Three Geometrizations in History

Juliano C. S. Neves writes a New paper:

There have been three geometrizations in history. The first one is historically due to the Pythagorean school and Plato, the second one comes from Galileo, Kepler, Descartes and Newton, and the third geometrization of nature begins with Einstein's general relativity. Here the term geometrization of nature means the conception according to which nature (with its different meanings) is largely described by using geometry. ...

Undoubtedly, the history of the geometrized nature begins in the ancient Greek period. ...

Then the third movement into the geometrization of nature begins with Einstein (1916) and general relativity, which I call geometrization 3.0. However, following Lehmkuhl (2014), contrary to the common opinion in physics, it is worth emphasizing that Einstein did not consider general relativity as the theory that geometrizes gravity. But, as we will see, general relativity brings a lot of geometrical concepts to describe the phenomena.

For attempts to take geometry out of general relativity, see Anderson 1999 and Brown 2009.

So if Einstein did not geometrize gravity, who did? Everyone else accepted relativity as a geometric theory.

Brown explains that Einstein took decades to come around to the geometric view that Poincare and Minkowski had in 1905-8.

Why this lapse on Einstein’s part? I wonder if it was not because of the misgivings he had about the way he formulated his 1905 paper, misgivings which grew throughout his life. First, there is little doubt that right from the beginning he was aware of the limited explanatory power of what he called “principle theories” like thermodynamics. Secondly, when he confessed in 1949 to having committed in 1905 the “sin” of treating rods and clocks as primitive entities, and not as “moving atomic configurations” subject to dynamical analysis, he was merely repeating a point of self-correction he made in 1921. Finally, it is fairly clear that Einstein was increasingly unhappy with the central role that electrodynamics, and in particular the behaviour of light, played in his 1905 paper.

This last aspect of Einstein’s reasoning brings us to the main point of this subsection. Einstein wrote in 1935:

The special theory of relativity grew out of the Maxwell electromagnetic equations. ... [but] the Lorentz transformation, the real basis of special-relativity theory, in itself has nothing to do with Maxwell theory. (Einstein 1935).

Similarly, in a 1955 letter to Born, Einstein would write that the “Lorentz transformation tran- scended its connection with Maxwell’s equations and has to do with the nature of space and time in general”. He went on to stress that “the Lorentz-invariance is a general condition for any physical theory.” (Born et al. 1971, p. 248). What is clear is that for the mature Einstein, the principle of Lorentz covariance, which applies to all the non-gravitational interactions, not just electrody- namics, is the heart of special relativity.8 In stressing this point, Einstein was distancing himself from his formulation of 1905 with its emphasis on fundamental phenomenological postulates (one of which being the “constancy” of the speed of light relative to the “rest” frame).

So Einstein finally adopted in 1935 the geometric view of relativity that Poincare published in 1905 and Minkowski improved and popularized in 1908. In that view, the Lorentz transformation is a symmetry of spacetime. It is a symmetry for any physical theory, and not just the Maxwell theory.

My theory is that the third geometrization occurred with Poincare and Minkowski in 1905-8. They both described it as a radical break from existing thinking. Poincare said that the new geometry was like Copernicus replacing Ptolemy, and Minkowski said that henceforth space and time will be united. The essence of special relativity is that there is a non-euclidean geometry on spacetime. Einstein missed it, but it is what make special relativity so popular with others.

Here is Sean M. Carroll describing the geometry of relativity, in his recent book:

But he didn’t go quite so far as to advocate joining space and time into a single unified space-time. That step was left to his former university professor, Hermann Minkowski, in the early 20th century. The arena of special relativity is today known as Minkowski space-time. ...

But, says relativity, just as the distance as the crow flies is generally different from the distance you actually travel between two points in space, the duration of time that you experience generally won’t be the same as the universal coordinate time. You experience an amount of time that can be measured by a clock that you carry with you on the journey. This is the proper time along the path. And the duration measured by a clock, just like the distance traveled as measured by the odometer on your car, will depend on the path you take. ...

The difference is this: In space, a straight line describes the shortest distance between two points. In space-time, by contrast, a straight path yields the longest elapsed time between two events. It’s that flip from shortest distance to longest time that distinguishes time from space.

Euclidean geometry has the shortest distance between two points is given by the Pythagorean theorem. Distances in the non-euclidean geometry of spacetime work differently.

Of course he does have to give some goofy reason to credit Einstein, as everyone else does:

The development of relativity is usually attributed to Albert Einstein, but ...

Einstein’s contribution in 1905 was to point out that the ether had become completely unnecessary, and that we could better understand the laws of physics without it.

No, this is a myth. What Einstein said about the aether was nearly identical to what Lorentz said ten years earlier. That is, they both rejected theories that depended on aether motion or motion against the aether, and their theories avoided mentioning the aether. Einstein could never explain how his theory was different from Lorentz's. Poincare and Minkowski did explain how their relativity theory was different, and the difference was non-euclidean geometry, not aether.