The Contributions of LaGrange and Gauss to GEORG BERNHARD REIMANN

Screen Shot 2018-04-30 at 6.31.18 AM.pngThe Google Doodle on the 241st anniversary of Gauss’ birth

If we look at mathematics as a process in motion – like calculus itself – we can see the development from LaGrange’s thinking and outcomes directly to Gauss.  It seems almost inevitable that a “Field Theory” would develop for electrostatics. Not so fast, you might say.  And you would be correct.

However, what was not known was the conception of that same ‘form of theory’ for gravitation or the motion of the planets in “conical shapes” or ellipses. And the use of field theory in the study of gravity and spacetime. Gauss saw past one discipline of study – mathematics – to others. THAT was his greatest contribution.

Today  Gauss, or Gauß depending how tight you are, is honoured by Google not on his 240th, but 241st Birthday.*

“The Prince of Mathematics”

Mathematician he was, and he spent most of his time in that field at Gottingen.   And as Gauss’ theories were only part-way understood, even by himself, and solved by Maxwell in four eloquent partials, it was left to Georg Bernhard Reimann to completely understand the need for a space and time independent of coordinate system and to resolve Gauss’  mysteries.

Physicists alway have need of mathematicians

The Greatest Mathematician of German origin in the 19th Century 

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Reimann understood what no other person understood.  That Euclidean Geometry had failed, its days were numbered, and at the end of the day it was wrong.  Gauss had struggled his whole life with that form and conception of coordinates.  But to free our conception of Spacetime and Gravity we had to free ourselves, even of the coordinate system and our thinking about its meaning, as Riemann showed us.  Einstein summarized this in 1949,

“Now it came to me…the independence of the gravitational acceleration from the nature of the falling substance, may be expressed as follows: In a gravitational field (of small spatial extension) things behave as they do in a space free of gravitation…This happened in 1908.  Why were seven years required for the construction of the general theory of relativity?  The main reason lies in the fact that it is not so easy to free oneself from the idea that coordinates must have an immediate metric meaning.” – A. Einstein (1949) as quoted by Kip Thorne, et al in GRAVITATION (pp5).

Later, Einstein remembered Gauss, but moved rapidly past his limited thought process to Reimann,

“[In 1912] I suddenly realized that Gauss’s theory of surfaces holds the key for unlocking this mystery.  I realized that Gauss’s surface coordinates had a profound significance. However, I did not know at that time that Riemann had studied the foundations of geometry in an even more profound way. I suddenly remembered that Gauss’s theory was contained in the geometry course given by Geiser when I was a student… I realized that the foundations of geometry have physical significance.  My dear friend the mathematician Grossmann was there when I returned from Prague to Zürich. From him I learned for the first time about Ricci and later about Riemann. So I asked my friend whether my problem could be solved by Riemann’s theory [Pais’s italics], namely, whether the invariants of the line element could completely determine the quantities I had been looking for?”

The breakthrough had come in Einstein’s thinking and only 2 years later he published his first work on General Relativity.  With the right mathematicians he rapidly moved forward.

Emmy Noether

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Finally, Emmy Noether appeared with Hilbert’s blessings and defended Einstein, and in the process blew us ALL AWAY!


* How did he add 1 to 100? 😉

 

 

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