6 4 D O C U M E N T 7 9 M A R C H 1 9 2 1

P. S. Currently Abraham is in Zurich. He filed an appeal in Rome at the highest

court to [annul] his dismissal as professor, and from what he said, he has prospects

of

success.[7]

– I read your fine speech in the Berl[iner] Ber[ichte] with great

pleasure.[8]

79. From Wilhelm Wirtinger

Vienna IX, 4 Strudlhofgasse, 4 March 1921

Highly esteemed Colleague,

Many thanks for your kind lines of 22 February

1921.[1]

Please forgive my over-

sight—with ; during the computation I had overlooked an index

commutation.[2]

I also see, as you pointed out, the tensor H in Weyl’s paper, which I had likewise

initially overlooked.[3] The question whether I is the only invariant of the sought

kind must be answered with no for n = 4, but I believe only for this number, because

the linear geometry immediately yields a second one differing from it.[4] For if you

calculate the index pairs 12/34, 13/42, 14/23 corresponding to the identity between

line coordinates that pair together with the same let-

ters, but that are differently accented , then with ,

also is a tensor. Thus = also

becomes a scalar L, which takes the factor when the ’s are substituted by

. In place of d , any differential form ds.M(I,L) can therefore appear, where

M is a homogeneous function of I and L of the dimension ¼, hence about

or . The relation I:L is then a function of the location only dependent on

the ’s and its first and second differential quotients, to which physical signifi-

cance will surely be ascribed. With spaces of constant radius of curvature, both

vanish identically with H, so at the boundary transition [to] them, other conditions

will simply have to be added.

For variation problems of the four-dimensional region, initially only

will probably come into consideration, where

5 differential identities are then valid between the Lagrangian equations. Perhaps

conditions of a very general type to our worldview can already provide a decision

here on and .

Your assumption [used] earlier for the transition to special problems, that space

at infinity was Euclidean or even just of const. curvature, would have the conse-

K K

p12p34 p13p42 p14p23 + + 0 =

H

g

/

1

gH -- -

/

= H g

1H

g

-- - H

a–2

g

ag

I L + 4

I

4

L

4

+

g

0 I L + g x1d d x2d x3dx4 =