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 =