Thursday, October 15, 2015

The geometrization of physics

The greatest story of 20th century (XXc) physics was the geometrization of physics. Special relativity caught on as it was interpreted as a 4-dimensional non-Euclidean geometry. Then general relativity realized gravity as curved spacetime. Quantum mechanics observations were projections in Hilbert space. Electromagnetism became a geometric gauge theory, and then so were the weak and strong interactions. By the 1980s, theorists were so sold on geometry that they jumped on the wildest string theories. The biggest argument for those theories was that the geometry is appealing.

Many people assume that Einstein was a leader in this movement, but he was not. When Minkowski popularized the spacetime geometry in 1908, Einstein rejected it. When Grossmann figured out to geometrize gravity with the Ricci tensor, Einstein wrote papers in 1914 saying that covariance is impossible. When a relativity textbook described the geometrization of gravity, Einstein attacked it as wrongheaded.

I credit Poincare and Minkowski with the geometrization of relativity. In 1905, Poincare had the Lorentz group and the spacetime metric, and at the time a Klein geometry was understood in terms of a transformation group or an invariant metric. He also implicitly used the covariance of Maxwell's equations, thereby integrating the geometry with electromagnetism. Minkowski followed where Poincare left off, explicitly treating world-lines, non-Eudlidean geometry, and covariance. Einstein had none of that. He only had an exposition of Lorentz's theorem, not covariance or spacetime or geometry.

For a historian's detailed summary of how Poincare and Minkowski developed the geometric view of relativity, see Minkowski, Mathematicians and the Mathematical Theory of
and The Non-Euclidean Style of Minkowskian Relativity by Scott Walter.

In 1900, physics textbooks were not even using vector notation. That is how far we have come. Today vectors are indispensable, but they are also more subtle than the average student realizes. Most physics books don't explain the geometry adequately.

Carlo Rovelli writes:

We will continue to use geometry as a useful branch of mathematics, but is time to abandon the longstanding idea of geometry as the description of physical space. The idea that geometry is the description of physical space is engrained in us, and might sound hard to get rid of it, but it is unavoidable; it is just a matter of time. Better get rid of it soon.

Geometry developed at first as a description of the properties of parcels of agricultural land. In the hands of ancient Greeks it became a powerful tool for dealing with abstract triangles, lines, circles, and similar, and was applied to describe paths of light and movements of celestial bodies with very great efficacy. ...

In reality, they are quantum entities that are discrete and fluctuating. Therefore the physical space in which we are immersed is in reality a quantum dynamical entity, which shares very little with what we call "geometry". It is a pullulating process of finite interacting quanta.
This essay is included in a new book, This Idea Must Die: Scientific Theories That Are Blocking Progress (Edge Question Series), has a curious collection of opinions. It was promoted on a Science Friday broadcast.

I don't know what he has against geometry. Geometry is central to most of XX century physics, and no good alternative has been found.

There is no evidence that physical space is discrete or fluctuating. We do not even have proof that electrons are discrete. Sure, some observables have discrete spectrum. But our theories for nature itself are continuous, not discrete, and make heavy use of geometry.

I have previously commented on Rovelli's defense of Aristotle, Einstein, and free will.


  1. Shinichi Mochizuki and Rovelli should team up and tour the world as a freak show.

    Why does Roger avoid the Mochizuki 500 page math proof madness and pick on the physicists?

  2. The first big problem I do have with geometry, is not with what is for, but how it is misused. Geometry was not created to be used with time originally, and when time is incorporated ala Minkowski, it is often done so foolishly. Minkowski space violates Einstein's own statements about how time is considered relativistically. If you ware zipping about the universe, you do not check your clock you left at home on earth, you check the clock you have with you. Time must always be measured locally, since to do otherwise you import all manner of problems as you attempt to pull in your time information from a very distant source with the limitation of the speed of light. Not only does the distance become critical, the velocity becomes a factor too, since the velocity can change things just as much as the distance can. There can be no instantaneous time clock over relativistic distance, ever. Think about it.

    The second big problem with Minkowski space is that the time variable IS NOT and HAS NEVER BEEN orthogonal to x y and z mechanically. How could it be? If your spaceship x is zipping away along an axis at one relativistic rate, and your spaceship y is zipping away along it's axis at a different relativistic rate, their clocks are aboard each of their respective ships...time is measured locally, it has to be. Time is internal to your x and y spaceships, not some other location far far way neither of them occupies. Can you consider each of your ships in an imaginary simultaneous period moment? Certainly. Can they actually be in any way measured simultaneously, no. Physicists and scientists are responsible for always acknowledging the limits or constraints of their measurements and how these limitations affect the limits of what they can measure. You can imagine anything in any way you like. You can not measure anything in any way you would like. imagine =/= measure.

  3. That 500 page paper has not be published in a journal. I have no idea whether it is a good work.

  4. Feynman said the geometrical explanation of gravity in GR is neither "necessary' nor "essential" to physics. ‘‘It is one of the peculiar aspects of the theory of gravitation, that it has both a field interpretation and a geometrical interpretation. ... the fact is that a spin-two field has this geometrical interpretation: this is not something readily explainable—it is just marvelous. The geometrical interpretation is not really necessary or essential to physics.’’