258 RELATIVITY, THE GENERAL THEORYlends an exalted quality to his paper, from the title, 'Die Grundlagen der Physik,'
The Foundations of Physics, to the concluding paragraph, in which he expressed
his conviction that his fundamental equations would eventually solve the riddles
of atomic structure. In December 1915, Einstein remarked that Hilbert's com-
mitment to Mie's theory was unnecessary from the point of view of general rela-
tivity [E53]. 'Hilbert's Ansatz for matter seems childish to me,' he wrote some
time later [E54]. Justified though these criticisms are, Hilbert's paper nevertheless
contains a very important and independent contribution to general relativity: the
derivation of Eq. 14.15 from a variational principle.
Hilbert was not the first to apply this principle to gravitation. Lorentz had done
so before him [L3]. So had Einstein, a few weeks earlier [E44]. Hilbert was the
first, however, to state this principle correctly:for infinitesimal variations g"(x) — g"(x) + f>g(x) such that 8g"" = 0 at the
boundary of the integration domain (R is the Riemann curvature scalar, L the
matter Lagrangian). It is well known that Eq. 14.18 leads to Eq. 14.15, including
the trace term, if L depends on gf" but not on their derivatives.
Hilbert's paper also contains the statement (but not the proof!!) of the following
theorem. Let / be a scalar function of n fields and let b$J\/~gdx = 0 for varia-
tions x" — x" + ^(x) with infinitesimal |". Then there exist four relations
between the n fields. It is now known that these are the energy-momentum
conservation laws (Eq. 14.17) if /= L and the identities (Eq. 14.16) if /= R,
but in 1915 that was not yet clear. Hilbert misunderstood the meaning of the
theorem as it applied to his theory. Let / correspond to his overall gravitational-
electromagnetic Lagrangian. Then / depends on 10 + 4 fields, the g^, and the
electromagnetic potentials. There are four identities between them. 'As a conse-
quence of... the theorem, the four [electromagnetic] equations may be considered
as a consequence of the [gravitational] equations.... In [this] sense electromag-
netic phenomena are gravitational effects. In this observation I see the simple and
very surprising solution of the problem of Riemann, who was the first to seek
theoretically for the connection between gravitation and light.' Evidently Hil-
bert did not know the Bianchi identities either!
These and other errors were expurgated in an article Hilbert wrote in 1924
[H5]. It is again entitled 'Die Grundlagen der Physik' and contains a synopsis of
his 1915 paper and a sequel to it [H6], written a year later. Hilbert's collected
works, each volume of which contains a preface by Hilbert himself, do not include
these two early papers, but only the one of 1924 [H7]. In this last article, Hilbert
*Scc the detailed discussion of variational principles in [W10] and [M5]. The tensor T" is defined
by SJL Vgd'x = }i-SV^T"(x)Sgl,(x)dtX.
"Here Hilbert referred to the essay 'Gravitation und Licht' in Riemann's Nachlass [R2].