222 RELATIVITY, THE GENERAL THEORYincorrect, is nevertheless quite important. Consider (he says) a four-dimensional
space-time domain divided into two parts, L, and L 2. The source 6^ of the grav-
itational field (see Eq. 12.34) shall be nonzero only in L,. Nevertheless, B^ deter-
mines the g,,, in all of L by means of Eq. 12.34. Make a generally covariant
transformation #„ -» xj, such that x^ — x'^ in L, while, at least in part of L 2 ,
x^ ¥= x'p. Then g^ ¥= g'^ in that part of L 2. The source 6^ remains unchanged:
®i» — &>in L, while, in L 2 , 6^ once it is equal to zero stays equal to zero. There-
fore, general covariance implies that more than one g^ distribution is possible for
a given 6^ distribution. 'If—as was done in this paper—the requirement is
adhered to that the g,,, are completely determined by the #„„ then one is forced to
restrict the choice of reference system' (my italics). (Note that the above transfor-
mation x^ — x\ is not allowed if the transformation is linear!) This reasoning is
quite correct. Then what had gone wrong?
Einstein's 'physical argument' is irrelevant. The gf, are not completely deter-
mined by #„,,. His predicament was, put most succinctly, that he did not know the
Bianchi identities. Let us consider the final form for Eq. 12.34, which he was to
derive in 1915:
where R^ is given by Eq. 12.20 and R = K"g^. The left-hand side satisfies the
four Bianchi identitiesBecause of these relations, Eq. 12.36 does not determine the g^1 " uniquely—just as
the Maxwell equations do not determine the electromagnetic potentials uniquely
[W9]. The gf, are determined only up to a transformation gf, —» g'^, correspond-
ing to an arbitrary coordinate transformation x^ —*• x'f. Einstein still had to under-
stand that this freedom expresses the fact that the choice of coordinates is a matter
of convention without physical content. That he knew by 1915—although even
then he still did not know the Bianchi identities (Chapter 15).
We now also understand Grossmann's difficulty with the Newtonian limit. Use
Eq. 12.29 and define h'^ = £„„-}£ v^**- Then Eq-^12 -^35 becomes
an intransparent relation. However, one is free to choose a coordinate frame in
which
In the static weak-field limit, all components of Rf, except R^ are negligible and
(see Eq. 12.31)