3 6 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
5. “Critical Comment on a Solution of the
Gravitational Field Equations Given by Mr. De Sitter”
[Einstein 1918c]
Submitted 7 March 1918
Published 21 March 1918
In: Königlich Preußische Akademie der Wissenschaften (Berlin). Sitzungsberichte (1918):
270–272.
Herr De Sitter, to whom we owe deeply probing investigations into the field of the
general theory of relativity, has recently given a solution for the equations of grav-
itation1
which, in his opinion, could possibly represent the metric structure of the
universe. However, it appears to me that one can raise a grave argument against the
admissibility of this solution, which shall be presented in the following.
The De Sitter solution of the field equations
(1)
is
(for all indices)
where are to be treated as coordinates .—
We shall have to take it as required by the theory that the equations (1) are valid
for all points in the finite domain. This can only be the case if both the and the
associated contravariant (including their first derivatives) are continuous and
differentiable; in particular, the determinant must vanish nowhere in the
1Proc.
Acad. Amsterdam, vol. 20 (June 30, 1917). Monthly Notices of the Royal Astro-
nomical Society, vol. 78, no. 1.
[p. 270]
[1]
Gμν λgμν κTμν –
1
2
--gμνκT - + = –
Tμν 0 =
ds2 dr2
–
R2
r
R
---[
2
dψ2 ψdθ2]
2
sin +
R
---c2dt2r
2
cos + sin – =
[2]
[3]
r, ψ θ, t , x1…x4) (
gμν
gμν
g gμν =
(2)
},
,