3 3 0 D O C U M E N T 3 8 7 C O M M E N T O N T R E F F T Z

387. “Comment on E. Trefftz’s[1] Paper:

‘The Static Gravitational Field of Two Mass Points

in Einstein’s

Theory’”(1)

[Einstein 1922r]

Presented 23 November 1922

Published 21 December 1922

In: Preußische Akademie der Wissenschaften (Berlin). Sitzungsberichte (1922): 448–449.

The author grounds his analysis on the field equations in vacuo,

, (1)

which are equivalent to the equations:

, (1a)

as is easily proved by reducing

(1a).[2]

The author believes he has found a solution

that has a spherical connection in space and except for the two masses no singular-

ity, also not containing any other

masses.[3]

In view of the importance of the problem to the cosmological issue, i.e., the

question of the large-scale geometrical structure of the universe, I was interested to

know whether the equations really did yield as a physical possibility a static uni-

verse whose material mass was concentrated in just two celestial

bodies.[4]

It

became apparent, however, that Trefftz’s solution does not permit this physical

interpretation at all. This will be demonstrated in the following.

Mr. Trefftz sets out on the assumption for the (four-dimensional) line element:

. (2)

This assumption corresponds to a space of spherical symmetry around the origin.

The special case = const; would correspond to the Euclidean-Galilean

isotropic and homogeneous space.

(1)Mathem.

Ann. 26, 317, 1922.

[p. 448]

Rik

1

4

--gikR - – 0 =

Rik

1

2

--gikR - – λgik – 0 =

ds2 f4(x)dt2 dx2 f2(x)(dϑ2 sin2ϑdφ2)]

+ + [ – =

f4 f2 x2 =