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]