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THE NEW DYNAMICS 277

well known, this is not a trivial problem. Equation 15.3 can equivalently be writ-

ten in the form

The second term—which accounts for the possibility of exchanging energy-
momentum between the gravitational field and matter—complicates the transition
from differential to integral laws by simple integration over spatial domains. Ein-
stein found a way out of this technical problem. He was the first to cast Eq. 15.6
in the form of a vanishing divergence [El6]. He noted that since the curvature
scalar R is linear in the second derivatives of the g^, one can uniquely define a
quantity R* which depends only on the g^ and their first derivatives by means
of the relation

Next define an object tff by

With the help of the gravitational field equations, it can be shown that Eq. 15.6

can be cast in the alternative form

Therefore, one can define

as the total energy-momentum of a closed system. Einstein emphasized that,
despite appearances, Eq. 15.9 is fully covariant. However, the quantity f^ is not
a generally covariant tensor density. Rather, it is a tensor only relative to affine
transformations.
These results are of particular interest in that they show how Einstein was both
undaunted by and quite at home with Riemannian geometry, which he handled
with ingenuity. In those years, he would tackle difficult mathematical questions
only if compelled by physical motivations. I can almost hear him say, 'General
relativity is right. One must be able to give meaning to the total energy and
momentum of a closed system. I am going to find out how.' I regard it as no
accident that in his October 1916 paper Einstein took the route from Eq. 15.9 to
Eq. 15.6 rather than the other way around! For details of the derivation of Eq.
15.9 and the proof that ^ is an affine tensor, I refer the reader to Pauli's review
article [P4] and the discussion of the energy-momentum pseudotensor by Landau
and Lifshitz [L5].