C O N S E RVAT I O N
Note Added in Proof. Further reflections of the subject matter have led me to the
conclusion that it is preferable for the formulation of the momentum-energy theo-
rem of a quasi-spherical world (but not a quasi-elliptical one) to use coordinates
obtained by stereographic projection of a sphere onto a (three-dimensional) hyper-
plane. In case of uniform distribution of matter, one then has
.
The singularity, apparently attributable to the choice of coordinates, is then
moved to spatial
infinity.9
Because of the symmetry in the three spatial coordinates,
this formulation appears more natural. The proof of the vanishing of the total mo-
mentum is even simpler than the one given in the text, because it is immediately
seen that the spatial substitutions
and
can be achieved by a continuous change of the coordinates (rotation of the coordi-
nate system), from which follow the equations in the text
9
The case of the quasi-spherical world (i.e., irregularly distributed, arbitrarily moving
matter) will then allow for an analogous choice of coordinates insofar as the apparent sin-
gularity of the field, corresponding to the choice of coordinates, is then moved to
and is of the same character as it is in the case of uniformly dis-
tributed matter at rest.
[p. 459]
[18]
ds2 dx4
2–
dx1
2
dx2
2
dx3
2
+ +
1
1
4R2
---------( + x1
2
x2
2
x3
2)
+ +
2
------------------------------------------------------------ - =
x1 x2 x3 ±∞ = = =
x1 ′ x1 – = x′
1
x1 =
x′
2
x2 – = x′
2
x2 – =
x′
3
x3 = x′
3
x3 – =
J′
1
–J1 =
J′
2
–J2 =
J′
3
J3 – =