Dear Edwin,
Thank you for your kind words. I will directly address some of your specific comments below:
"I very much like your explanation that Einstein's invariance of the laws of physics first focused on "the independence of the speed of light from the speed of its source" but this was then recast "in terms of the invariance of the speed of light." Thank you! These are not the same and you are the first I have observed to contrast them!"
You may find the discussion in the article by Baierlein which I cited of interest.
"In my essay The Fundamental Nature of Time I (as have others) propose local gravity as the medium through which light propagates, i.e., "the ether". In this case the independence from the speed of its source is preserved, but the invariance of the speed of light (in all frames) is not!"
Well, from my point of view, this is a rather delicate issue. As I mentioned in my paper, one cannot have Minkowski spacetime without at least some object with a non-zero proper time and that implies at least some object with non-zero mass, which, by the equivalence principle implies, non-zero gravity field. So I am sympathetic to the idea that gravity is lurking in the background even in Minkowski spacetime, and I have some of my own ideas of how I would make that explicit, but this is right now a backburner project for me.
"You then reconceptualize the Lorentz contraction in terms of dimensional abatement. As I understand this you are assuming that Lorentz contraction is physically real."
As real as anything in a 3-dimensional slice of spacetime, which, I mentioned in my paper, is the arena of our reality.
"In the literature the Lorentz transformation is always derived between two different inertial frames, each of which has its own 'universal time' dimension. I have in An Energy-Based Derivation of Lorentz Transformation in One Inertial Frame derived the Lorentz transformation in one inertial frame, and shown that any length contraction is only apparent. The corresponding 'time dilation' that is implied by space-time symmetry is re-interpreted in terms of energy-time conjugation, and the result is that "the relativity of simultaneity" vanishes and time regains its meaning as universal simultaneity."
Well, I would have to look at how you implement this, e.g. whether you are claiming that relativity of simultaneity is not "real" in some sense but still necessary in our mathematical description of spacetime events. But let me just say that eliminating simultaneity altogether is in conflict with the very geometric structure of spacetime.
"In short, our assumptions differ, but we both treat the Lorentz-based relativity in novel ways. I hope you enjoy my essay as much as I have enjoyed yours, and I welcome any comments you might have."
I will be happy to look at your paper.
"You consider a body moving with v = c. But is this limiting case really possible?"
Well, certainly it is not possible for any spacetime observer to attain that speed in space in any frame.
"If I understand you correctly, you then attribute "redundant dimensionality" to such a hypothetical frame. You trace this to the assignment of a 4D coordinate frame."
The third bullet point claimed that in a speed of light frame it is *spacetime* which has redundant dimensionality. I do not trace this to the assignment of a 4D coordinate system but to the fact that in such a frame both the timelike and the spacelike direction of motion become lightlike, and therefore linearly dependent.
The mention of "assignment of a 4D coordinate frame" comes in when I examine how this conundrum can be easily solved: You cannot assign a 4D frame to such objects, but that does not mean you cannot assign a coordinate frame whatsoever (as it is commonly believed today)! The what seems to me in retrospect obvious solution is that you assign a 3D frame, which in spherical coordinates wholly contains a lightcone without any matter in it.
"My Lorentz derivation in one frame supports only one time dimension. The 4D Minkowski rotates time into space, but if only one time dimension exists, it only projects time onto itself, leaving 3D objects to rotate and translate in 3D."
The orthodox explanation for the existence of two distinct kinds of time parameters is that this simply reflects the fact that observers in relative motion "slice up" spacetime differently. When we get right down to it, according to special relativity there are no objects in space for which "time passes". There are only extremely long and narrow and often branching 4-dimensional tubes. However, I can understand that this may not be such a satisfying answer, and that worries that special relativity implies an infinite number of time dimensions may persist.
In part 2, the companion paper to this one, I address this issue in a novel way: Proper time is reconceptualized in terms of duration of existence in spacetime. In that way, time dilation becomes reframed in terms of a comparison of the observed durations of existence of objects during an interval of the observer's own duration of existence. In this way, time dilation becomes an entirely about a comparison of different local phenomena, as opposed to different global phenomena: There is only one time dimension, but different objects may differ on their relative durations of existence in spacetime. I know these ideas are very unfamiliar, hopefully my paper will be finished soon and, if you like, you can read the details there.
"You link the dimensional reduction to invariance of the speed of light. Does reduction occur in ether - either gravity or 'quantum vacuum'?"
Well, this is another delicate subject which I cannot adequately answer at the moment because the distinctions which are necessary to give a satisfactory answer have not been defined yet. However, the companion paper will address this issue: The second part shifts from special relativity to quantum mechanics.
"When discussing photons, it's fascinating to note that the Maxwell-Hertz equations are Galilean invariant. I only recently learned this, and I believe it is significant."
Perhaps...I will have to look at it.
I've not yet understood how this relates to your demonstration that "magnetic fields are line integrals of dimensionally reduced versions of electric fields". I will try to study this until I understand it. Nor have I understood whether this depends upon the Lorentz transformation. It does not seem to, at first glance, but this may conflict with your appendix.
Well, since I am not familiar with the Maxwell-Hertz equations, I cannot tell, either (yet?). But there is a clear relationship with the Lorentz transformations which I did not have enough space in my paper to point out. In some textbooks, magnetic fields are explained as a "relativistic effect" by imagining the following scenario: a point charge near a wire in which the uniformly drifting electrons are exactly canceled by the stationary positive charges in the cable material "observes" a net zero electric field. Now, another point charge moving parallel to the wire will observe the wire to be contracted, which means that the charge density increases, but because the relative motion of the positive charges is different from the relative motion of the negative ones, the charge densities increase by different amounts and therefore no longer cancel, so that the moving charge "sees" a net electric field, which exerts a force that, when transformed back to the original charge has exactly the form we attribute to the magnetic force.
What this shows is that Lorentz contraction plays a key role in generating magnetic fields, but the usual ways of conceptualizing it are indirect, involving a transformation to a moving frame and then transforming back. My analysis is direct: Lorentz contraction can be described as dimensional abatement, and if it causes some phenomenon, then that phenomenon must also have a description in terms dimensional abatement. Fields are infinitely extended objects, so dimensional abatement there cannot occur by means of length contraction since an infinitely long object contracted by any finite factor is still infinitely long. Rather, it occurs by superimposing E and B fields (notice how it makes no sense to say that an E field is "length contracted", we can only talk about the field strength of components of the field at a point in space).
Incidentally, I feel that this point should just be visually very obvious, but the feedback I have gotten so far makes it seem as if it may not be? So let me ask you: Imagine a Coulomb field; all the force arrows are aligned radially in 3 dimensions. Now imagine the magnetic field of an infinitely long straight wire. All the force arrows are aligned radially in 2 dimensions, and to get a 3D description you have to integrate over the length of the wire. Does this not strike you as a visually obvious example of my claim that B-fields are line integrals of dimensionally reduced analogs of E-fields?
Of course, there are more complicated examples, like magnetic dipole fields etc. but these complications do not negate the underlying simple structural relationship between E and B fields. As I mentioned in my paper, this is mathematically implied by the fact that to transform from a frame with a pure E-field to a frame with a pure B-field requires v=c, which indicates dimensional reduction.
"You conclude that dimensional abatement is a more fundamental concept of nature than Lorentz contraction. In my essay Lorentz transformation is interpreted as an energy-time interpretation of reality, not a space-time reality. I wonder if you will see if our papers make sense together, or are sent down different paths by being based on different assumptions."
I should have a better idea once I read your paper.
Thank you again for reading my paper and for your extensive essay comments.
All the best,
Armin
