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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.
This is to be found at
"CONTRIBUTIONS OF 20TH CENTURY WOMEN TO PHYSICS." CWP
< http://www.physics.ucla.edu/~cwp >
.....
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 Göttingen 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 the theory 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.
....
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