Monday, January 25, 2016

Stewart on beauty and truth in relativity

I expect physicists to under-appreciate the fine points of a theory, and to under-credit the mathematical contributions. But I just stumbled across a 2007 book by a real mathematician who does a great job of explaining a lot of great mathematics, and then recites the same stupid Einstein myths as the physicists.

The book is Why Beauty Is Truth: The History of Symmetry, by Ian Stewart. Chap. 11 is on "the clerk from the patent office, and has a lot of Einstein idol worship. In particular:
The common theme here is symmetry. Changing from one frame of
reference to another is a symmetry operation on space-time. Inertial frames are about translational symmetries; rotating frames are about rotational symmetries. Saying that Newton's laws are the same in any inertial frame is to say that those laws are symmetric under translations at uniform speed. For some reason, Maxwell's equations do not have this property. That seems to suggest that some inertial frames are more inertial than others. And if any inertial frames are special, surely it should be those that are stationary relative to the aether.

The upshot of these problems, then, was two questions, one physical, one mathematical. The physical one was, can motion relative to the aether be detected in experiments? The mathematical one was, what are the symmetries of Maxwell's equations?

The answer to the first was found by Albert Michelson, a US Navy officer who was taking leave to study physics under Helmholtz, and the chemist Edward Morley. They built a sensitive device to measure tiny discrepancies in the speed of light moving in different directions, and concluded that there were no discrepancies. Either the Earth was at rest relative to the aether - which made little sense given that it was circling the Sun -- or there was no aether, and light did not obey the usual rules for relative motion.

Einstein attacked the problem from the mathematical direction. He didn't mention the Michelson-Morley experiment in his papers, though he later said he was aware of it and that it had influenced his thinking. Instead of appealing to experiments, he worked out some of the symmetries of Maxwell's equations, which have a novel feature: they mix up space and time. (Einstein did not make the role of symmetry explicit, but it is not far below the surface.) One implication of these weird symmetries is that uniform motion relative to the aether - assuming that such a medium exists - cannot be observed.

Einstein's theory acquired the name "relativity," because it made unexpected predictions about relative motion and electromagnetism. [p.189] ...

Einstein was not the only person to notice that the symmetries of spacetime, as revealed in Maxwell's equations, are not the obvious Newtonian symmetries. In a Newtonian view, space and time are separate and different. Symmetries of the laws of physics are combinations of rigid motions of space and an independent shift in time. But as I mentioned, these transformations do not leave Maxwell's equations invariant.

Pondering this, the mathematicians Henri Poincaré and Hermann Minkowski were led to a new view of the symmetries of space and time, on a purely mathematical level. If they had described these symmetries in physical terms, they would have beaten Einstein to relativity, but they avoided physical speculations. They did understand that symmetries of the laws of electromagnetism do not affect space and time independently but mix them up. The mathematical scheme describing these intertwined changes is known as the Lorentz group, after the physicist Hendrik Lorentz.

Minkowski and Poincare viewed the Lorentz group as an abstract expression of certain features of the laws of physics, and descriptions like "time passing more slowly" or "objects shrinking as they speed up" were thought of as vague analogies rather than anything real. But Einstein insisted that these transformations have a genuine physical meaning. Objects, and time, really do behave like that. He was led to formulate a physical theory, special relativity, that incorporated the mathematical scheme of the Lorentz group into a physical description not of space and a separate time, but of a unified space-time.

Minkowski came up with a geometric picture for this non-Newtonian physics, now called Minkowski space-time. It represents space and time as independent coordinates, and a moving particle traces out a curve - which Einstein called its world Line - as time passes. Because no particle can travel faster than light, the slope of the world line can never get more than 45° away from the time direction. The particle's past and future always lie inside a double cone, its light cone. [p.192]
No, it was Minkowski who introduced "world-point" for a spacetime point in his 1907 paper, and "world-line" in his more famous 1908 paper. Einstein did not use any of this spacetime terminology until after Minkowski's famous 1908 paper gained widespread acceptance.

On the bigger picture of who discovered special relativity, let's look at what Stewart gets right. The physical basis was the Michelson-Morley experiment, and the mathematical basis was the symmetry group of Maxwell's equations. Stewart gets this right, and acknowledges that Einstein did not explicitly mention either basis.

By comparison, Lorentz explicitly relied on Michelson-Morley in his papers of 1895, 1899, and 1904, Poincare did so in his 1904, short 1905, and long 1905 papers, and Minkowski in his 1907 and 1908 papers. Poincare and Minkowski explicitly described the Lorentz group as the symmetries of 4-dimensional spacetime and Maxwell's equations.

Thus Lorentz, Poincare, and Minkowski explicitly had the physical and mathematical bases of special relativity, and Einstein did not.

(I should note that the Michelson-Morley experiment by itself did not prove relativity. There were other possible explanations, such as a stationary Earth, aether drift, and emitter theory of light. Those explanations were rejected for other reasons. Historically, Michelson-Morley was the crucial experiment.)

So why does Stewart insist on crediting Einstein? The argument is that Poincare and Minkowski understood spacetime "on a purely mathematical level", and their transformations "were thought of as vague analogies rather than anything real."

Stewart is wrong. As you can see from the above papers, Lorentz, Poincare, and Minkowski are explicitly using their transformations to explain the Michelson-Morley experiment. There is no vague analogy. They are better grounded in real experiments than Einstein.

Minkowski's famous paper begins:
Gentlemen! The concepts about time and space, which I would like to develop before you today, have grown on experimental physical grounds. Herein lies their strength. Their tendency is radical. Henceforth, space for itself, and time for itself shall completely reduce to a mere shadow, and only some sort of union of the two shall preserve independence.
He is most emphatically saying that the Michelson-Morley experiment along with Maxwell's equations lead to a new understanding about the reality of space and time. He is transforming physical coordinates for space and time, and not just some vague mathematical non-real analogy.

Stewart says that Poincare and Minkowski did not understand that the symmetries mix up space and time. He is obviously wrong, as Minkowski emphasized that space and time were inseparable.

Again it is hard to understand how a mathematician like Stewart can get this so badly wrong. I can only assume that he never looked at any of the original papers, and relied on Einstein-idolizing accounts by physicists.

It is somewhat true that "Einstein attacked the problem from the mathematical direction." Philosophers who credit Einstein do so largely because he ignored experiments, and was just trying to give an alternative formulation of Lorentz's theory. As Lorentz later said, Einstein's 1905 innovation was to postulate what was previously proved. But Einstein failed to get to the mathematical heart of the theory that Poincare and Minkowski found -- that relativity is a 4D spacetime non-Euclidean geometry theory.

Covariance is an essential concept to XX century theoretical physics. Physicists commonly misunderstand this, and textbooks usually do not explain it correctly. It is what makes symmetry one of the most important concepts in all of physics. To Poincare and Minkowski, the heart of relativity is the covariance of Maxwell's equations under the Lorentz group. Einstein did not understand or appreciate the concept until 1915, as he wrote a paper against it in 1914.

The facts of relativity history are not in any serious dispute. It is amazing how many people get the facts and theory essentially right, and then idolize Einstein for work done by others, not Einstein.

Stewart is right that the history of relativity is a symmetry story. The Michelson-Morley experiment demonstrated symmetries in the physical world that were difficult to reconcile with the mathematical symmetries of Maxwell's equations. That is the issue that Lorentz squarely addressed and partially solved in 1895. This work ultimately led to Poincare and Minkowski discovering that all of these symmetries can be explained by a non-Euclidean geometry on 4-dimensional spacetime. That is the essence of special relativity. Einstein played no part in this story, except to help popularize work that had been published and accepted by experts years earlier.


  1. ``Stewart is right that the history of relativity is a symmetry story.''

    Roger, you are being just as much of a fraud as those you routinely criticize on your blog, to various quantitative extents.

    Symmetry is so simple a concept---and by itself, so *bereft* of much knowledge by itself---or of endowing knowledge even in other contexts---that, in the context of the progress of man's knowledge, it couldn't have had too much a position of prestige had it not been for Deductive sort of guys and Western Women (trying) to occupy the centerpiece (and/or Superiority over Men's Being (mostly through fradulent means including their own selves.)).

    Sorry. You might have simultaneously defended a Berkeley co-graduate, a woman's tradition, and a favorite at Princeton, but, no, to a knowledge-seeker like me, it adds nothing but a non-cognitive burden.

    Yours sincerely [whatever that phrase means [my childhood schoolteachers [mostly men!]] meant it as a polite way to end letters [thought I have no desire to pull them back into the game, just the way, I have no desire to pull the late Dr. Kajale [a non-Brahmin, non-Westerner, non-White, and, best of all, already dead!] into the game either.]]


  2. Roger,

    Why did my above comment go through so fast?

    You owe me an explanation.

    Make it as brief as possible, so that it looks as ``heTaaLaNavaadi'' (ask the Dalit/Christian Pradnya Walhekar) as possible.

    Your friend,


  3. Ajit,
    I'm not certain what language you think you are speaking, but it isn't coherent English. Roger is not fond of Einstein idolatry in academia, I get it. I don't always agree with it, but I get his point.

    You on the other hand, I think you might have some kind of grudge against men? white people? Western civilization in general? I'm not sure you are sure what you are talking about. Please clarify your point if you have one.

  4. There is some lack of logic in Einstein's original 1905 derivation:

    (1) He does not use the 1st postulate to derive Lorentz transforms though he uses the 2nd postulate (c=const).
    (2) Then he shows that Lorentz transforms indeed render E-M equations invariant. In some sense he proves the validity of the 2nd postulate that it holds for the E-M equations and Lorentz transform.

    If he really used the 2nd postulate he would have derived Lorentz transforms as the ones that makes E-M equations invariant. Then he would not really need the 2nd postulate concerning the constancy of the speed of light. I think, this is how Lorentz himself got his transforms and Poincare confirmed them and proved that they constitute a group. Certainly it was Lorentz who was puzzled with the outcome of the M-M experiment and this lead him, I think, to derive his transforms. However I am not sure if he directly used wave equation to look for transform that renders it invariant. This would the most logical course for somebody who is dealing with M-M experiment. BTW, Einstein never said that M-M was the main reason for his discovery. He mentioned Fizeau and stellar aberration and on some occasions he claimed he did not know of M-M result.

    I suspect that the reason Einstein derivation is somewhat idiosyncratic (structurally/epistemologically not logical) it is because he had to present "his theory" as original and different from the work of Lorentz and Poincare to cover up his borrowing from them.

    Also I found it strange that he mentions that the transform and relativistic addition of velocities constitute a group in a very off hand manner in passing. He does not show the derivation of it while he shows derivation of much more trivial points in the paper. Did he even know what group (in algebraic sense) was? I suspect that Lorentz did not know math at this level of abstraction. It was Poincare who brought it in his last paper or papers but for him it was natural. By tracing when Poincare used it for the first and last time we could estimate to which Poincare's papers Einstein had access to.

    I have to admit however that Einstein's paper reads much better than papers of Lorentz or Poincare perhaps part of it is because we have been taught STR using Einstein's concepts and notations.

    Do you know under what circumstances Einstein could have had access to the last two Poincare papers (the 2nd of them is Palermo paper) before submitting his paper? Or would consider the French hypothesis that the work was actually done in Gottingen by the Hilbert group?

  5. Poincare's first 1905 paper says that the Lorentz transformations form a group, and Einstein's library received a copy about 2 weeks before his 1905 paper was submitted. My theory is that Einstein added that sentence about a group without understanding what a group is.

    Einstein only shows that the E-M equations are invariant in the same sense that Lorentz had already shown.

    Yes, Einstein's paper is easier to read, as he made an attempt to make it self-contained. But a lot of details are very confusing.

    Einstein said that M-M was crucial for special relativity, but not for his own contribution to the subject.

    Some books explain STR like Einstein. But many others present it as M-M experiment, Lorentz transformations, and Minkowski space. This approach has very little to do with anything Einstein did.