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THE EINSTEIN-GROSSMANN COLLABORATION 221

12d. The Stumbling Block

Clearly, Einstein and Grossmann were in quest of a tensor !"„, of such a kind that

the Newton-Poisson equation

(see Eqs. 12.17 and 12.18). There remained the last question: what are the field

equations of gravitation itself? Einstein guessed correctly that 'the needed gener-

alization [of the Newtonian equations] should be of the form

where ... F,,,, is a ... tensor of the second rank which is generated by differential

operations from the fundamental tensor gf,.'

Then the trouble began.

would emerge as a limiting case. This, Einstein said, was impossible as long as

one requires, in the spirit of Eq. 12.35 that F^ be no higher than second order in

the derivatives of the g^. Two arguments are given for this erroneous conclusion.

The first one, found in Einstein's part, can be phrased as follows. One needs a

generalization of div grad 0. The generalization of the gradient operator is the

covariant differentiation. The generalization of 0 is g^. But the covariant deriv-

ative of g,,, vanishes (Eq. 12.16)! In Einstein's words, 'These operations [the cov-

ariant version of div grad] degenerate when they are applied to ... g^. From this,

it seems to follow that the sought-for equations will be covariant only with respect

to a certain group of transformations ... which for the time being is unknown to

us.'

The second argument, contained in Grossmann's part, is also incorrect. As was

mentioned above, Grossmann saw that the Ricci tensor (Eq. 12.20) might well be

a candidate for T^ in Eq. 12.34. However, according to Grossmann, 'it turns out

... that this tensor does not reduce to A0 in the special case of the weak gravita-
tional field.' Reluctantly, the conclusion is drawn in EG that the invariance group
for the gravitational equations has to be restricted to linear transformations only
(dx"/dx'f is independent of x"), since then, it is argued, d/dx"(gvdgp,,/dx'') does
transform like a tensor, which, moreover, reduces to Q^ in the weak-field limit
given by Eq. 12.29. 'If the field is static and if only g^ varies [as a function of
x], then we arrive at the case of Newton's gravitation theory.' The troublesome
Eq. 12.16 had been evaded!
Einstein also gave a 'physical argument' for the impossibility of generally
covariant equations for the gravitational field. This argument, though of course