D O C . 5 C O M M E N T O N D E S I T T E R S O L U T I O N 3 7
finite domain. But this statement requires both a closer determination and a limita-
tion. A point is called “a point in the finite domain” when it can be connected by
a curve with a fixed, chosen point , so that the distance integral
has a finite value. Furthermore, the continuity condition for the and the
must not be interpreted to mean that it must be possible to choose coordinates such
that the conditions are satisfied in the entire space. Obviously, one must only de-
mand that for the neighborhood of every point one can select coordinates such that
the continuity conditions are met within this neighborhood. These limitations on
the demands for continuity follow quite naturally from the general covariance of
equations (1).
For the De Sitter solution one has according to (2),
.
Now, g vanishes first of all for and for . But with a suitable choice
of coordinates it can be easily shown that this violation of the continuity condition
is only imaginary. However, g vanishes also for , and it seems that no
choice of coordinates can remove this discontinuity. It is also clear that the points
on the surface have to be viewed as points in the finite domain, if we take
point at because, taken at constant , , and , the integral
is finite. Until the opposite is proven, we have to assume that the De Sitter solution
has a genuine singularity on the surface in the finite domain; i.e., it does
not satisfy the field equations (1) for any choice of coordinates.
P
P
°
[p. 271]
ds
P
°
P
∫
gμν
gμν
[4]
g
4
r
R
--- ψcos
r
R
---
2 2
sin
sin4
–R =
r 0 = ψ 0 =
r
π
2
-- -
R =
r
π
2
-- -
R =
P
°
r t 0; = = ψ θ t
dr
°
π
2
-- -
R
∫
r
π
2
-- - R =
[5]