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)

},

,