9780192806727.pdf

THE NEW DYNAMICS 275

physics of Einstein's relativity principle' [H6].) As for Einstein, in 1914 [E13]

and again on November 4, 1915, [E14] he had derived the field equations of grav-

itation from a variational principle—but in neither case did he have the correct

field equations. In his paper of November 25, 1915, [El 5] energy-momentum

conservation appears as a constraint on the theory rather than as an almost

immediate consequence of general covariance; no variational principle is used.

I repeat one last time that neither Hilbert nor Einstein was aware of the Bian-

chi identities in that crucial November. Let us see how these matters were straight-

ened out in subsequent years.

The conservation laws are the one issue on which Einstein's synopsis of March

1916 [E6] is weak. A variational principle is introduced but only for the case of

pure gravitation; the mathematics is incorrect;* matter is introduced in a plausible

but nonsystematic way ([E6], Section 16) and the conservation laws are verified

by explicit computation rather than by an invariance argument ([E6], Section 17).

In October 1916 Einstein came back to energy-momentum conservation [E16].**

This time he gave a general proof (free of coordinate conditions) that for any

matter Lagrangian L the energy-momentum tensor T^1 " satisfies

as a consequence of the gravitational field equations. I shall return shortly to this
paper, but first must note another development.
In August 1917 Hermann Weyl finally decoded the variational principle (Eq.
15.2) [W10]. Let us assume (he said) that the £* are infinitesimal and that f and
its derivatives vanish on the boundary of the integration domain. Then for the case
that / = L, it follows that Eq. 15.3 holds true, whereas if / = R we obtainf

A correspondence between Felix Klein and Hilbert, published by Klein early in

1918 [K4], shows that also in Goettingcn circles it had rapidly become clear that

the principle (Eq. 15.2), properly used in the case of general relativity, gives rise

to eight rather than four identities, four for / = L and four for / = R.

Interestingly enough, in 1917 the experts were not aware that Weyl's derivation

of Eq. 15.4 by variational techniques was a brand new method for obtaining a

long-known result. Neither Hilbert nor Klein (nor, of course, Einstein) realized

that Eq. 15.4, the contracted Bianchi identities, had been derived much earlier,

first by the German mathematician Aurel Voss in 1880, then independently by

*As Bargmann pointed out to me, Einstein first specializes to the coordinate condition \Jg = 1 and

then introduces a variational principle without a Lagrange multiplier for this condition.

**An English translation of this paper is included in the well-known collection of papers by Einstein,

Lorentz, Minkowksi, and Weyl [S7].

fFor this way of deriving Eqs. 15.3 and 15.4, see [Wll]. Other contributions to this subject are

discussed in [P3]. For the relation of Weyl's results to those of Klein, see [K3].