276 RELATIVITY, THE GENERAL THEORYRicci in 1889, and then, again independently, in 1902 by Klein's former pupil
Luigi Bianchi.* The name Bianchi appears neither in any of the five editions of
Weyl's Raum, Zeit, Materie (the fifth edition appeared in 1923) nor in Pauli's
review article of 1921 [PI]. In 1920, Eddington wrote in his book Space, Time
and Gravitation, 'I doubt whether anyone has performed the laborious task of
verifying these identities by straightforward algebra' [El7]. The next year he per-
formed this task himself [E18]. In 1922 a simpler derivation was given [Jl], soon
followed by the remark that Eq. 15.4 follows fromnow known as the Bianchi identities, where R^, is the Riemann curvature tensor
[H7].** Harward, the author of this paper, remarked, 'I discovered the general
theorem [Eq. 15.5] for myself, but I can hardly believe that it has not been dis-
covered before.' This surmise was, of course, quite correct. Indeed, Eq. 15.5 was
the relation discovered by the old masters, as was finally brought to the attention
of a new generation by the Dutch mathematicians Jan Schouten and Dirk Struik
in 1924: 'It may be of interest to mention that this theorem [Eq. 15.5] is known
especially in Germany and Italy as Bianchi's Identity' [S9].
From a modern point of view, the identities 15.3 and 15.4 are special conse-
quences of a celebrated theorem of Emmy Noether, who herself participated in
the Goettingen debates on the energy-momentum conservation laws. She had
moved to Goettingen in April 1915. Soon thereafter her advice was asked. 'Emmy
Noether, whose help I sought in clarifying questions concerning my energy law
...' Hilbert wrote to Klein [K4], 'You know that Fraulein Noether continues to
advise me in my work,' Klein wrote to Hilbert [K4]. At that time, Noether herself
told a friend that a team in Goettingen, to which she also belonged, was perform-
ing calculations of the most difficult kind for Einstein but that 'none of us under-
stands what they are good for' [Dl]. Her own work on the relation between
invariance under groups of continuous transformations and conservation theorems
was published in 1918 [N5]. Noether's theorem has become an essential tool in
modern theoretical physics. In her own oeuvre, this theorem represents only a
sideline. After her death, Einstein wrote of her, 'In the judgment of the most com-
petent living mathematicians, Fraulein Noether was the most significant creative
mathematical genius since the higher education of women began' [E19].
Let us return to Einstein's article of October 1916. The principal point of that
paper is not so much the differential as the integral conservation laws. As is now
*For more historical details, see the second edition of Schouten's book on Ricci calculus [S8].
**Equation 15.4 follows from Eq. 15.5 by contraction and by the use of symmetry properties of the
Riemann tensor [W12].