It is a rare writer who can describe an historical episode in
mathematical development that makes me sit up, rather than fall to sleep.
I offer this sample from Nina Byers's narrative, which I hope you
find as riveting as I did.
An extract from
"E. Noether's Discovery of the Deep Connection Between Symmetries
and Conservation Laws" by Nina Byers.
.....
II.Chronology of the Events Leading to the Discovery
In 1915, Emmy Noether was invited to join the team of mathematicians
assembled in Göttingen by David Hilbert. Hermann Weyl reports [7]:
"To both Hilbert and Klein, Emmy was welcome as she was able to help them
with invariant-theoretic knowledge." She was thirty-three at that time,
having received a doctorate in mathematics from the University of Erlangen
seven years earlier, and written the eleven papers listed in Appendix A.
The first in the list is her thesis which was done under the supervision
of Paul Gordan. For her thesis she calculated all the 331 invariants of
ternary bi-quadratic forms! Shortly thereafter she took the abstract
approach to algebra following Hilbert's 1888 basis theory paper.
She worked unpaid in Erlangen supervising students and sometimes lecturing
for her ailing father. After her father died, she joined Hilbert and his
team in Göttingen. [8]
In June-July 1915, shortly after Noether arrived, Albert Einstein gave
six lectures in Gottingen on the general theory of relativity. At that time
the theory was not yet finished; he had not yet found the complete field
equations. However, the basic ideas were clear and his audience found
them compelling. After giving the lectures, Einstein said [9]:
"To my great joy, I completely succeeded in convincing Hilbert and Klein."
He had been working to generalize the special theory of relativity to
include gravity since 1905. In 1907 he discovered the importance of the
equality of gravitational and inertial mass and formulated the equivalence
principle, but it took another eight years to complete the theory.
Finally in November 1915, having found the complete field equation, he
submitted the famous paper [10] that gives thetheory in its final form.
Remarkably, in the same month, Hilbert submitted a manuscript [11] in
which the same field equations are obtained as the solution to a
variational problem. Hilbert and Einstein had independently found the
field equations at about the same time [9]. In November 1915, Emmy Noether
wrote to Ernst Fischer: "Hilbert plans to lecture next week about his
ideas on Einstein's differential invariants,and so .. [we] had better
be ready" [8].
It would seem, therefore, that she began to study relativity theory then.
Out of that study came two papers which Hermann Weyl characterized as
giving "the genuine and universal mathematical formulation of two of
the most significant aspects of general relativity theory: first,
the reduction of the problem of differential invariants to a purely
algebraic one by use of normal coordinates; and second, the identities
between the left sides of Euler's equations of a problem of
variation ..." [7].
Regarding the first paper referred to by Weyl [6], Einstein wrote
to Hilbert: "Yesterday I received from Miss Noether a very interesting
paper on invariant forms. I am impressed that one can comprehend these
matters from so general a viewpoint. It would not have done the old guard
at Göttingen any harm had they picked up a thing or two from her. ..." [12].
The second paper referred to by Weyl is the I.V. paper we will discuss at
length.
....