274 RELATIVITY, THE GENERAL THEORYknew already that one may obtain two pointlike images of the far star if the align-
ment is imperfect [E12a]. In any event, to Mandl's delight [M5] Einstein went
on to publish a calculation of the dependence of the image intensity upon the
displacement of the observer from the extended line of centers of the two stars
[E12b].* He believed that 'there is no hope of observing this phenomenon.' How-
ever, in 1979 it was shown that the apparent double quasar 0957 + 561 A,B is
actually the double image of a single quasar [W8a]. An intervening galaxy acts
as the gravitational lens [Yl].15c. Energy and Momentum Conservation; the Bianchi Identities
The collected works of Felix Klein contain a set of papers devoted to the links
between geometry on the one hand and group theory and the theory of invariants
on the other, his own Erlangen program. The last three articles of this set deal
with general relativity. ('For Klein ... the theory of relativity and its connection
with his old ideas of the Erlangen program brought the last flare-up of his math-
ematical interests and mathematical production' [W9].) One of those three, com-
pleted in 1918, is entitled 'On the Differential Laws for the Conservation of
Momentum and Energy in the Einstein Theory of Gravitation' [K2]. In its intro-
duction Klein observed, 'As one will see, in the following presentation [of the con-
servation laws] I really do not any longer need to calculate but only to make use
of the most elementary formulae of the calculus of variations.' It was the year of
the Noether theorem.
In November 1915, neither Hilbert nor Einstein was aware of this royal road
to the conservation laws. Hilbert had come close. I recall here some of his conclu-
sions, discussed in Section 14d. He had derived the gravitational equations from
the correct variational principle
for variations g^ —*• g^ + dg^, where the dg,,, are infinitesimal and vanish on
the boundary of the integration domain. Without proof, he had also stated the
theorem that if / is a scalar function of n fields and if
then there exist four identities between the n fields. He believed that these ident-
ities meant that electromagnetism is a consequence of gravitation and failed to see
that this theorem at once yields the conservation laws [H4]. In a sequel to his
work of 1915, presented in December 1916 [H5], his interpretation of Eq. 15.2
had not changed. (In view of the relations between Hilbert and Einstein, it is of
interest to note that in this last paper Hilbert refers to his subject as 'the new
*For references to later calculations of this effect, see [S6a].