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252 RELATIVITY, THE GENERAL THEORY

Having described this splitting of the Ricci tensor, Einstein next proposed his

penultimate version of the gravitational equations:

covariant under local unimodular transformations. They are a vast improvement

over the Einstein-Grossman equations and cure one of the ailments he had diag-

nosed only recently: unimodular transformations do include rotations with arbi-

trarily varying angular velocities. In addition, he proved that Eqs. 14.8 can be

derived from a variational principle; that the conservation laws are satisfied (here

the simplified definitions Eq. 14.3 play a role); and that there exists an identity

where T is the trace of T^. He interpreted this equation as a constraint on the

g^. A week later, he would have more to say on this relation.

In the weak-field limit, g^ = t}^ + h^ (Eq. 12.29), one recovers Newton's law

from Eq. 14.8. Einstein's proof of this last statement is by far the most important

part of this paper. 'The coordinate system is not yet fixed, since four equations

are needed to determine it. We are therefore free to choose* [my italics]

Then Eqs. 14.8 and 14.10 yield

which reduces to the Newton-Poisson equation in the static limit.
The phrase italicized in the above quotation shows that Einstein's understand-
ing of general covariance had vastly improved. The gravitational equations do not
determine the h^ (hence the g^) unambiguously. This is not in conflict with caus-
ality. One may choose a coordinate system at one's convenience simply because
coordinate systems have no objective meaning. Einstein did not say all this explic-
itly in his paper. But shortly afterward he explained it to Ehrenfest. 'The appar-
ently compelling nature of [my old causality objection] disappears at once if one
realizes that ... no reality can be ascribed to the reference system' [E43].

3. November the Eleventh. A step backward. Einstein proposes [E46] a
   scheme that is even tighter than the one of a week earlier. Not only shall the
   theory be invariant with respect to unimodular transformations—which implies
   that g is a scalar field—but, more strongly, it shall satisfy

*'Wir diirfen deshalb willkiirlich festsetzen.. .'. Equation 14.10 is the harmonic coordinate condi-

tion in the weak-field limit [W3].