# Talk:Special relativity

Special relativity was one of the Natural sciences good articles, but it has been removed from the list. There are suggestions below for improving the article to meet the good article criteria. Once these issues have been addressed, the article can be renominated. Editors may also seek a reassessment of the decision if they believe there was a mistake.
Article milestones
DateProcessResult
December 16, 2004Featured article candidateNot promoted
January 1, 2005Good article nomineeListed
February 12, 2006Featured article candidateNot promoted
October 30, 2006Good article reassessmentKept
August 26, 2009Good article reassessmentDelisted
Current status: Delisted good article
Wikipedia Version 1.0 Editorial Team / v0.5 / Supplemental (Rated C-class)
C  Quality: C-Class
???  Importance: not yet rated

## Thought experiments

In his popular and semi-popular writings, Einstein was well-known for illustrating basic concepts of relativity with the aid of thought experiments.

Am I simply missing it, or does there not exist an article in Wikipedia devoted to "Special relativity thought experiments"?

Would creation of such an article be desirable? Or would such an article violate wp:NOTTEXTBOOK?

Prokaryotic Caspase Homolog (talk) 03:24, 5 April 2018 (UTC)

I think that it would be a good article to have, if it is framed as an article about the history of relativity and limited to sourced thought experiments devised by Einstein himself. JRSpriggs (talk) 04:24, 5 April 2018 (UTC)
Definitely it needs to be a sourced article. If we wish to make it an historical article strictly about Einstein's unique approach to conceptualizing complex scientific ideas, then the article name could be "Einstein's thought experiments", that would describe the ones that he devised not just for special relativity, but also ones that he devised for general relativity and for quantum mechanics. Prokaryotic Caspase Homolog (talk) 10:11, 5 April 2018 (UTC)
I have created Einstein's thought experiments. I hope you find it decent. Prokaryotic Caspase Homolog (talk) 17:14, 28 April 2018 (UTC)
Yes, thank you. JRSpriggs (talk) 01:39, 29 April 2018 (UTC)

## Measurement versus visual appearance

Triggered by recent edits ... While I have no (perceived) problem in the original, probably terse version of identifying the "measured shape" of an object as a collection of 3d-space-coordinates, obtained from a section of spacetime coordinates, and appropriately associated to corresponding object-inherent coordinates, revealing the length contraction in the direction of the velocity, I am unsure about the term "snapshot" in the current version. I think "snapshot" is "taking a picture", and induces inherently propagation of light, which is carefully excluded in "observing", i.e. taking spacetime coordinates.

I must admit that the notion of "visual appearance" is a bit bewildering to me in both versions. I think this is now about taking a "snapshot", which involves a central projection, including dependencies on distance between the object and the observer, the direction of the velocity, and what not.

I think that the presented material is excellent, but the presentation is not fully rigorous and sufficiently explicative, and I am unsure, whether the edits constitute an improvement. 12:29, 10 September 2018 (UTC)

Can you suggest a better wording? Now that you bring it up, I can see the problem that you might have with the word "snapshot" as a means of describing the "measured shape" of an object. Prokaryotic Caspase Homolog (talk) 02:14, 12 September 2018 (UTC)
I am sorry, but my reservations, and only sometimes direct suggestions for marginal improvements, are all I can provide. I am an intuition-less non-expert in STR, heavily suffering from the total collapse of the concept of rigid body in STR already in 1 dimension (rockets with string). Additionally, I disagree with certain adhered to concepts (necessity of talking about moving observers vs. light sources) claiming to be based on Einstein, and I do not feel adequately versed in the use of this non-native tongue, to express such delicate matters. Purgy (talk) 08:02, 12 September 2018 (UTC)
Hmmm... You bring up a variety of issues unrelated to your original concern. Born rigidity and Bell's spaceship paradox are not covered in the article as presently written, but one could argue that they need to be covered. One could also argue that coverage of those topics would represent unnecessary digressions, given the article's other deficiencies. The collective authorship paradigm that Wikipedia follows, while very good for developing articles in history, biography, etc. has not proven itself very well adapted to the development of technical articles like special relativity. In common with most other technical articles, the current article is a hodgepodge of parts with widely differing levels of difficulty. It needs a thorough overhaul by a person with a clear vision of how the article should be structured and what the target audience is supposed to be.
However, giving this article a thorough overhaul is beyond my competency. I can only focus on the little bits and pieces that I myself have added. The best that I can promise is that I'll continue to think about the points that you raised. Maybe somebody else will find a better wording. Prokaryotic Caspase Homolog (talk) 23:39, 12 September 2018 (UTC)
I really had no intention to bring up these topics as issues of this article (in need of covering), but only as prominent in causing me troubles in developing a good intuition about STR. I feel quite similar to the description of your last paragraph, just additionally handicapped by the necessity to use a non-native language.
Triggered by your remarks, I want to mention a thorough attempt -not too long ago- to deal with this article in the perspective you mentioned, which seems to have failed the target, but certainly has brought about significant improvements. BTW, I strongly object to the collective authorship paradigm being any good for questionable articles in history or biography. All the best, Purgy (talk) 07:13, 13 September 2018 (UTC)
This article? The last really major revamping that I recollect was the decision in mid-2015 to delete Introduction to special relativity as being an even worse hodgepodge than the main article. Prokaryotic Caspase Homolog (talk) 07:52, 14 September 2018 (UTC)
I am deeply concerned by me sloppily mixing up this article with Spacetime, which encountered heavy efforts of targeted improvement in 2017. My attention here was by far too focused on the "snapshooting" of "spacetime vectors", i.e., just on the local changes, being related to STR. Pardon! Purgy (talk) 09:36, 14 September 2018 (UTC)
The strength (and weakness!) of Spacetime as currently written was the principal editor's determination to adhere, as much as possible, to a purely geometric approach to presenting the material. There are neither trains nor lightning bolts in Spacetime. For the most part, the geometric demonstrations are logically presented, but by their very nature, the demonstrations are somewhat divorced from intuitive understanding. Most people, including myself, are rather more comfortable with a kinematic approach, i.e. with railway cars and spaceships. The problem is, how to add this introductory material? Most "Introduction to" articles get only a few percent of the readership of their associated main articles. Instead of trying to resurrect Introduction to special relativity (which needs to stay dead), I wonder if such material could be added as an extended introductory section to the current article? Against this idea would be the following objections:
1) Such material could very easily violate wp:NOTTEXTBOOK.
2) Such an introductory section could easily double the size of this article.
3) A featured wikibook exists on Special Relativity which has the merits of being principally authored by a single knowledgeable editor. It has a consistent presentation and relatively clear focus, and as a wikibook, it was allowed to take on textbook aspects. Despite this, I'm not very happy with it. Could somebody like myself do any better? Absolutely not.
Thoughts? Prokaryotic Caspase Homolog (talk) 15:17, 14 September 2018 (UTC)
... thinking ... Purgy (talk) 08:35, 15 September 2018 (UTC)

───────────────────────── I take back part of what I said about the wikibook. I'm very unhappy with it. If you're going to write a textbook on special relativity, you need problems with solutions, or at least lots of example scenarios. Prokaryotic Caspase Homolog (talk) 11:10, 15 September 2018 (UTC)

To start with the result of my thinking: I have none. I agree on your verdict the wikibook not making me happy, I do not cling to the WP:NOTTEXTBOOK beyond not allowing for collections(!) of examples (paradigmata are a core necessity in WP! imho), yes, the danger of doubling the length is dangling, and finally, given my engagement and eruditeness on this matter, I am convinced I could not do half as good as you.
As an aside, I am very skeptical about the usual intuition on kinetics. All this rubbish about "moving observers" stems imho from "intuitively" "observing" TWO reference frames, thereby silently introducing a third frame, leaving the uninitiated confused.
Sorry, I think the best I can do, is commenting from the off sometimes. Please, do never assume any malevolence from my side. Purgy (talk) 07:36, 18 September 2018 (UTC)

## Rearranging the sections, and now I'm stuck

I've been rearranging the sections of this article so as to put them into a more rational order, and now I'm stuck.

There are a variety of approaches to teaching relativity:

• The dominant approach found in most college textbooks is begin with the "two postulates" (almost always starting with a stronger, less intuitive form of the second postulate than that adopted by Einstein in his 1905 paper) and to proceed through relativity of simultaneity, time dilation, length contraction etc. to the Lorentz transformations. While traditional, this principle-based approach has many issues. As Miller has noted, "Teaching STR that way is especially problematic because, unlike the case of classical thermodynamics which is also taught as a principle theory, the two postulates or principles in the case of STR are strongly counterintuitive when taken together."
• Several textbooks begin with Minkowski spacetime as the central focus, often approaching Minkowski spacetime through constructive arguments. This, for instance, is the approach adopted by Taylor & Wheeler's Spacetime Physics. The article Spacetime attempts consistently to follow this approach, how successfully, I'm not sure.
• Some authors advocate beginning with the Lorentz transformations as the first principle. I know of no introductory college textbook that teaches special relativity this way.

This article starts off as if it were following a two-postulates presentation, and then suddenly switches over to presenting the Lorentz transformations as the first principle, from which everything else derives.

How should I go from here? Any suggestions? Prokaryotic Caspase Homolog (talk) 08:53, 28 October 2018 (UTC)

I think that I've managed a kludgy fix by adding some transitional commentary about different approaches to presenting special relativity. Prokaryotic Caspase Homolog (talk) 10:00, 28 October 2018 (UTC)
I personally am not happy with basing special relativity on the single postulate of universal Lorentz covariance, but that's the way the article appears to have been written. Prokaryotic Caspase Homolog (talk) 15:36, 28 October 2018 (UTC)

## I need help here

Does this section really provide a comprehensible explanation of why FTL is impossible? All it does is state that "one can show" that causal paradoxes can be constructed. Prokaryotic Caspase Homolog (talk) 03:24, 29 October 2018 (UTC)

It makes sense to me. It explains how FTL travel would violate causality. There’s no proof of causality but it’s intuitively very appealing as without causality paradoxes arise, so is widely accepted as being true. And if you accept causality then FTL travel must be impossible.--JohnBlackburnewordsdeeds 03:47, 29 October 2018 (UTC)
I restored the section, but I still don't like it. At the very least, it needs a second Minkowski diagram showing how, through the exchange of FTL signals, one can generate a causality-violating scenario. As currently written, it demands an act of faith on the part of the reader. I suppose I could draw the necessary diagram and modify the text to work with the new figure. I don't see a ready-made figure on Commons that will do. Prokaryotic Caspase Homolog (talk) 04:57, 29 October 2018 (UTC)
It’s alright to me. It’s the sort of thing that’s hard to draw as it quickly gets cluttered, but if you’ve looked at enough such diagrams you can visualise it in your head. Or follow the logic of the text which does not really depend on the diagram except to initially establish the relationship between A, B and C.
I’m removing the text too now it’s back in the article; it’s still in the page history if there’s any need to refer to it.--JohnBlackburnewordsdeeds 09:44, 29 October 2018 (UTC)
I suggest to get rid of one spatial dimension in the relevant pic. An (x/t + x'/t')-diagram would do the job (there is already a comment about this in the article text) better than this fancy x/y/t-cone. Maybe it is helpful to hint to the trivial fact that any line in the upper half of the first quadrant through the origin and an arbitrary event represents a t'-axis, with an appropriate x'-axis (symmetrically to x = t), whereas any lines through an event in the lower half can only be an x'-axis, because reverse interpretations have no real solutions within the Lorentz transformation (for the less mathy inclined: there is no meaningful place to put the respective other axis ).
As an aside: maybe Occham's razor can be considered applicable not only in reducing the number of dimensions in diagrams but also for reducing the number of premises to derive STR from. ;) Purgy (talk) 15:44, 29 October 2018 (UTC)
I was already considering replacing the fancy x-y-t light cone diagram with an x-t diagram. The extra dimensions don't add anything to the presentation, and the current figure occupies a disproportionate amount of real estate. John makes a good point about how cluttered a spacetime diagram illustrating causality violation can be. Use of color can be helpful, but I have to be careful not to rely too much on color. Also, since we have already established that the article begins with universal Lorentz covariance as the central principle for its development of special relativity, it would be better to show this using the LTs rather than with a spacetime diagram. However, I have to mind wp:NOR. Anything I add has to be sourced, and the references that I have in mind to use to source my additions all use spacetime diagrams as the simplest way to illustrate their discussion. Prokaryotic Caspase Homolog (talk) 16:12, 29 October 2018 (UTC)

───────────────────────── I am heavily concerned that I cause so much sighs, really. I was aware that my Spacetime effort would have some superior formulation, but I was convinced that the "1800s" are no good either.

... and now I gave some more chance to sigh deeply about this here article, but again I am convinced that I implemented some hints to a substantial improvement of the status quo ante. BTW, I left the latter part of the paragraph untouched. Simply throw it all away ... I do not mind too much. Purgy (talk) 08:49, 30 October 2018 (UTC)

Let's talk about your proposed changes. I'm also working on changes at the same time. Prokaryotic Caspase Homolog (talk) 12:45, 30 October 2018 (UTC)

### Causality and prohibition of motion faster than light

Figure 10-4. Light cone

In Fig. 10‑4 the interval ${\displaystyle {\text{AB}}}$ is 'time-like'; i.e., the line connecting ${\displaystyle {\text{A}}=(x=0,ct=0)}$[note 1] and ${\displaystyle {\text{B}}=(x=x_{\text{B}},ct=t_{\text{B}})}$[note 2] can be taken as a ${\displaystyle ct'}$-axis, that establishes with the line symmetric to ${\displaystyle ct=x}$ an ${\displaystyle x'/ct'}$-frame,[note 3] in which events ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ occur in the primed frame at the same spatial coordinate ${\displaystyle x'=0}$, separated by a time interval of length ${\displaystyle t'_{\text{B}}.}$[note 4] The event ${\displaystyle {\text{A}}}$ precedes ${\displaystyle {\text{B}}}$ in all frames possible under Lorentz transformation (${\displaystyle ct'}$-axis within the light cone).[note 5] It is feasible to observe from the ${\displaystyle x/ct}$-frame a matter-/information-transport from ${\displaystyle {\text{A}}}$ to ${\displaystyle {\text{B}}}$[note 6] at some speed smaller than lightspeed,[note 7] so the event ${\displaystyle {\text{A}}}$ can cause the event ${\displaystyle {\text{B}}}$, if this speed can be achieved by the transport.[note 7]

The interval ${\displaystyle {\text{AC}}}$ is 'space-like'. Since the Lorentz transformation prohibits a ${\displaystyle ct'}$-axis within the shaded cone, the line connecting ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{C}}}$ cannot be taken for this, but only as an ${\displaystyle x'}$-axis.[note 3] The suitable ${\displaystyle x'/ct'}$-frame is again symmetric to the ${\displaystyle ct=x}$-line,[note 3] and in this frame the events ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{C}}}$ occur at the same temporal coordinate ${\displaystyle ct'=0.}$ So for all events ${\displaystyle {\text{E}}}$ within the shaded cone there exists a primed frame in which ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{E}}}$ are simultaneous, separated by some spatial distance.[note 4] Besides this frame with simultaneity, there are frames in which ${\displaystyle {\text{A}}}$ precedes ${\displaystyle {\text{C}}\;(t'_{\text{C}}>0),}$ but also frames in which ${\displaystyle {\text{C}}}$ precedes ${\displaystyle {\text{A}}\;(t'_{\text{C}}<0).}$ Naively, some speed above lightspeed (determined by the slope of the line connecting ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{C}}}$) would allow for the latter frames a transport between the spatial coordinates of ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{C}}}$ that triggers an event there, prior to the transport's depart, as observed in the ${\displaystyle x/ct}$-frame, thereby violating causality.[note 4] However, the Lorentz transformation does not yield a solution for such frames.[note 8]

Notes

1. ^ Unnecessary to explain that A is at the center of the unprimed coordinate system
2. ^ Unnecessary to explain that B is at the coordinates of B.
3. ^ a b c Not illustrated. You force the reader to have to draw the scenario, either in his/her head or on paper.
4. ^ a b c Verbose restatement of the original.
5. ^ Unnecessary additions to "A precedes B in all frames"
6. ^ Jargonese rewording of "It is hypothetically possible for matter (or information) to travel from A to B"
7. ^ a b Are you implying here that FTL is possible?
8. ^ Why not? What happens to the LT that prohibits this?
I care about my wordings, but I do not add much to a stranded investment, I just stand prepared to answer any follow up. I plan for more global remarks in the reply to the second set of notes.
1. It's about making a coordinate explicit, even when it is 0, not about explaining the origin.
2. It's (again) about making the coordinate ${\displaystyle x_{B}}$ explicit, soon afterwards there will be an ${\displaystyle x'_{B},}$ too. It's a flaw of math to rely on microscopic differences in notation.
3. I hoped for getting it illustrated.
4. (a) I'd say it's about different frames. (b) No, E is a totally new event. (c) ??? It's the first occurrence of "causality".
5. They're intended as an introduction to the non-existence of ${\displaystyle ct'}$-axes in the shaded cone.
6. Jargon in math is at the core of unique meaning, avoiding the "lyrics" of popular introduction (lyrics = much emotion, no precision, just a good feeling for high volume intuition pumping)
7. By no means! Should I have mentioned that worldlines in the first quadrant with slopes greater 1 represent speeds below lightspeed, and that ${\displaystyle \infty }$ is rest = observer?
8. No real solutions exist, because ${\displaystyle \gamma }$ isn't a real factor anymore.
I'll take some time for the announced 2. part, since I want to avoid a TL;DR, but want to say sooo much. :) Purgy (talk) 17:36, 31 October 2018 (UTC)

### I'm concurrently working on additions to the text based on this diagram

Figure 10-5. Causality violation

The narrative will go more or less like this: C and D are on a high speed train. A and B are on the ground. D passes B just as the lottery winning numbers are announced. B tells D the winning numbers. D uses his ansible to instantaneously inform his partner, C, of the lottery results. C, who is passing A at that moment, informs A of the winning numbers. A user her ansible to instantly flash the numbers to B, who writes the numbers on his lottery ticket, submitting it before the drawing. Prokaryotic Caspase Homolog (talk) 13:26, 30 October 2018 (UTC)

For the sake of simplification, you could just omit B and C, and have the whole thing go between A and D. Here I made a rather crude simplified version of the diagram, adding a marker for an example location of the lottery draw. --uKER (talk) 14:02, 30 October 2018 (UTC)
Looking at your diagram, the lottery event needs to be placed on D's world line, otherwise there would be a communications delay.
There is another issue, however, and that is wp:NOR. I based my diagrams on a published reference. Too much deviation from the diagrams and narrative that I used as my source would constitute original research. Prokaryotic Caspase Homolog (talk) 14:45, 30 October 2018 (UTC)
The location of the lottery is fine as long as 1. it's in the past light cone of the moment the moving subject sends the info backwards, and 2. it's in the future light cone of the moment when the stationary subject receives it. About keeping your diagrams, yeah, you probably have a point. --uKER (talk) 18:56, 30 October 2018 (UTC)
There are lots of spacetime diagrams to choose from to illustrate the paradox. I've seen this one in multiple contexts. I first came across it several years ago in "The Einstein Paradox and other Science Mysteries Solved by Sherlock Holmes" by Colin Bruce, in "The Case of the Faster Businessman". Then I encountered the same diagram in in David Morin's book, and now, doing a web search, I see it in the lecture notes for a course taught at Virginia Tech. Each source accompanies what is essentially the same diagram with a different narrative. On Quora, I remember a soldier being warned just in time not to step on an IED after the driver of a passing troop carrier witnesses the soldier getting his foot blown off. Maybe a Sherlock Holmes mystery would be more appealing and less gory? Prokaryotic Caspase Homolog (talk) 22:49, 30 October 2018 (UTC)

### Purgy's comment

• I trimmed my thoughts above and removed the paragraphs at the end to which I deny any comment. I hope you did allow for this.
• I cannot comment much on my suggestion, beyond what I stated already: I am convinced that it is by far more consistent than the status quo, and I concede that it maybe hard to read for the details,[note 1] which I consider to be necessary for a thorough understanding.[note 2] To my taste there is way too much lyrics around about STR.[note 3]
• I would gladly defend my proposition, but do not know against what and I also would enjoy seeing it improved, I would clarify all I can[note 4] and I can even accept it being ignored, BUT:
• May I plead for reconsidering the inclusion of -say- fictional devices in an explanation, intended to be serious? I object with all my argumentative strength against including this "ansible"-story. This is explosion (ex falso quodlibet), but no serious argumentation. I cannot accept a claim being refuted because some contradictory device had lead to a contradiction. This is abuse of space time diagrams, rape of LT, cheap baiting with fraudulent gambling, ...

Primarily, I fully accept and support your prerogative on this article. Purgy (talk) 16:01, 30 October 2018 (UTC)

Thought experiments invoke particulars that are irrelevant to the generality of their conclusions.
You object to the use of these fictional devices. However, it is precisely the invocation of these particulars that give thought experiments their experiment-like appearance. A thought experiment can always be reconstructed as a straightforward argument, without the irrelevant particulars. John D. Norton, a well-known philosopher of science, has noted that "a good thought experiment is a good argument; a bad thought experiment is a bad argument."[1]
When effectively used, the irrelevant particulars that convert a straightforward argument into a thought experiment can act as "intuition pumps" that stimulate readers' ability to apply their intuitions to their understanding of a scenario.[2]
I could use the spacetime diagrams to support a straightforward argument demonstrating that FTL communications implies violation of causality, but the ensuing description would be verbose and relatively nonintuitive.Prokaryotic Caspase Homolog (talk)

References

1. ^ Norton, John (1991). "Thought Experiments in Einstein's Work". In Horowitz, Tamara; Massey, Gerald J. Thought Experiments in Science and Philosophy (PDF). Rowman & Littlefield. pp. 129–148. ISBN 9780847677061. Archived from the original (PDF) on June 1, 2012.
2. ^ Brendel, Elke (2004). "Intuition Pumps and the Proper Use of Thought Experiments" (PDF). Dialectica. 58 (1): 89–108. Archived from the original (PDF) on 28 Apr 2018. Retrieved 28 April 2018.

Notes

1. ^ VERY hard to read!
2. ^ In general, encyclopedia entries are expected to provide a sketch of a proof, not attempt to provide the entire proof in detail, which is usually impossible within the space limitations of an article.
3. ^ What do you mean by "lyrics"?
4. ^ Excessive detail does not clarify, but obscures.
Well, protest as announced: I neither buy the necessity of "irrelevant particulars", not even their usefulness, nor do I believe that they "always" can be removed later on, leaving something that is a real argument. I resort to the opinion this thought experiment is a bad argument (Norton is fine), and I do not want to see "intuition pumps" (Brendel is rubbish) included in WP, but rather -especially in scientific articles- valid arguments, doubly checked for their validity. However, since you seem to be petrified to include this subspace communications, ... I do not bother for your arguments how to render that "ansibles" as irrelevant for the story or how to reconstruct it as a straightforward argument, when it is just a silly story (there are far more respectable time travels in the pertinent literature), as well as I do not ask any longer for any specific leaks or obscurities in my formulations (that you wanted to discuss!?). May you find a fake, that looks like an experiment that helps others. Purgy (talk) 19:03, 30 October 2018 (UTC)
This obviously cannot be a matter of my strong opinion against your strong opinion, but will require consensus with others' inputs.
By the way, I am looking closely at your revised proposal. I just can't pay a lot of attention to it right at the moment, since I work for a living. I'll return to examining it tonight.
The results of our previous discussions have always been improvements in the articles in question. I value your comments, even when I disagree wholeheartedly! Prokaryotic Caspase Homolog (talk) 21:07, 30 October 2018 (UTC)

───────────────────────── If you are against colorful descriptions with multi-million dollar lotteries, Sherlock Holmes mysteries, soldiers on tour in Iraq and the like, how about this:

Consider the spacetime diagrams in Fig. 10‑5. A and B stand beside railroad tracks. A high speed train passes by with C riding in the last car of the train and D riding in the leading car. The world lines of A and B are vertical reflecting the stationary position of these observers on the ground, while the world lines of C and D are tilted forwards, reflecting the rapid motion of these observers in the train.
1. Fig. 10‑5a. B flashes a message to D as the leading car passes by.
2. D passes the message back to C using an instantaneous communication device. The signal follows along the ${\displaystyle x'}$ axis, which is a line of simultaneity between C and D.
3. Fig. 10‑5b. C flashes the message to A who is standing by the railroad tracks.
4. A passes the message forwards to B via the instantaneous communication device. The signal follows along the ${\displaystyle x}$ axis, which is a line of simultaneity between A and B.

Such a description is not very far off from a straightforward argument with none of the "irrelevant particulars" to which you take offense. Prokaryotic Caspase Homolog (talk) 00:49, 31 October 2018 (UTC)

Edit conflict with the creation of the newest section, will reply there separately.
It's easy to refer to single notes:
1. Yes, it does not belong to my strengths to write directly to the heart, I'm more the nitpicking type, but I am convinced that deep understanding needs deep arguments.
2. I'm not requesting formal proofs (Four color theorem), but I oppose, as strong as is possible to me, to pseudo-explanations for "some thing", like

The assumption of the "existence of ansibles" (a logical constant 'False' in the theory)
explains "some thing".

3. See previous comment of mine.
4. See #1.: "deep"
As regards my general impression about explanations of counter-intuitive consequences in STR, I perceive a desire to flesh these out with most spectacular details (twins, pole in a barn, rockets and ropes, ansibles, Holmes, Iraq war, riches, ... ), even when the coexistence of arbitrarily many observers (~frames), all of them at rest in their respective frames, is not fully appreciated by a good deal of the audience. Oblique coordinate systems, unacquainted by themselves, and, additionally, describing a spatially just one-dimensional world and its temporal sections, should be treated with greater care (=detail!), imho. I appreciate the remark that an ansible works along the x-axis, but I miss the emphasis that it is about simultaneity within the -say- unprimed frame, only, and that there is no connection possible to the primed frame supported by LT.
I understand that me being satisfied by the non-existence of a world line from A to C does not pertain to all readers, but I am really convinced that this is the core of the story, and all involvements of additional frames and actors, stories and whatnot impossibilities, only blur the core fact: NO worldline from A to C! As is usual in logic, any assumption to the contrary (e.g.: existence and use of ansibles) allows derivation of all claims, true ones and false ones (I referred to this in my proposal by using the word "naively"). I deeply regret that my idiomatic abilities cannot provide a text with the necessary ease of readability.
To my understanding an encyclopedia might (should!) contain information about the most wide spread and most surprising, most funny, ..., stories of counter-intuitivities, but should not(!) involve them in attempts of explanations. Here I am with my personal POV that I only alter for arguments, but not for just "consensus in WP". However, as said already, this won't cause any effects in WP-texts. No edit-warring for me, just TP-skirmishes.
You may pump as you might, I won't develop any intuition about ansibles. I do however enjoy any of your appreciations. Purgy (talk) 09:24, 1 November 2018 (UTC)

## LT vs. two postulates vs spacetime approaches to understanding SR

Mathematically, it is extremely simple to establish the non-existence of a world line from A to C, end of story, no need to go any further. But for most, the pure mathematical demonstration doesn't satisfy the need for an intuitive understanding of why that should be so. That is why I am so unhappy about the decision of the original authors of this article to develop special relativity starting with the single postulate of universal Lorentz covariance.

The appeal of the two-postulates development of special relativity is how, starting with these intuitive principles, one can arrive at all sorts of fantastic results, including the Lorentz transformation. But many people just don't get the deductive style of the two-postulates approach. They get lost at the very start trying to understand relativity of simultaneity, and if one gets stuck there, there is no going forwards.

Then there is the spacetime approach, which is frequently taught from a constructive standpoint through analogies with Euclidean geometry. If you buy the analogies and accept the results of experiment, that is the best approach for many people.

What I am trying to get at is that your reservations seem, to me anyway, mostly that you are most comfortable with a pure mathematics approach to understanding special relativity, which is why you are so dead set against irrelevant particulars, intuition pumps etc. You are not a thought experiments sort of guy.

Prokaryotic Caspase Homolog (talk) 23:14, 1 November 2018 (UTC)

Using the LTs, Fig. 10-5 can be explained in just three lines. Let S and S' be two frames in in standard configuration, and let ${\displaystyle {\text{E}}}$ be the event corresponding to the crossing of the B and D world lines. Then ${\displaystyle \beta =ct_{E}/x_{E}=v/c.}$ The event coordinates ${\displaystyle (x_{E},ct_{E})}$ in frame S transform to ${\displaystyle (x_{E}/\gamma ,0)}$ in frame S'. An instantaneous signal from ${\displaystyle {\text{E}}}$ in frame S' intersects ${\displaystyle (0,0),}$ and an instantaneous signal from the origin intersects ${\displaystyle (x_{E},0)}$ in frame S, preceding event ${\displaystyle {\text{E}}}$ by ${\displaystyle t_{E}.}$ This is totally trivial math, but it does not leave me with any sense of satisfaction that I understand what is going on. It's just symbol manipulation. The spacetime diagram, however, is different. I get a visual handle on the transformation that I simply do not get working the symbols. I can see the effect of increasing the speed of the train, and I understand visually why even a speed infinitesimally greater than ${\displaystyle c}$ can result in causality violation. Prokaryotic Caspase Homolog (talk) 03:22, 2 November 2018 (UTC)

I frankly admit to be "most comfortable with a pure mathematics approach" (especially as long as the math is simple enough to my abilities), but I strongly refuse not being accessible by thought experiments. Any of the many indirect proofs I valuate much are such (Let blabla, then ..., therefore ¬blabla.), and I often see essential gain in accessibility by affixing a funny hat (irrelevant particular) to some entity (Four color theorem - not very funny, but famous). For reasons given already by elementary formal logic, I am, however, strongly averse to introducing evident antinomies, like ansibles, in any proof of any claim. (Assume "False", then "Anything". –is a tautology.) I would agree to disproving the existence of ansibles, relying on geometry or LT and causality, but I disallow for calling any disquisition on anything, which involves the use of an ansible, a proof of anything (repeating: ex falso quodlibet). I accept the path, leading from the assumption of a speed, infinitesimally (yuck) larger than lightspeed, to violation of causality, but I am in serious doubt, whether this path is easier to describe and(!) to follow, than the attempt of getting familiar with ct'-axes being restricted to the light cone, and x'-axes to the dark cone, converging with increasing speed to the useful limiting case of ct' and x' coinciding along the propagation of light in all frames (with common origin), the ubiquitous simultaneity. Personally, I perceive the introduction of two additional comoving frames as making things more complex (BTW, the Twin paradox in STR gains its life from the frame change of the twin.)
The following is less apodictic and more personal: Dealing with verbally formulated principles is a language game, hard to join in non-native languages, so my primary effort is to create formal tools, independent of natural language, applicable also in hard to understand, in counter intuitive, in surprising, ... situations. To me these tools are the LT. I confess getting lost along so called derivations –often enriched with irrelevant particulars, intuition pumps, and other distracting stuff, just there to hide the leaks or even flaws (like using ansibles!) within logic– but with the help of LT I am able to overcome my resentments and arrive at a stable understanding(?)/manipulation, reinforcible at any time by some calculation, even of matters like lost simultaneity. (As an aside, I am not sure, if an "intuitive" understanding of "relative simultaneity" is possible, at all.)
Imho, there is no "Euclidean" geometry in the spacetime diagrams: it's about the difference of squares and not their sum. The connecting line of two events in these diagrams is quite misleading wrt their spanned spacetime interval. (I recall to have had a hard time myself to get rid of the Euclidean intuition.) E.g.: Summing the two legs in the twin paradox is "shorter" than the direct connection. I would rather fight Euclidean associations within the x/ct-frame than further them.
As for your second part, I am perfectly aware that neither my scaring efforts nor your discouraging formulations are very invitational to read, but I rather accept frightening hardship than logical disaster. Maybe there is an intuitively viable path from the slopes of the axes representing the speed and its reciprocal, converging at light propagation, where I started with "Naively, some speed above lightspeed ..."? Anyhow, you decided to take your road.
Anticipating your objection regarding NOR, may I report that I was moved almost to tears by the sad comment by Krea. For heavens sake, what is done here, in this context, at this level, is NO research, so it cannot be original research. All this is just a "making explicit" of reproducible, trivial math, not to be published for higher academic merit, but to aid the interested reader in his struggle to understand physics beyond falling apples. Are the publishers of books about popular interest topics afraid of losing their clientele to WP? There is so much published rubbish, why shouldn't WP contain some coherent information, without calling it research. Purgy (talk) 16:30, 2 November 2018 (UTC)
Your opinion is valuable, even if we disagree a lot. Note now the narrative to the FTL spacetime diagram does not mention lotteries or Sherlock Holmes or soldiers getting their feet blown off by an IED. Your doing! And I agree that the section is better because of your pushing.
In regards to NOR, see my reply to Krea here. Prokaryotic Caspase Homolog (talk) 22:57, 2 November 2018 (UTC)
It's not the first time while being around in WP that I enjoy disagreement, but it's a rare moment still - thanks. Thanks also for correcting my link; I am sorry not to have checked it myself. I would not have bothered to notify Krea myself, but I understand that it could be considered appropriate, and misspelling the name was certainly inappropriate. Just for completeness sake, I assume that we also disagree about the level, above which even well-written paragraphs in scientific articles are to be removed for being not properly sourced. Finally, I announce some boldness of mine, and humble ask for it being kindly checked. Cheers, Purgy (talk) 09:01, 3 November 2018 (UTC)
Looks fine to me. You've achieved the greater precision that you wanted without loading it down with the excessive parenthetical digressions that, to my mind, made your previous attempt unreadable.
By the way, the other reason for pushing Krea's contribution to Talk was that it was not written in any sort of encyclopedic style, violating wp:NOTTEXTBOOK in a rather extreme fashion. I also disagreed with some sections that, while not technically incorrect, made somewhat misleading points. You can read his writing here and judge for yourself. Prokaryotic Caspase Homolog (talk) 11:52, 3 November 2018 (UTC)

## Proposed section revision

The proposed section revision below represents a distinct issue from Purgy's proposed rewording of the opening two paragraphs for greater precision at the cost of lesser clarity, which Purgy considers a good trade-off. Hence, this can be deployed separately from any decision regarding Purgy's proposed revisions.

Issues concerning rewording have been resolved. Prokaryotic Caspase Homolog (talk) 14:13, 3 November 2018 (UTC)

I do have a question about placement. The article as written uses the single postulate of universal Lorentz covariance as its basic starting principle.

• In terms of subject matter, it belongs in Other consequences.
• However, it uses Minkowski diagrams to perform the demonstration rather than Lorentz transforms. Therefore, in terms of presentation, it belongs in Spacetime.

Where should it go? Prokaryotic Caspase Homolog (talk) 08:27, 1 November 2018 (UTC)

• Spacetime: "No FTL", even when less important, is similarly elementary as contraction and dilation. The current "Other consequences", while certainly worth mentioning, are by no means elementary in the same way.
This is not say that I perceive the structure of this article as perfect. The mass-energy-equivalence is most certainly not derived from the LT, what is the difference between "derived" and "other" consequences, the LT takes good care about the whole spacetime-space, ... Purgy (talk) 14:38, 1 November 2018 (UTC)
OK. I'll keep it there. I could easily have recast the whole argument in terms of the LT, except that would have constituted original research. There is already too much original research in this article that I have been reluctant to throw away. Prokaryotic Caspase Homolog (talk) 15:03, 1 November 2018 (UTC)

Also, it appears that there is plenty of real estate if we want to reinstate the 3-dimensional light cone diagram that was originally Fig. 10‑4. Do we want to revert? Prokaryotic Caspase Homolog (talk) 08:31, 1 November 2018 (UTC)

• No revert: A flat spatial geometry in two dimensions offers no additional effects relevant to the question of FTL, when compared to a one-dimensional geometry. Since any picture is a projection to two spatial dimensions, the thereby induced spatial ambiguities do not pay the rent, and a higher artificial appeal is just cheating. The troubles of staying aware of a temporal dimension with different metric properties being projected to a spatial dimension is sufficiently bewildering. Purgy (talk) 14:38, 1 November 2018 (UTC)
OK. I had no preference for the 2D drawing just because I drew it. Rather, I want what is best for the article. Prokaryotic Caspase Homolog (talk) 15:03, 1 November 2018 (UTC)

### Causality and prohibition of motion faster than light

Section has been transferred to the article via this edit.

Besides the reservations rolled out in previous comments, this version has run to fat for explaining that no infinite speed is necessary to derive a contradiction from a contradiction. Sorry, I had to. ;) Purgy (talk) 14:38, 1 November 2018 (UTC)
It is not obvious from the figure that a slightly greater than light-speed signal would lead to paradox. Either I had to draw a new figure, or I had to explain how a revised figure would look. Since how a revised figure would look is documented in the supplied reference, that appeared to be the preferable route. Prokaryotic Caspase Homolog (talk) 15:03, 1 November 2018 (UTC)

## I'm going to have to squeeze in an "Introduction to spacetime diagrams" somewhere

Although everything in special relativity can be derived from the Lorentz transforms, spacetime diagrams are a highly useful tool for visualization.

The two big elephants in the room are the two sections Geometry of spacetime and Physics in spacetime, neither of which were written at a level appropriate for what I deem the prime target audience for this article, high school through lower division college students. The two sections are inadequately sourced, and some of the writing may represent original research. For these reasons, I pushed these sections to the end. I have, however, been exceedingly reluctant to delete them, since technically I have found nothing wrong with them.

Somehow or other, I'm going to have to squeeze in a quick "Introduction to spacetime diagrams", since five spacetime diagrams are used in this article without adequate explanation about how they may be interpreted, and as I continue to edit this article, I may introduce more spacetime diagrams. Yes, there will be redundant overlap with the Spacetime section, but that seems an unavoidable evil with the article in its current state. (Despite my efforts so far, I personally would rate the article, in its current state, as C-class because it does not meet the "Readers are not left wanting" criterion necessary to meet B-class.) Prokaryotic Caspase Homolog (talk) 23:13, 5 November 2018 (UTC)

Some fringe thoughts, triggered by the above concerns.
- There is an article Minkowski diagram, a redirect from "spacetime diagram", that also does not emphasize that the diagrams are just visualizations of the LT.
- I dispute the general pedagogic value of deriving STR from principles beyond deriving the LT, as well as of a "constructive" approach to STR. E.g., only the most hardcore intuitive physicists develop an intuition of squished EM-fields and how to express them, imho.
- WP is not very apt to address a specific cross-section/level-set of its readership, so I would shed some tears over shooting the elephants. Maybe a title, referring to greater advancedness, and leaving them to the end of the article would help, already.
- Classifying any article with a scientific topic is hard (WP-rules are contradictory, incoherent, rudimetary, ... rubbish). I do not argue for or against any capital letter: to suggest the worst, let WMF decide. Purgy (talk) 10:49, 6 November 2018 (UTC)
It is possible to develop STR starting from the single postulate of Minkowski spacetime, treating the LTs as a derived principle. Given the emphasis of this article, starting with universal Lorentz covariance as the fundamental principle underlying STR, the approach taken by Minkowski diagram would be completely incorrect.
I fully intend to mention that practical computations in STR usually start with the LTs and/or the fundamental effects immediately derived from the LTs. Minkowski diagrams are most useful as a tool for visualization. They are less useful as a tool for computation.
Spacetime diagrams will be treated as derived from the LTs. I intend only a bare minimum introduction, with wikilinks to other articles developing them in greater detail. Thanks for the reference to Minkowski diagram, by the way!
Shuttling legacy and/or limited-interest advanced topics beyond freshman-sophomore college level to the end and clearly labeling them as "advanced" is the strategy employed in a variety of articles on Wikipedia. For example, see Quadratic equation#Advanced topics and Spacetime#Technical topics. Rather than euthanization, that would probably be my choice of what to do with these sections, except that Special relativity#Causality and prohibition of motion faster than light is not an advanced topic. This section should not be caged with the others. It needs a separate home.
Special_relativity#Consequences derived from the Lorentz transformation and Special_relativity#Other consequences are illogical separations of topic. I'm thinking of the following reorganization, taking some inspiration from Rindler:
Kinematics: RoS, TD, LC, Thomas rotation, causality and prohibition of FTL
Optical effects: Doppler, measurement vs visual appearance
Dynamics: mass-energy equivalence, how far can one travel
That allows me to cage the two old beasts separately from the others. I can only do this reorganization after developing the "Introduction to spacetime diagrams" section, because of how much use both the RoS and Causality sections make of spacetime diagrams.
Different people learn differently. Constructive and deductive approaches are both important.
Prokaryotic Caspase Homolog (talk) 12:25, 6 November 2018 (UTC)

## Doubting ...

I am disturbed about the last preparatory edits. I did not like the previous setting either, but I sense further blurring.

- The transformation equations do not relate arbitrary 'measurements', but strictly spacetime coordinates. Hopefully, the measurements are 'covariant', and the appropriately formulated laws of physics confirm this.

- May I suggest to get rid of the "relatively moving observers", at least in new edits? I see an effort to introduce a "standard frame", that is exactly the observer (at rest!) and his frame. It is this frame, in which further frames move and can be said to move relatively wrt each other. (I tried to emphasize this less ambiguous POV in my last edit of a caption). Referring to an observer in one of these further frames, makes this frame the new standard frame, in which coordinates wrt the former standard frame are calculated via the inverse LT of the LT transforming from the old standard to the new (embarrassing).

- It should be made explicit that the parallel orientation of the spatial axes and the orientation of the velocity along the x-axis is a simplifying assumption. (I would not dig deep in the hyperbolic rotation, imho applicable only under this restriction.)

- BTW, what is the origin in spacetime diagrams? Is it the spatial origin, only? Does it make sense to talk about "whereabouts" of an origin at some time?

Just noise from the off. Purgy (talk) 08:24, 8 November 2018 (UTC)

I made some changes in the wording.
Re your other comment, I thought I was already being very explicit that use of standard configuration represents a simplifying assumption, which with care would allow simpler math without invalidating the generality of the conclusions.
Prokaryotic Caspase Homolog (talk) 08:59, 8 November 2018 (UTC)
Spacetime diagrams usually compare two frames in standard configuration. So the origin represents ${\displaystyle t=t'=0}$ where all the spatial coordinates line up. The preparatory work is to (1) enable a bare minimum introduction to spacetime diagrams, since they are currently used in the article with no explanation; (2) allow derivation of the invariant interval from the LTs for the simple case of frames in standard configuration. Prokaryotic Caspase Homolog (talk) 09:22, 8 November 2018 (UTC)
I was primarily triggered by the diff-display and expected some immediate treatment of spacetime diagrams (x/ct), and so I think I misunderstood not only the term "standard configuration", in wrongly binding it to a frame with orthogonal temporal and spatial axes, but also the term "origin", as the event with full blown coordinates (0,0,0,0), and no worldline of (0,0,0). Looking at the whole section, I understand your hint to "already", nevertheless, I still think that S and S' are depicted as spatial coordinates, whereas the section deals with spacetime coordinates. I experience this as potentially misleading. Let me know, please, when I get a nuisance. Purgy (talk) 13:42, 8 November 2018 (UTC)
The treatment of Minkowski spacetime diagrams in progress begins with the spatial diagram as a starting point. It probably won't be ready to upload until the weekend. I have been delayed by libsvg bugs. You wouldn't believe how much trouble I had trying to draw a simple green line! I finally gave up on green lines, in favor of another color. Prokaryotic Caspase Homolog (talk) 15:41, 8 November 2018 (UTC)

## Let's discuss

I don't mind adding in motivations, etc. but your additions need a bit of work. Will add edits with notes in a bit. Mostly considerations of language and target audience, etc. Prokaryotic Caspase Homolog (talk) 10:39, 10 November 2018 (UTC)

Well, I don't mind being given the chance to learn on improvements to my writing. I will abuse the tq-template in inserting my remarks directly into your structure. If this intrusion is considered or turns out as inappropriate for some reason, do not hesitate to simply revert my edit. Purgy (talk) 19:50, 10 November 2018 (UTC)
• Edit 1: In everyday experience, people do not routinely measure or think about SQUARED distances or times. A bit verbose.
I was aware of having been wordy, but I am convinced that hammering on the facts that spacetime coordinates have both spatial and temporal components pays the rent for newbie readers. Maybe, hammering that the ${\displaystyle \Delta }$'s (pls, allow for the idiotic apostrophe) in ${\displaystyle x}$ and ${\displaystyle t}$ are squared differences and not differences of squared values, but with ${\displaystyle s}$ it's about the difference of squares, and the "square" has here the meaning of, hmm, what?, is too much. I'm always unsure how to be wordy enough, but not too much. I definitely want the items ${\displaystyle \Delta t^{2}}$ and ${\displaystyle \Delta x^{2}}$ to appear separately, to contrast these two independent Galilean invariants to the only one remaining STR invariant ${\displaystyle \Delta s^{2}.}$
• Edit 2: I believe that a majority of working physicists consider classical physics to mean non-quantum physics, i.e. relativity would be a classical theory.
I give in to "classic" being wrong, but I do want some serious physics in there, to have some contrast to the non-scientific "everyday" experience: Galilean, pre-Einsteinian, non-relativistic, ...
• Edit 3: Counterintuitive to whom? Let's not scare the audience too soon. Also, how new is new? I think 100+ years is not new. :-)
We are acquainted to two Galilean (near)-invariants, and the spacetime interval is in a non-trivial way a new one. I gave two reasons,at least, for counterintuitivity. I thought nowadays they scream for trigger-warnings, mine should scare them away? ;)
• Edit 4: Most students who have learned about Cartesian coordinates are perfectly comfortable with positive and negative distances, times, temperatures etc. Reference to "positive definite", "imaginary time" and "metric signature" should be relegated to a Note, since no attempt is made to provide an inline definition for the reader. They are just "throwaway" terms.
In math, "distances" are non-negative, it takes "pesudo-distances" (like pseudo-Riemannian) to let them be negative. I have no precise notion covering "throwaway term", I introduced these terms for the possibly rare species that wants to connect their acquaintance with STR with rigorous math, or to satisfy their free associations (i² = -1).
Pushing to a note preserves your thoughts on this while not interrupting the flow for most readers.
• Edit 5: I'll try putting this chunk and the last chunk of text that I deleted into Notes, to see how putting them into Notes work for you. They are obviously matters that are important to you.
I prefer the "Pythagoras" to "The complete form", because it gives a reason (I like to hammer on), and it strengthens the relation to coordinate values. (less wordy?)
How about "an expanded form"? "Pythagorean" is confusing because of the minus sign, and I'm not sure I want to spend the time to explain here its significance. It is certainly an important topic, but I don't want to digress too much. Maybe expand the note?
• Edit 6: I like the overset def. I presume this is a standard form in the math literature?
It is one of the notations I have seen, I prefer it to the \equiv for the latter's many other meanings, I myself used the ":=" which is deprecated in WP, I think for the computer scientists sake. However, I think it is not appropriate at the second occurrence. It is a def the first time ("along a straight line"!), then it is just a consistent repetition in a detailed section.
Two dimensions versus four dimensions does not seem overmuch a repetition.
• Edit 7: Restoring deleted chunk of text as a note.
Note is probably fine.
• Edit 8: Restoring deleted chunk of text as a note.
See #5
• Edit 9: Minor copyedits, both in my language and in yours.
"as such" = "as an invariant": This constitutes the content of the derivation.
I understand what you were trying to say now.

## Latest suggestion

In pre-relativistic physics, measured distances (${\displaystyle \Delta x}$) and time lapses (${\displaystyle \Delta t}$) between events were assumed to be independent invariants, and there were just, then only recently, emerging ideas that these measurements could change when taken in another frame. In special relativity the intrinsic interweaving of spatial and temporal coordinates fundamentally destroys this separate invariance, supported from everyday life, leaving just the difference of the squares of these quantities, denoted as ${\displaystyle \Delta s^{2}}$, as invariant. The invariance of this quantity can be deduced in a straightforward manner from the Lorentz transform.

${\displaystyle \Delta s^{2}\;{\overset {def}{=}}\;c^{2}\Delta t^{2}-\Delta x^{2}.}$[note 1]

Expanding the linear spatial distance ${\displaystyle \Delta x}$ with Pythagoras' theorem makes the invariant interval applicable to the general transformation between any two Cartesian inertial frames, which may include, in addition to the standard Lorentz transformation, rotations, translation in space, and translations in time (i.e. a Poincaré transformation).[1]:33–34

${\displaystyle \Delta s^{2}=c^{2}\Delta t^{2}-(\Delta x^{2}+\Delta y^{2}+\Delta z^{2}).}$

References

1. ^ Rindler, Wolfgang (1977). Essential Relativity (2nd ed.). New York: Springer-Verlag. ISBN 0-387-10090-3.

Notes

1. ^ This concept is counterintuitive at least for the fact that, in contrast to usual concepts of distance, it may assume negative values (is not positive definite for non-coinciding events), and that the square-denotation is misleading. This negative square lead to, now not broadly used, concepts of imaginary time. It is immediate that the negative of ${\displaystyle \Delta s^{2}}$ is also an invariant, generated by a variant of the metric signature of spacetime.

Rationale:

• Making ${\displaystyle \Delta x}$ and ${\displaystyle \Delta t}$ explicit makes it easier to hint to the special feature of "minus"(!) in the Minkowski metric. I think this remains hard enough to keep in mind when looking at spacetime diagrams, when the "obvious sum" of two triangle sides is "shorter" than the "longest" side. Furthermore, I believe reasons to assume non-invariance did emerge then.
• Recalculating the Euclidean "unique linear spatial distance" via "Pythagoras" is no "re-definition", superseding the previous one. I inserted the parens and exchanged the "-"s for making Pythagoras more obvious. I settle on agreeing on disagreement.

May I ask that you include as much as is to your liking, I consider in any case my ideas as sufficiently considered. Purgy (talk) 09:54, 11 November 2018 (UTC)

Mentioning asides: Shouldn't this section be moved up to Consequences ..., too. More distant: I was very proud about me writing in the lead the "causing - caused" play on words, because I hoped someone were reminded of Wheeler's "how to move - how to curve", but I fully understand that it might not be good English. :) Sorry, Purgy (talk) 11:04, 11 November 2018 (UTC)

Much improved! You've bypassed my objections to ${\displaystyle \Delta x^{2}}$ and ${\displaystyle \Delta t^{2}}$, but I still think it is a bit verbose and overly coy in hinting at (but not actually describing) the "emerging ideas" (i.e. Lorentz, Poincaré). Let me re-read some relevant chapters in Arthur I. Miller's Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905-1911) before responding. You make me WORK! I'm still uncertain about Pythagoras. Prokaryotic Caspase Homolog (talk) 12:23, 11 November 2018 (UTC)

───────────────────────── OK, how about this? There is absolutely no need to (hint, hint) at the "emerging ideas that these measurements could change when taken in another frame" since those ideas had already been quickly suggested in the Introduction, and a link to History of special relativity had been provided.

In pre-relativistic physics, measured distances (${\displaystyle \Delta x,\Delta y,\Delta z}$) and time lapses (${\displaystyle \Delta t}$) between events were assumed to be independent invariants. In special relativity, the intrinsic interweaving of spatial and temporal coordinates fundamentally destroys these separate invariances, supported from everyday life, leaving just the difference of the squared time lapse and the summed squares of the spatial quantities, denoted as ${\displaystyle \Delta s^{2}}$, as invariant:

${\displaystyle \Delta s^{2}\;{\overset {def}{=}}\;c^{2}\Delta t^{2}-(\Delta x^{2}+\Delta y^{2}+\Delta z^{2}).}$

The invariance of this quantity can be deduced from the Lorentz transform.[note 1] This invariant interval, related to the Pythagorean theorem, is in fact applicable to the general transformation between any two Cartesian inertial frames, which may include, in addition to the standard Lorentz transformation, rotations, translation in space, and translations in time (i.e. a Poincaré transformation).[1]:33–34

For simplified scenarios such as in the analysis of spacetime diagrams, a reduced-dimensionality form of the invariant interval is often employed:

${\displaystyle \Delta s^{2}\,=\,c^{2}\Delta t^{2}-\Delta x^{2}.}$

Demonstrating that the interval is invariant is straightforward for the two dimensional case and with frames in standard configuration:[2]

References

1. ^ Rindler, Wolfgang (1977). Essential Relativity (2nd ed.). New York: Springer-Verlag. ISBN 0-387-10090-3.
2. ^ Cite error: The named reference Morin2007 was invoked but never defined (see the help page).

Notes

1. ^ This concept is counterintuitive at least for the fact that, in contrast to usual concepts of distance, it may assume negative values (is not positive definite for non-coinciding events), and that the square-denotation is misleading. This negative square lead to, now not broadly used, concepts of imaginary time. It is immediate that the negative of ${\displaystyle \Delta s^{2}}$ is also an invariant, generated by a variant of the metric signature of spacetime.

Prokaryotic Caspase Homolog (talk) 08:44, 13 November 2018 (UTC)

Only because you asked for it explicitly:
Omitting the (hint, hint) is fine if hinting was already done (note the "are" in the text now); Euclidean metric (= Pythagoras = SUM of squares) is well known, but I would avoid mentioning Pythagoras, when the spacetime metric (= DIFF of squares) is addressed nearby. How about renaming the first occurrence of ${\displaystyle \Delta x}$ to ${\displaystyle \Delta r}$? I simply like the ultimately terse definition of the spacetime metric, in contrast to the Euclidean metric, as the difference of ${\displaystyle c^{2}\Delta t^{2}}$ and ${\displaystyle \Delta r^{2}}$ much better than the one with a boring list of components (x,y,z). Orthogonal decomposition of a strictly 1-dim property into three components is no rocket science. Stripped by all refs, and cramming in some wild beasts to put into footnotes or to annihilate totally, this would look like:

In pre-relativistic physics, measured distances (${\displaystyle \Delta r}$) and time lapses (${\displaystyle \Delta t}$) between events are independent invariants. In special relativity, the intrinsic interweaving of spatial and temporal coordinates radically destroys these separate invariances, supported by everyday life experience; just the difference of the squared time lapse and the squared spatial distance, denoted as ${\displaystyle \Delta s^{2}}$, remains as the invariant spacetime interval, demonstrating a fundamental discrepancy between Euclidean and spacetime distances.

${\displaystyle \Delta s^{2}\;{\overset {def}{=}}\;c^{2}\Delta t^{2}-\Delta r^{2}.}$

In three spatial dimensions ${\displaystyle \Delta r^{2}}$ can be expanded, according to Pythagoras' theorem, to

${\displaystyle \Delta s^{2}=c^{2}\Delta t^{2}-(\Delta x^{2}+\Delta y^{2}+\Delta z^{2}),}$

but in simplified 1-dimensional scenarios, as in the analysis of spacetime diagrams, the reduced-dimensionality form from above is often employed.

The invariance of the quantity ${\displaystyle \Delta s^{2}}$ is a property of the general Lorentz transform (also Poincaré transformation), making it an isometry of spacetime. The general Lorentz transform between any two Cartesian inertial frames covers, in extension to the standard Lorentz transform, which deals just with translations (Lorentz boosts) in x-direction, all other translations and reflections in space and in time, and also all transformations that keep the origin fixed (rotations).

Demonstrating ...

Objective measures of readability rank the various revised versions of the section, both yours and mine, about three grade levels more difficult to read than what is currently in place. Now, I believe that the first paragraph in any section needs to be maximally accessible, and the difference in difficulty level between the suggestions offered here and what is in place is quite dramatic. As I've stated before, a successful compromise leaves neither person completely happy. I will push as much as I can of your suggestions into main article space to achieve the additional precision of expression that you desire, but I insist on the readability of the first paragraph. Neither of us will get entirely what we want, but that's the nature of compromise... Prokaryotic Caspase Homolog (talk) 23:18, 13 November 2018 (UTC)
Who were I, if I dared to discuss the readability of my English babbling? There is just a faint doubt about the feasibility of "deep learned" programs significantly judging roughly five(!) sentences for their readability. I myself wrote about me cramming in topics I consider interesting for a reader passing by (sometimes with the intent to trivialize high brow lingo). I really feel honored by any small phrase of mine that makes it into an accepted version. I see my cramming in as just offering on topic window shopping for things I find didactically valuable and generally interesting.
Besides being quite satisfied as it stands, there is just this
${\displaystyle \Delta s^{2}=c^{2}\Delta t^{2}-(\Delta x^{2}+\Delta y^{2}+\Delta z^{2}),}$
or worse in its original form
${\displaystyle \Delta s^{2}=c^{2}\Delta t^{2}-\Delta x^{2}-\Delta y^{2}-\Delta z^{2},}$
being preferred to
${\displaystyle \Delta s^{2}=c^{2}\Delta t^{2}-\Delta r^{2}}$
that I do not understand. The latter version easily focuses on the important NEW temporo-spatial metric (with its minus), emphasizes the one-dimensionality of distance and is way shorter and therefore way more clearly laid out. I am convinced that the embedding of one-dimensional spatial distances in 3-dimensional Euclidean spaces via Pythagoras as
${\displaystyle -\Delta r^{2}=-(\Delta x^{2}+\Delta y^{2}+\Delta z^{2})}$
convinces even the faintest hearted. I admit that I am not a fan of this "standard configuration", with its dragging along of two parasitic spatial dimensions, adding nothing to substantial understanding at this level.
Please, do not bother to proselytize me to opinions in reliable sources, I really accept the versions that you find fit for WP. Purgy (talk) 10:10, 14 November 2018 (UTC)

## Next steps

### New sections under consideration

After adding the section on Graphical representation of the Lorentz transformation, it became possible for me to move Causality and prohibition of motion faster than light from where it had been stuck to a more rational location in the article, and then to fence off the old legacy sections behind a "Warning! There be lions and tigers and bears (oh my) beyond this point!" sign.

There is obviously a lot left to be done. Optical effects ought to include relativistic aberration and maybe the Fizeau experiment??? Dynamics, of course, covers force, energy and momentum, collisions, and relativistic mass (and why the majority of physicists consider it to be a deprecated concept). Relativistic mass is a concept that most lay persons have heard about, and I'm sure that many visitors to this article page have been disappointed not seeing any mention of it.

I'm hoping that my reorganization should make it easier to add these additional topics. Prokaryotic Caspase Homolog (talk) 09:33, 11 November 2018 (UTC)

Hmmm... it looks like I need to cover the magnet and conductor thought experiment as a preparatory step before covering relativistic aberration. Prokaryotic Caspase Homolog (talk) 06:48, 18 November 2018 (UTC)

### Should experimental results be blended with the main narrative or kept separate?

Currently, almost all discussion of the experimental justification for SR is sequestered in the Status section, not that there is very much of it.

Is this a desirable organization? Prokaryotic Caspase Homolog (talk) 20:49, 14 November 2018 (UTC)

## Pre-relativistic understanding of length contraction and time dilation

Please comment on the proposed revised statement+notes+references and suggest changes as necessary. Prokaryotic Caspase Homolog (talk) 07:47, 15 November 2018 (UTC)

• I had written the following:
In pre-relativistic physics, distance and time were considered to be independent measurements, and despite some puzzling experimental results, physicists had no inclination to believe that measured distance or time between events should change as a result of a shift in frame from which measurements are made.
• You commented, "distance between events DOES change in prerelativistic physics as a result of a change in the reference frame, unless the time of the events is the same" and changed the wording to the following:
In pre-relativistic physics, distance and time were considered to be independent measurements, and despite some puzzling experimental results, physicists had no inclination to believe that measured time between events should change as a result of a shift in frame from which measurements are made.
• Pre-relativistic views of length contraction and time dilation are rather complex to describe, and my initial phrasing was an (apparently futile) attempt to avoid going into extensive discussion of Lorentz's and Poincaré's speculations. I propose the following revision with notes and references:
In pre-relativistic physics, distance and time were considered to be independent measurements, and despite some puzzling experimental results, physicists had no inclination to believe that any "true" measured distance[note 1] or time[note 2] between events should change as a result of a shift in frame from which measurements are made.
• I am quite aware that the final form of Lorentz ether theory predicts, within its domain of applicability, results which are identical to those of special relativity. LET, however, underwent extensive development between 1892 through 1905, and meant quite different things at different times. Just because the final form of the theory does not contradict Newton's third law, does not invalidate my statement in the notes that earlier versions had difficulties in conforming with classical mechanics.

Notes

1. ^ The results of the Michelson–Morley experiment led George Francis FitzGerald and Hendrik Lorentz independently to propose the phenomenon of length contraction. Lorentz believed that length contraction represented a physical contraction of the atoms making up an object.[1]:62–68 In his view, length contraction should result in compressive strains in an object that should result in measurable effects. Such effects would include optical effects in transparent media, including optical rotation[p 1] and induction of double refraction,[p 2] and the induction of torques on charged condensers moving at an angle with respect to the aether.[p 2] Lorentz was perplexed by experiments such as the Trouton–Noble experiment and the experiments of Rayleigh and Brace which failed to validate his theoretical expectations.[1]
2. ^ For mathematical consistency, Lorentz proposed a new time variable, the "local time", which depended on the position of a moving body following the relation ${\displaystyle t'=t-vx/c^{2}}$.[p 3] Lorentz considered local time not to be "real"; rather, it represented an ad hoc change of variable. Impressed by Lorentz's "most ingenious idea", Poincaré saw more in local time than a mere mathematical trick. It represented the actual time that would be shown on a moving observer's clocks. On the other hand, Poincaré did not consider this measured time to be the "true time" that would be exhibited by clocks at rest in the aether.[2] The multiplication of hypotheses led to disturbing conflicts with classical mechanics, including violation of Newton's third law of action and reaction.[1]:39–42

Primary sources

1. ^ Lorentz, H.A. (1902). "The rotation of the plane of polarization in moving media" (PDF). Huygens Institute - Royal Netherlands Academy of Arts and Sciences (KNAW). 4: 669–678. Retrieved 15 November 2018.
2. ^ a b Lorentz, H. A. (1904). "Electromagnetic phenomena in a system moving with any velocity smaller than that of light" (PDF). Huygens Institute - Royal Netherlands Academy of Arts and Sciences (KNAW). 6: 809–831. Retrieved 15 November 2018.
3. ^ Lorentz, Hendrik (1895). "Investigation of oscillations excited by oscillating ions". Attempt at a Theory of Electrical and Optical Phenomena in Moving Bodies (Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern). Leiden: E. J. Brill. (subsection § 31).

References

1. ^ a b c Miller, Arthur I. (1998). Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905-1911). Mew York: Springer-Verlag. ISBN 0-387-94870-8.
2. ^ Darrigol, Olivier (2005). "The Genesis of the Theory of Relativity" (PDF). Séminaire Poincaré. 1: 1–22. Retrieved 15 November 2018.
• You are over-thinking this. I was not talking about length contraction or any such thing. The sentence which I changed said "... measured distance or time between events should change as a result of a shift in frame ..." (emphasis added). Before special relativity, we had Galilean relativity according to which the transformation law for frames of reference was:
{\displaystyle {\begin{aligned}t'&=t\\x'&=x-vt\\y'&=y\\z'&=z,\end{aligned}}}.
The "− v t" term means that the location of an event depends on the time that its position is measured. So if you subtract two such locations to get the x-component of the distance, you will get a value which depends on the times of the events. That is my whole point. JRSpriggs (talk) 20:10, 15 November 2018 (UTC)
Well, that does not represent what Purgy and I intended. Will have to do a major re-write. Prokaryotic Caspase Homolog (talk) 23:49, 15 November 2018 (UTC)
More precisely one could perhaps write about coordinatizing two events
${\displaystyle {\text{E}}_{i},\quad i\in \{1,2\}}$
in two Galilean ${\displaystyle (t'=t;\;x'=x-vt;\;y'=y;\;z'=z)}$ inertial frames in standard configuration
${\displaystyle E_{i}\mapsto \{(t_{i},x_{i},y_{i},z_{i}),\;(t'_{i},x'_{i},y'_{i},z'_{i})\},}$
leaving separately both their time lapse
${\displaystyle \Delta t=t_{2}-t_{1}=t'_{2}-t'_{1}}$
and their contemporal ${\displaystyle (t_{1}=t_{2})}$ spatial distance
${\displaystyle \Delta r^{2}=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}=(x'_{2}-x'_{1})^{2}+(y'_{2}-y'_{1})^{2}+(z'_{2}-z'_{1})^{2}}$
invariant.
Not talking about the boring coordinates, this could be also more detailed to
${\displaystyle \Delta r=r_{2}-r_{1}=(r_{2}-vt_{2})-(r_{1}-vt_{1})=r'_{2}-r'_{1}.}$
The space—time interweaving puts an end to identical time as well as to contemporality across non comoving frames. Sorry, I missed from the diff-view the suggestion below, and also had no edit conflict. Use to your liking. Purgy (talk) 09:17, 16 November 2018 (UTC)

• Let's try this:
In Galilean relativity, length (${\displaystyle \Delta r}$)[note 1] and temporal separation between two events (${\displaystyle \Delta t}$) are independent invariants, the values of which do not change when observed from different frames of reference.[note 2][note 3]
In special relativity, however, the interweaving of spatial and temporal coordinates generates the concept of an invariant interval, denoted as ${\displaystyle \Delta s^{2}}$:
${\displaystyle \Delta s^{2}\;{\overset {def}{=}}\;c^{2}\Delta t^{2}-(\Delta x^{2}+\Delta y^{2}+\Delta z^{2})}$[note 4]
The interweaving of space and time revokes the implicitly assumed concepts of absolute simultaneity and synchronization across non-comoving frames.

Notes

1. ^ In a spacetime setting, the length of a rigid object is the spatial distance between the ends of the object measured at the same time.
2. ^ The results of the Michelson–Morley experiment led George Francis FitzGerald and Hendrik Lorentz independently to propose the phenomenon of length contraction. Lorentz believed that length contraction represented a physical contraction of the atoms making up an object. He envisioned no fundamental change in the nature of space and time.[1]:62–68
Lorentz expected that length contraction would result in compressive strains in an object that should result in measurable effects. Such effects would include optical effects in transparent media, such as optical rotation[p 1] and induction of double refraction,[p 2] and the induction of torques on charged condensers moving at an angle with respect to the aether.[p 2] Lorentz was perplexed by experiments such as the Trouton–Noble experiment and the experiments of Rayleigh and Brace which failed to validate his theoretical expectations.[1]
3. ^ For mathematical consistency, Lorentz proposed a new time variable, the "local time", called that because it depended on the position of a moving body, following the relation ${\displaystyle t'=t-vx/c^{2}}$.[p 3] Lorentz considered local time not to be "real"; rather, it represented an ad hoc change of variable.[2]:51,80
Impressed by Lorentz's "most ingenious idea", Poincaré saw more in local time than a mere mathematical trick. It represented the actual time that would be shown on a moving observer's clocks. On the other hand, Poincaré did not consider this measured time to be the "true time" that would be exhibited by clocks at rest in the aether. Poincaré made no attempt to redefine the concepts of space and time. To Poincaré, Lorentz transformation described the apparent states of the field for a moving observer. True states remained those defined with respect to the ether.[3]
4. ^ This concept is counterintuitive at least for the fact that, in contrast to usual concepts of distance, it may assume negative values (is not positive definite for non-coinciding events), and that the square-denotation is misleading. This negative square lead to, now not broadly used, concepts of imaginary time. It is immediate that the negative of ${\displaystyle \Delta s^{2}}$ is also an invariant, generated by a variant of the metric signature of spacetime.

Primary sources

1. ^ Lorentz, H.A. (1902). "The rotation of the plane of polarization in moving media" (PDF). Huygens Institute - Royal Netherlands Academy of Arts and Sciences (KNAW). 4: 669–678. Retrieved 15 November 2018.
2. ^ a b Lorentz, H. A. (1904). "Electromagnetic phenomena in a system moving with any velocity smaller than that of light" (PDF). Huygens Institute - Royal Netherlands Academy of Arts and Sciences (KNAW). 6: 809–831. Retrieved 15 November 2018.
3. ^ Lorentz, Hendrik (1895). "Investigation of oscillations excited by oscillating ions". Attempt at a Theory of Electrical and Optical Phenomena in Moving Bodies (Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern). Leiden: E. J. Brill. (subsection § 31).

References

1. ^ a b Miller, Arthur I. (1998). Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905-1911). Mew York: Springer-Verlag. ISBN 0-387-94870-8.
2. ^ Bernstein, Jeremy (2006). Secrets of the Old One: Einstein, 1905. Copernicus Books (imprint of Springer Science + Business Media). ISBN 978-0387-26005-1.
3. ^ Darrigol, Olivier (2005). "The Genesis of the Theory of Relativity" (PDF). Séminaire Poincaré. 1: 1–22. Retrieved 15 November 2018.
• This edit rephrased the footnote defining "length" as a prima vista triviality. A prominent intent in me writing a "bulky" definition for this everyday notion was to introduce the notion of events in their fundamental role of establishing "length", the value of which will turn out as varying from frame to frame, because of varying coordinates of these events. I think the ladder paradox is directly connected here, and a caveat of carrying forward a sloppy notion of length into the realm of STR is appropriate. Purgy (talk) 11:10, 17 November 2018 (UTC)
I know what you meant, but I found your original wording confusing. "In a spacetime setting the length of a rigid object is defined by the spatial distance of the two events made up of the ends of this object at the same time." The ends of the object are not events, but follow world lines. So you intended that the length of the object should mean the spatial distance between two events, having the same time coordinates, selected from the world lines of the two ends. This spatial distance would, in general, be less than the length measured in the rest frame of the object (i.e. its proper length). In the rest frame of the object, however, it is not at all necessary that the selected events have the same time coordinate.
All this amounts to a lot of superfluous detail, especially in a footnote for the Galilean scenario for which length is an invariant. Prokaryotic Caspase Homolog (talk) 13:45, 17 November 2018 (UTC)
To me an end of an object at some point in time makes up a perfect event ("two events made up of the ends of this object at the same time"), and varying time traces out a world line, which is a certain collection of events; and I perceive no problem with getting acquainted to the notion of a time-varying lenght, as observed from different frames, never getting longer than some distinguished length, one might call a proper length. I do not know how fruitful it is to continue measuring lengths at different times, which only works in the rest frame of the object, and I expressed my doubts about leaving the notion of length in its "trivial" setting of Galileian spacetime.
I did not suggest a footnote pertaining to the Galileian notion of length, but a footnote to a sustainable notion of length in general, and to focus the attention to the upcoming change in the notion of length. I think the method of "how to measure length" should not be changed/restricted during migration to STR, and thus I am unaware of a "lot of superfluous detail", however, I consider a footnote, missing to warn about the upcoming subtleties, to be really superfluous. Lifting the wording above confusion (which I miss) to your standards is beyond me, disagreeing is not. :) Purgy (talk) 17:01, 17 November 2018 (UTC)
In terms of English language usage,
• I do not know what a "spatial distance of two events" means. I do know what a spatial distance between two events means.
• I do not know what "two events made up of the ends of the object at the same time" means. I do know what two "ends of the object measured at the same time" means.
• Applying the two "Englishian phrase transformations" essentially converts one phrase to the other, except for your use of "is defined by."
Prokaryotic Caspase Homolog (talk) 18:33, 17 November 2018 (UTC)
An apology for lack in English idiomology is certainly nonsensical, I just regret it. In German the use of a genitive ("of") instead of a preposition ("between") is common habit, I do ask for improving my construction of "making up" events from their spatial and temporal coordinates (but leaving "events" in place), while talking about their "spatial distance", and, thirdly, after applying the EPT (see above) I do miss not only the rigor-effusing "is defined by", but also the mentioned "event", which is at the heart of post-Galilean spacetime.
I repeat the heart of my complaint: "prima vista triviality". If it were not for the last, innocuously sounding phrase ("measured at the same time"), the whole footnote would be absolutely "superfluous". My concern is to transport to the reader this essential condition not as a small closing phrase, but as an essential, concept transforming information, and I tried to do this via the demonstrative use of "defining" and "event". As usually, take what you like. Purgy (talk) 21:13, 17 November 2018 (UTC)

───────────────────────── No, it is not trivial. It is at the heart of the "brain freeze" that led to my oversight. I should have known better. I thank JRSpriggs for correcting me, and I thank you for introducing the original version of the footnote.

Pushed to main space. We can work on improving it later, but there are other topics to add. Prokaryotic Caspase Homolog (talk) 04:19, 18 November 2018 (UTC)

I am afraid of being misunderstood. I never wanted to state that the footnote, as is stands now, were trivial. I want to express my perception that its current formulation lacks emphasis on the changing settings in Galilean and -say- Einsteinian spacetime. To my taste, it evokes a prima vista(=sometimes wrong!)-impression of not being that fundamental as it is, by putting a condition, generating the decisive difference, into a small appendix of the sentence. My pleadings just ask for more emphasis on the change, suggesting the use of the words "define" and "event". There is absolutely no need, for a statement to be correct, of conforming to my taste. Purgy (talk) 09:24, 18 November 2018 (UTC)