Briefly, Lorentz explained the contraction (and Lorentz transformation LT) as motion causing an electromagnetic distortion of matter and fields. ESR deduces the LT as a logical consequence from either the Michelson-Morley experiment (as FitzGerald 1889 and Lorentz 1992 did) or from postulates that Lorentz distilled from that experiment and Maxwell's equations (as Einstein 1905 did). MSR recasts the LT as symmetries of a 4-dimensional geometry (as per Poincare 1905 and Minkowski 1908).
In case you think that I am taking some anachronistic view of MSR, I suggest The Non-Euclidean Style of Minkowskian Relativity and Minkowski’s Modern World (pdf) by Scott Walter, and Geometry and Astronomy: Pre-Einstein Speculations of Non-Euclidean Space, by Helge Kragh. Non-Euclidean geometry was crucial for the early acceptance and popularity of special relativity in 1908, and dominated the early textbooks.
While the Lorentzian constructive view is unpopular, it has some explanatory advantages, and is a legitimate view. I tried to add a sentence last year saying that to the Wikipedia article on length contraction, but I was overruled by other editors.
Dan Shanahan recently wrote a paper on explanatory advantages to LET, and asks:
You say:To answer this, you have to understand that Poincare and Minkowski were mathematicians, and they mean something subtle by a geometrical space.
"In the preferred (Minkowski) view, the LT is not really a transmutation of space and time. Minkowski spacetime has a geometry that is not affected by motion. The LT is just a way to get from one set of artificial coordinates to another."
"The physics is all built into the geometry, and the contraction seems funny only because we are not used to the non-Euclidean geometry."
I would understand, and could accept, both passages if you were describing the non-Euclidean metric of the gravitational equation where we have the intuitive picture of a curved spacetime. But I cannot see what changes could actually be occurring in flat spacetime in consequence of the rotation described by the LT. You say that ”Minkowski spacetime has a geometry that is not affected by motion” and it is here in particular that I am not sure of your meaning.
I would say myself that whatever space might be, neither it nor its geometry could be affected (except in the sense described by general relativity) either by the motion through it of an observer or other object, or (and this is the important point) by changes in that motion. But that cannot be what you are saying for that is the Lorentzian constructive approach.
Roughly, a geometrical space is a set of points with geometrical structures on it. The subtle part is that it must be considered the same as every other isomorphic space. So a space could be given in terms of particular coordinates, but other coordinates can be used isomorphically, so the coordinates are not to be considered part of the space. The space is a coordinate-free manifold with a geometry, and no particular coordinates are necessarily preferred.
The concept is so subtle that Hermann Weyl wrote a 1913 book titled, "The Concept of a Riemann Surface". The concept was that it could be considered a coordinate-free mathematical object. I can say that in one sentence, but Weyl had a lot of trouble explaining it to his fellow mathematicians.
Minkowski space can be described as R4 with either the metric or LT symmetry group (or both), but that is not really the space. It is just a particular parameterization of the space. There is no special frame of reference in the space, and many frames can be used.
Poincare introduced Minkowski space in 1905, and gave an LT-invariant Lagrangian for electromagnetism. The point of this is that the Lagrangian is a (covariant) scalar function on the coordinate-free Minkowski space. He then introduced the 4-vector potential, and implicitly showed that it was covariant. Minkowski extended this work in 1907 and 1908, and was more explicit about using a 4-dimensional non-Euclidean manifold, and constructing a covariant tensor out of the electric and magnetic fields. With this proof, the fields became coordinate-free objects on a geometrical space.
This is explained in section 8 of Henri Poincare and Relativity Theory, by A. A. Logunov, 2005.
Thus the physical variables are defined on the 4D geometrical space. The LT are just isomorphisms of the space that preserve the physical variables.
This concept is like the fact that Newton's F=ma means the same in any units. To do a calculation, you might use pounds, grams, or kilograms, but it does not matter as long as you are consistent. The physical variables do not depend on your choice of units.
Changing from pounds to grams does not affect the force, and the LT does not transmute spacetime. They are just changes of coordinates that might make calculations easier or harder, but do affect any physical reality.
To answer Dan's question, motion is just a change of coordinates, and in MST, the physics is built on top of a geometry that does not depend on any such coordinates. Motion by itself is not a real thing, as it is always relative to something else.
To me, the essence of special relativity is that it puts physics on a 4D non-Euclidean geometry. That is why it bugs me that Einstein is so heavily credited, when he only had a recapitulation of Lorentz's theory, and not the 4D, symmetry group, metric, covariance, and geometry. Many physicists and historians blame Poincare for believing in an aether with a fixed reference frame. The truth is more nearly the opposite, as Poincare said that the aether was an unobservable convention and proved that symmetries make all the frames equivalent in his theory.
Dan gives this thought experiment, which is his variant of the twin paradox:
Consider two explorers, who we will call Buzz and Mary. They had been travelling, in separate space ships, side by side, in the same direction. But Buzz has veered away to explore a distant asteroid. Mary concludes from her knowledge of the LT that time must now be running more slowly for Buzz and that he and his ship have become foreshortened in the direction that Buzz is travelling relative to her. Buzz observes no such changes either in himself or in his ship. To Buzz, it is in Mary and her ship that these changes have occurred. Buzz is also aware that events that he might previously have regarded as simultaneous are no longer so.Some books try to avoid this issue by saying that ESR only applies to inertial frames like Mary, not Buzz. But historically, Lorentz, Poincare, Einstein, and Minkowski all applied special relativity to accelerating objects. General relativity is distinguished by gravitational curvature.
But what has actually changed? ...
For Buzz the LT will describe very well his altered perspective. But it would be as inappropriate to explain length contraction, time dilation and loss of simultaneity as resulting from a physical transformation of space or spacetime as it would be to describe the rotation of an object in 3-space as a rotation of space rather than a rotation in space.
Dan argues in his paper that LET gives a physical explanation for what is going on, and ESR does not. I agree with that. Our disagreement lies with MST.
I say that there are two ways of explaining this paradox, and they are the same that Poincare described in 1905. You can follow Lorentz, say that everything is electromagnetic, and attribute the LT to distortions in matter and fields. Dan's paper explains this view.
His second explanation was to say that relativity is a theory about “something which would be due to our methods of measurement.” The LT is not a contraction of matter or of space. It is just a different parameterization of a geometric spacetime that is not affected by motion at all.
Let me give my own thought example. Suppose Mary and Buzz are on the equator, traveling north side-by-side to the North Pole. Assume that the Earth is a perfect sphere. Along the way, Buzz takes a right turn for a mile, then two left turns for a mile each, expecting to meet back up with Mary. Mary waits for him, but does not alter her northerly path. Then they discover that they are not on the same course. What happened?
You can apply Euclidean geometry, draw the great circles, and find that they do not close up. Or you can apply spherical geometry, where Mary and Buzz are traveling in straight lines, and note that squares have angles larger than 90 degrees. The problem is not that matter or space got transmuted. The problem is that Buzz took a non-Euclidean detour but returned with Euclidean expectations.
In Dan's example, Buzz takes a detour in spacetime. It does not do anything strange to his spaceship or his clocks. The strangeness only occurs when he uses Euclidean coordinates to compare to Mary. Nothing physical is strange, as long as you stay within Minkowski's geometry.
Thus there are two physical explanations for the special relativistic effects. One is the FitzGerald Lorentz LET, and one is Poincare Minkowski geometry. Einstein gives more of an instrumentalist approach to LET, and does not really explain it.