Getting the Lorentz transformations without requiring an invariant speed (2015)
arxiv.org
arxiv.org
Einstein's original paper on special relativity, while mentioning the two famous postulates, also later assumes linearity of the transformations (allowing non-linear takes it from special relativity to general relativity). However, the paper did not explicitly call it out as a postulate. Also, Einstein derived a particular equation in the paper from two paths and mentions that there's consistency. This was actually a missed opportunity -- He could have forced consistency as an input assumption, worked backwards, and would then have seen that invariant speed is not needed as an input assumption anymore.
With this OP submission, I have now learnt that this has been known for more than a century now, and agree that this should be more popular in books.
https://www.goodreads.com/quotes/78897-since-the-mathematici...
>Strangely enough no personal contacts resulted between his teacher of mathematics, Hermann Minkowski, and Einstein. When, later on, Minkowski built up the special theory of relativity into his 'world-geometry', Einstein said on one occasion: 'Since the mathematicians have invaded the theory of relativity, I do not understand it myself any more'. But soon thereafter, at the time of the conception of the general theory of relativity, he readily acknowledged the indispensability of the four-dimensional scheme of Minkowski Einstein, philosopher-scientist, Schilpp
And later Einstein expressed admiration for geometry again and again
>This problem remained insoluble to me until 1912, when I suddenly realized that Gauss's theory of surfaces holds the key for unlocking this mystery. I realized that Gauss's surface coordinates had a profound significance. However, I did not know at that time that Riemann had studied the foundations of geometry in an even more profound way. Einstein, Kyoto Lecture, 1922
>I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot Hentschel (1998). The Collected Papers of Albert Einstein, Vol. 8 (English): The Berlin Years: Correspondence, 1914-1918
What he complained about, the Physics textbook approach was the path taken by Einstein who started from the "invariant speed" assumption to derive relativity:
https://sites.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/...
>"introduce another postulate... namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body."
There's also nothing really "wrong" with Einstein's approach either, it's motivated by physical constraint from Maxwell's electromagnetism.
Mathematician's approach is motivated by imposing logical constraint of an abstract mathematical structure (group of transformations from one coordinate system to another)
The two turn out to be connected (physical constraint selects a math model from alternatives). Nothing really strange.
When I was working out the maths myself, I had exactly the same thought.
>if A sees B at speed v, then B sees A at the same speed v
is directly derivable from the Lorentz velocity addition formula which is the result of the usual two postulates. Why would you want that as an extra assumption...
https://en.wikipedia.org/wiki/Velocity-addition_formula#Spec...
vAA = (vAB + vBA)/(1+(vAB*vBA/c^2) -> 0 = vAB + vBA
so velocity of A relative to B and B relative to A both have magnitude v.
The point is that in the derivation of relativity from elementary postulates, the “reflectivity of velocities” assumption is used. For instance the linked paper uses it between equations (12) and (13). All such derivations use it at some stage.
Intuitively, “reflectivity of velocities” is why time dilation leads to length contraction with same gamma factor.
I don’t think it follows from absence of a preferred frame since one could imagine some complicated group structure relating the velocities of the various observers relative to each other in such a way that all are on equal footing and yet v_AB is not exactly -v_BA.
As mentioned, this is certainly a reasonable assumption but who knows what whacky alternatives are out there? After all, in the different context of quantum mechanics, such “obvious” relationships as xp = px no longer hold true.
So, from the standpoint of pure math, isotropy of the speed of light is purely a convention. The argument can only be made based on, so to speak, philosophical reasoning.
- You derived the theory correctly (end result should match textbook result) and thought what you were doing were wrong.
- No physicist you asked could explain where you were wrong
- No physicist you encountered could see the obvious connection of this and the usual approach
It's not really a mysterious or novel fact that you can have these shift of perspectives (at least since Kant who heavily influenced Einstein) e.g. Newton took his three laws as the base, derived relativity as a theorem. Einstein took (modified version) relativity (together with the invariant speed assumption which comes from electromagnetism) as the base and rebuilt the rest of theory of motion that reduces to Newton's theory in the limit. Nothing is amiss here.
What the paper is actually saying: assuming there isn’t a privileged reference frame, you can show that either the Galilean transformations hold (i.e. simultaneity is invariant, suggesting instantaneous propagation of forces, a special case of Lorentz transformations where c is infinity) or Lorentz transformations hold (i.e. a single invariant speed in all reference frames, suggesting propagation of forces at that speed).
Empirically, we know from measuring the speed of light that we live in the latter world. But we can’t prove that a priori as the title suggests.
The paper's abstract says:
>> "... though the finite value of this speed must, of course, be obtained from experiment"
It should state, "the value of this speed..." (removed 'finite'). That the speed is finite does not come from the proof but from measurement.
So what have we really gained? Since we still need the extra assumption that an invariant speed actually exists, we could've just gone the other way and done the light clock calculation to get the Lorentz transformation instead.
I agree that the paper's title is somewhat misleading, since you still do need to assume an invariant speed to rule out the Galilean transformations.
However, this derivation does greatly narrow things down before the invariant speed comes in: at the point where the invariant speed is assumed, you already know that there are only two alternatives: an invariant speed (Lorentz transformations) or Galilean transformations. So it's much easier to see why you would assume an invariant speed; the assumption isn't just pulled out of thin air at the start, it is seen to be one of only two alternatives that are compatible with the principle of relativity.
Sadly, I feel like SR can only really be "understood" as a complete theory. All the individual phenomena (time dilation, length contraction, relativity of simultaneity, constant speed of light etc.) are very hard to understand, because you cannot just take one of them and add it to classical relativity without immediately running into paradoxes. Only once the whole picture is known you see that all the pieces beautifully imply each other. This problem applies to every approach to the subject I've seen.
In my experience approaching the same idea from many different perspectives yields a deeper, richer understanding of the underlying concept, and offers a path for someone to reach the big picture understanding you mention.
This isn't a postulate, it's a derived theorem. That's true no matter what axiomatic formulation you use.
> All the individual phenomena (time dilation, length contraction, relativity of simultaneity, constant speed of light etc.) are very hard to understand, because you cannot just take one of them and add it to classical relativity without immediately running into paradoxes. Only once the whole picture is known you see that all the pieces beautifully imply each other.
This is all true, but all of these things you talk about (except the speed of light) are not postulates; they are derived theorems. No approach to relativity that I'm aware of has ever tried to start with any of these things as postulates. Even Einstein's original 1905 paper didn't start with any of these things as assumptions. He started with the principle of relativity and the speed of light being invariant. This paper is just showing how to derive at least a part of the second assumption from the first.
"Let us consider two events, E1 and E2, at the same spatial location in frame O, but separated by a time difference τ. In O′ the two events are separated by a time lapse T."
If you didn't know special relativity, you would never get this idea.
You're looking at it backwards. The paper is not assuming anything here; in fact it is explicitly refusing to assume that we know the correct transformation law between frames. That means we have to leave open the possibility of the time difference changing, not because we know SR, but because we are being logically rigorous.
This is because you implicily used the (wrong) postulate that "phenomena of special relativity can be iteratively added to a a lassical description of a non-relativistic theory".
It's not out of thin air, it's from a very empirically successful theory: Maxwell's electrodynamics. The problem back then was that this theory was not relativistic, i.e. the speed of electromagnetic wave in Maxwell's equations was the same under all reference frames. So you either abandon the idea that laws of physics remaining the same under all reference frames, OR abandon Galilean velocity addition.
Einstein's approach was modifying the latter so that it fits with the former, by imposing invariant speed. This was written in Einstein's original paper. It's not a mystery assumption.
It's also a very common procedure: two empirically successful theories have conflict and you need to resolve them by building something larger than both and reducing to both under limit.
I also agree we have gained insight into how kinematic structure is derived from algebra + physical constraint. Though you still need the physical insight to choose which physical constraint.
That was the source of the assumption, yes--as you point out, Einstein said so in his original paper. But from the standpoint of mechanics, as opposed to electrodynamics, it was pulled out of thin air. There was no reason based on mechanics to make any such assumption. In fact, everyone else except Einstein that was working on the problem was looking at ways to modify electrodynamics, not mechanics--in other words, to come up with a theory of electrodynamics that was Galilean invariant, rather than to come up with a theory of mechanics that was Lorentz invariant.
> you either abandon the idea that laws of physics remaining the same under all reference frames, OR abandon Galilean velocity addition
Or, as above, you look for a Galilean invariant theory of electrodynamics. Of course we know today that that is a dead end, but that wasn't known then.
https://news.ycombinator.com/item?id=28402527 :
/? electrodynamic engineering in the time domain, not in the 3-space EM energy density domain https://www.google.com/search?q=electrodynamic+engineering+i...
"Electromagnetic forces in the time domain" (2022) https://opg.optica.org/oe/fulltext.cfm?uri=oe-30-18-32215&id... :
> [...] On looking through the literature, we notice that several previous studies undertook the analysis of the optical force in the time domain, but at a certain point always shifted their focus to the time average force [67–69] or, alternatively, use numerical approaches to find the force in the time domain [44,70–75]. To the best of our knowledge, only a few publications conducted analytical studies of the optical force evolution. Very recent paper employs the signal theory to derive the imaginary part of the Maxwell stress tensor, which is responsible for the oscillating optical force and torque [76]. The optical force is studied under two-wave excitation acting on a half-space [40] and on cylinders [77], and a systematic analytical study of the time evolution of the optical force has not yet been reported.
Undergrads will not appreciate the latter, as much as they will the first. Especially, because in the third semester, where relativity is first introduced, most physics undergrads don't know any group theory.
Both the paper and the video then finish off by ontologically tying this maximum speed, notation 'c', to the interesting physical E&M phenomenon we happen to call 'light'. My question - in this framework where we are deconstructing everything (which I applaud), then why make that leap to associate this 'c' with the phenomenon known as light?
Inference, sure why not until any future evidence suggests otherwise, but why do so in the intentionally limited scope of the paper & video's derivations?
That's an excellent question! And there are two answers.
1. If you apply the principle of relativity to Maxwell's equations, that's the result you get. Maxwell's equations predict a wave phenomenon that propagates at a constant speed in a vacuum. If the laws of physics, including the laws of electromagnetism, are the same for all inertial observers, then the speed at which those waves propagate have to be the Lorentz-invariant speed.
2. Experiments bear out this prediction! :-)
Note that it did not have to be this way, but if it were not, then EM experiments would give different results for observers in different inertial frames, which would falsify the relativity assumption. The different results for EM experiments would allow you to identify a privileged inertial frame. For a long time, physicists thought this would turn out to be the case, that there was a medium, a "luminiferous aether", through which EM waves propagated, and this medium would define a privileged frame (at least for electromagnetism). It just turns out by experiment that this is wrong, at least in our universe.
Yes, of course. First principles only leads you to the conclusion that there is a reference speed. It tells you nothing about its actual value. In fact, first principles cannot rule out the possibility that the reference speed is infinite. They can only lead you to suspect that it might not be.
Therefore, once you show that the speed of light in a vacuum is Lorentz-invariant, then it has to be the unique Lorentz-invariant speed (i.e., the universal speed limit).
I mean, historically yes. But you can define the fields as purely postulates if you like. The theory actually requires very little input to derive.
I’ve always had a problem with the speed of light being the ultimate speed limit in the universe. I would think “who granted the photon this authority?”. It wasn’t until I watched a lot of pbs space time YouTube’s that I learned what people really meant was the speed of causality is the ultimate speed limit. That makes more sense to me.
> Science educators love to make videos about relativity, but surprisingly few of them seem to understand what this stuff really means.
The two examples the video gives of science educators that don't "know what this stuff means" are Don Lincoln and Matt O'Dowd. Both have PhDs in physics. Lincoln was on the team that discovered the Higgs boson. O'Dowd is a professor at CUNY. They know what they're talking about.
Then the video gives as an example of a bad explanation the PBS Space Time on the speed of light. But that video makes pretty much the same argument. That the speed of light/causality falls out logically even if you don't postulate it. The author clearly didn't watch that video carefully https://www.youtube.com/watch?v=msVuCEs8Ydo
I would have subscribed to this channel, but this tone is a bad look.
Clickbait as in "Only this channel author Knows The Truth".
There's a YouTube channel called dialect that has a few videos on this subject
> The 19th century Lorentzian ether theory was ruled out by the famous Michelson-Morley experiment, which ruled out the possibility that the Earth moves through the ether. What we propose here is that the Earth (and everything else) is made of ether. No experiment so far ruled out that possibility, so such a neo-Lorentzian ether theory is a viable possibility
The interpretation of their result is totally wrong although the result is correct, because excluding Newtons' relativity (the Galilean group) is why electromagnetism or c is necessary.