D O C U M E N T 3 9 0 D E C E M B E R 1 9 2 2 3 3 3

390. From Alexander Friedmann[1]

Petrograd, Central Physics Observatory, Vassili OstroV, 23rd Line, 6 December 1922

Esteemed Professor,

In a letter from one of my friends who is currently abroad, I had the honor of

learning that a brief comment by you had been submitted to press in the 11th vol-

ume of Zeitschrift für

Physik,[2]

which points out that from assumptions ( ) and

(C) of my article on “The Curvature of Space” [Die Krümmung des Raumes] and

from the cosmological equations you had postulated it should follow that the radius

of the curvature of the universe was a time-independent quantity. You obtain this

result by exploiting the circumstance that one obtains as a necessary consequence

of the universal equations a vanishing of the divergence of tensor . From the

vanishing of the divergence of tensor , you obtained the relation:

(*)

.

Such a relation naturally demonstrates the continuity of the radius of curvature R,

however, and consequently also the incorrectness of my article.

I did not, of course, manage to obtain relation (*) from the vanishing of tensor

; the result I obtained does not contradict the case of a nonstationary world.[3]

In view of the definite interest attached to the problem of a possible existence of a

nonstationary world, I permit myself to submit to you the calculations of the diver-

gence of tensor I carried out for your evaluation and consideration. Let be

the kth component of the contragradient tensor that represents the divergence

and then, according to the formula, we shall have for the divergence:

;

interests us, as , , turn into zero and specifically as a consequence of

the circumstance that we have the unstationary world expressed in my article by the

formula under the conditions (C):

For we shall have:

D3

Tik

Tik

∂x4

∂ρ

0=

Tik

Tik Qk

Tik

Qk

1

g

------ -----------------------------

∂

ggασTαk

∂xσ

kσ

s

gασTαs – =

Q4 Q1 Q2 Q3

D3) (

41

4

0, =

42

4

0, =

43

4

0, =

44

4

0. =

Q4