THE FIELD EQUATIONS OF GRAVITATION 25!
heart in the course of my collaboration with my friend Grossmann three years
earlier.' (It should be said that in matters of science a heavy heart never lasted
very long for Einstein.)
For the last time, I recall that Einstein and Grossmann had concluded [El7]
that the gravitational equations could be invariant under linear transformations
only and that Einstein's justification for this restriction was based on the belief
that the gravitational equations ought to determine the g^ uniquely, a point he
continued to stress in October 1914 [E16]. In his new paper [E44], he finally
liberated himself from this three-year-old prejudice. That is the main advance on
November 4. His answers were still not entirely right. There was still one flaw,
a much smaller one, which he eliminated three weeks later. But the road lay open.
He was lyrical. 'No one who has really grasped it can escape the magic of this
[new] theory.'
The remaining flaw was, of course, Einstein's unnecessary restriction to uni-
modular transformations. The reasons which led him to introduce this constraint
were not deep, I believe. He simply noted that this restricted class of transfor-
mations permits simplifications of the tensor calculus. This is mainly because
Vg is a scalar under unimodular transformations (cf. Eq. 12.14). Therefore the
distinction between tensors and tensor densities no longer exists. As a result, it is
possible to redefine covariant differentiation for tensors of rank higher than 1. For
example, instead of Eq. 12.13, one may use [E45]Equation 12.17 can be similarly simplified. 'The most radical simplification' con-
cerns the Ricci tensor given in Eq. 12.20. Write*where [W2]Up is a vector since yg is a scalar; s^ is the covariant derivative of v^. Therefore,
under unimodular transformations, R^ decomposes into two parts, r^ and s^, each
of which separately is a tensor.*The quantities /?,,„, ?•„„, $„„ correspond to Einstein's G^, Rf,, Sf, in [E44].