[HN Gopher] Getting the Lorentz transformations without requirin...
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Getting the Lorentz transformations without requiring an invariant
speed
Author : lisper
Score : 61 points
Date : 2023-11-11 16:47 UTC (6 hours ago)
(HTM) web link (arxiv.org)
(TXT) w3m dump (arxiv.org)
| fuzzieozzie wrote:
| In 1987 my physics professor presented this proof to our class --
| I always wondered why this elegant derivation of the Lorentz
| transform was not more widely used in undergrad physics
| education.
| abdullahkhalids wrote:
| As a first introduction, its more instructive to develop
| physical theories from physical postulates (like speed of light
| is constant), rather than constraining possible theories using
| more abstract mathematical principles.
|
| 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.
| LudwigNagasena wrote:
| The contradiction between Maxwell's electrodynamics and
| Newtonian mechanics was the raison d'etre of the relativity
| foundations laid out by Lorentz, Poincare and Einstein. And
| since those subjects are usually studied before the theory of
| relativity, I think it makes sense to piggyback on them and use
| the historical reasoning to build the intuition behind
| relativity.
| alok-g wrote:
| A short personal story: While trying to understand special
| relativity from books, and failing to, I tried doing maths myself
| based on whatever I had understood till then. I ended up doing
| what this paper has mentioned. I was surprised. For the next
| twelve years, I was asking various physicists I knew about what
| was I doing wrong, however none could give me the time needed. I
| then saw basically the same proof appear on Wikipedia, so finally
| had the answer.
|
| 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.
| alexmolas wrote:
| It's curious that this simple proof -which don't require
| invariant speed, aka Maxwell equations- was discovered after
| Einstein's proposal which depends on invariant speed assumption.
| I wonder how the history of physics would have been if someone
| proposed this before Einstein. The maths needed for this
| derivation are quite simple, so I guess Newton or some
| mathematician before Einstein could have proposed special
| relativity.
| westurner wrote:
| (nonlinear) retrocausality:
| https://news.ycombinator.com/item?id=38047149
|
| 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._
| Hydraulix989 wrote:
| Why doesn't the requirement of a privileged reference frame imply
| the requirement of an invariant speed?
| lisper wrote:
| You have it backwards. The assumption is that there is _no_
| privileged reference frame. And that assumption (along with a
| few others) _does_ imply an invariant speed. That 's the whole
| point here.
| wangii wrote:
| Let's assume there exists one or more privileged ref system(s),
| and deal the two cases one by one. If there was one privileged
| reference system, and in our deduction process it's not
| involved,then the original deduction still holds. therefore the
| special ref system is an redudunt assumption. If there were
| many such ref systems, we would have to carefully get them
| involved in any possible deduction process, which is
| impossible. QED
| lisper wrote:
| A nice video that explains the paper:
| https://www.youtube.com/watch?v=UHKZMGdj7cI
| ricksunny wrote:
| Both the paper and the video (both of which i've skimmed but
| not gone through the math substantially) make a case for a
| maximum speed, notated as 'c'. Let me assume they make a very
| good mathematic case, and there is a maximum speed that
| satisfies the fundamental assumptions provided.
|
| 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?
| lisper wrote:
| > why make that leap to associate this 'c' with the
| phenomenon known as light?
|
| 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.
| ricksunny wrote:
| That's all fine, but from the moment Maxwell's eq's are
| invoked, we enter an empirical framing, not a first
| principles one. (as Maxwell's results follow from
| observations of the behavior of charged phenomena, current,
| magnets, & electromagnets). The paper & video seek to
| postulate the existence of a universal speed limit from
| first principles. They do so, and without changing the
| framing from a first principles one to an empirical one,
| associate their notational 'c' with the observed (i.e.
| empirical) speed of light. I find this element of their
| rhetorical strategy at odds with their intentions.
| lisper wrote:
| > That's all fine, but from the moment Maxwell's eq's are
| invoked, we enter an empirical framing, not a first
| principles one.
|
| 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.
| ricksunny wrote:
| Yes, this one also strikes me as an accurate framing of
| the situation's epistemology, Ty for clarifying :)
| akalin wrote:
| I think what's left unsaid in the paper is that if a
| Lorentz-invariant speed exists, then it is unique. (This
| is easy enough to show.)
|
| 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).
| ricksunny wrote:
| Yes, this strikes me as an accurate framing of the
| situation's epistemology.:)
| chasd00 wrote:
| Saying c as "the speed of light" is an easier thing to
| comprehend than c as "the speed of causality" which is really
| what it is. It so happens that photons travel at the speed of
| causality so we can get away with saying the speed of light.
|
| 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.
| light_hue_1 wrote:
| I don't understand why the video needs be so negative. Why dump
| on other videos? And unfairly so!
|
| > 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.
| whatshisface wrote:
| "Ignatowsky showed that the only admissible transforma- tions
| consistent with the principle of inertia, the isotropy of space,
| the absence of preferred inertial frames, and a group structure
| (i.e., closure under composition), are the Lorentz
| transformations, in which c can be any veloc- ity scale, or the
| Galilei transformations"
|
| 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.
| jkafjanvnfaf wrote:
| The mathematics are sound, but the reasoning around is unclear to
| me. The derivation shows that Lorentz transformations and
| Galilean transformations are the only ones that allow for the
| equivalence of all inertial frames, which is a nice result. But
| it clearly _does_ require the additional assumption of an
| invariant speed to conclude that Lorentz transformations are
| anything more than a mathematical curiosity.
|
| 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.
| pdonis wrote:
| _> what have we really gained?_
|
| 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.
| jkafjanvnfaf wrote:
| That is true. Still, we are kind of trading one unintuitive
| postulate (an invariant speed) for a different one: Why would
| we ever think that the time interval between two events can
| depend on the reference frame?
|
| 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.
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