[HN Gopher] Getting the Lorentz transformations without requirin... ___________________________________________________________________ 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. ___________________________________________________________________ (page generated 2023-11-11 23:00 UTC)