Wednesday, November 13, 2013

How non-Euclidean geometry entered physics

Danish historian Helge Kragh wrote Geometry and Astronomy: Pre-Einstein Speculations of Non-Euclidean Space in 2012. This appears to be a free copy of what was published in a journal under the title, The first curved-space universe, altho this version seems somewhat different.
This paper examines in detail the attempts in the period from about 1830 to 1910 to establish links between non-Euclidean geometry and the physical and astronomical sciences, including attempts to find observational evidence for curved space. Although there were but few contributors to "non-Euclidean astronomy," there were more than usually supposed. The paper looks in particular on a work of 1872 in which the Leipzig physicist K. F. Zoellner argued that the universe is closed in accordance with Riemann's geometry. ...

Whereas non-Euclidean geometry flourished as a mathematical research field in the last half of the nineteenth century (see the figure on p.8), its connection to the real space inhabited by physical objects was much less cultivated. The large majority of mathematicians did not care whether real space was Euclidean or not; and those who did care only dealt with the subject in a general and often casual way, avoiding to deal seriously with the possibility of determining a space curvature different from zero. After all, that was supposed to be the business of the astronomers.
This paper has a lot of good info, but some things are conspicuously missing.

By far the most important development in this subject is the 1905-8 formulation of special relativity as a 4-dimensional non-Euclidean geometry. That theory gave a geometrical spacetime interpretation to the FitzGerald contraction, Lorentz local time, Maxwell's equations, and various experiments. Poincare had the metric, symmetry group, and covariance, and Minkowski elaborated on those with diagrams and world-lines. This theory was one of the biggest breakthrus in the history of physics, and is in all the textbooks today. There is no evidence that Einstein or anyone else had these ideas independently.

Perhaps Kragh omits this because he is more interested in curved space. But I doubt it because he also says, "The present consensus view, in part based on the inflationary scenario, is that we live in a flat or Euclidean space". Minkowski space is a flat non-Euclidean geometry. He never explains that geometries can be Euclidean or non-Euclidean, and non-Euclidean geometry can be flat or curved. What he says is that he is interested in non-Euclidean geometry entering physics, and Minkowski space did exactly that.

I am not sure who introduced curved space into relativity. That is, I don't know who first had the idea that Minkowski space might be curved. It was probably Marcel Grossmann. He was an expert in non-Euclidean geometry and he proposed such a theory in 1913, with the condition that a gravitational field in empty space has Ricci tensor zero. Einstein published papers denying that such a non-Euclidean geometrical theory was possible, and suggesting less geometrical theories. Grossmann turned out to be exactly correct, until the recent discovery of dark energy. It appears that Levi-Civita and Hilbert eventually convinced Einstein that Grossmann was correct.

According to recent scholarship, Einstein never really accepted the non-Euclidean geometrization of gravity.

Kragh only says this about Grossmann:
In Zurich, Fiedler taught geometry to, among others, Einstein and his friend Marcel Grossmann, who wrote his doctoral thesis under Fiedler. As well known, Einstein’s development of the general theory of relativity relied crucially on Grossmann’s mathematical expertise.
Specifically, what Einstein got from Grossmann was the metric, stress-energy, and Ricci tensors, the gravitational field equations for empty space, and covariant geometrical formulation of the theory.

The story of how non-Euclidean geometry because essential to modern physics is an important one, as it underlies much of 20th century physics from relativity to particle interactions, and everyone gets it wrong. I cannot explain how Kragh overlooks the elephant in the room. Kragh is a well-respected historian and does excellent work. But somehow all of these professors have blinders on when it comes to crediting Einstein, as I have criticized Kragh in 2010 and 2011.

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