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Mathematical structures

Spacetime is a mathematical model that stands in opposition to absolute space and time. The melding of two categories in mathematical physics is accomplished with linear algebra, first with a spatial-temporal plane (x, ct), and more fully as a (t, x, y, z) plenum. Special relativity (SR) and general relativity (GR) are as seed and flower. Attention to the germination of the essential spacetime concept (the meter second) sustains flowering of spacetime into cosmology. The spatial-temporal plane is best represented by D = split-complex numbers as events. Indeed, the identity component of D represents the future events, and a branch of the unit hyperbola represents frames of reference indexed by rapidity. Arithmetic of D is used to determine hyperbolic orthogonality, which provides the simultaneous events for a given rapidity. For 4D flat spacetime, B = biquaternions has been used to identify a real 4-subspace of B to model spacetime, with rotations and boosts provided by products with elements of the quasi-sphere of B. Differential geometry grew out of surface theory as necessitated by Maxwell’s equations. Manifold concepts are central , so treatment of spacetime along the lines of differential geometry would necessitate laying a foundation for manifolds. To that end, nothing outdoes the vector spaces like ℝ² and ℂ⁴ that provide the points of the algebras of split-complex numbers and biquaternions. Readers here are more likely to appreciate a bit of linear algebra to grasp SR, while efforts to approach GR with differential geometry are likely too ambitious. — Rgdboer (talk) 23:03, 28 June 2017 (UTC)

Quoting Rgdboer:

To that end, nothing outdoes the vector spaces like ℝ² and ℂ⁴ that provide the points of the algebras of split-complex numbers and biquaternions.

Oh, indubitably. Indubitably. Bring on the biquaternions.

@Rgdboer: On a serious note, I’ve felt for a few weeks that the earliest parts of this article could benefit from a super-simple (the most simple one can make) version of a Minkowski diagram. A pure effort of “introduction to Minkowski diagrams.” What do you think? Greg L (talk) 01:07, 29 June 2017 (UTC)

Figure 1-4? ----

@Stigmatella aurantiaca: I was hoping for something terribly basic where the entire “Ah-Haa” can be conveyed within a semi-lengthy caption. Not possible? Greg L (talk) 05:01, 29 June 2017 (UTC)

@Rgdboer: In applications to physics of mathematics, it is never necessary to enter the zoo of various "fields" (division rings, etc) beyond the familiar complex numbers. Actually, real numbers suffice where complex numbers are used, but the complexes possess all of the nice properties, and none of the bad. Please keep the zoo in the history section. YohanN7 (talk) 10:04, 3 July 2017 (UTC)

Editors may be interested to read the dialogue at Talk:Rapidity#Comments where YohanN7 defends reference to Lie algebras in that article. Above he argues for "complexes possess all of the nice properties, and none of the bad", but there he delves in obscurantism. Creating conflict between editors degrades discussions. Derogatory remarks such as referring to abstract algebra as a "zoo" are just name-calling. Editors are invited to read my Talk contribution above, read the links, and see that relativity needs linear algebra and wins with split-complex numbers. — Rgdboer (talk) 21:06, 3 July 2017 (UTC)

It is not about what "wins" in anyone's personal opinion. It is about what is in the literature. Actually, it is what is in the modern literature. There is, to my knowledge, no modern mathematics, physics or engineering introductory text that treats SR from the perspective of any of animals in the zoo. Even if there were one, it is not enough to overthrow standard texts as references. There is a reason for this: Present day mainstream treatment wins over the quaternionic 1800s era – by definition – because the present physics community has chosen modern mathematics. About the thread Rgdboer refers to: He violently objected me correcting an erratic description of rapidity (it was confused with velocity). In that thread, like in just about every thread on SR, he turns the discussion into a discussion about zoo animals.

That said, it is interesting (definitely) and historically correct (perhaps) that some aspects of relativity coincides with some properties of some animals in the zoo. This is coincidental, and not unique for this case. Some respected researchers, like John Baez, have looked to see whether octonions are mathematically (beyond their own cage) and even physically relevant (google for octonions and black holes). Anyone really interested in quaternions and beyond and their role in physics should read Road to reality by Roger Penrose. The conclusion is that these "numbers" have not been found relevant yet, but Penrose hopes still. I personally believe they are a dead end. If there is any relevance to string theory (and its spacetime), you have to deal with generalized Lorentz groups. These are beyond the scope of zoo animals. Likewise, few of the other Lie groups relevant to physics can be handled within the zoo. (The group SU(2) is a notable exception. It can be represented by the unit quaternions, see classical group.)

This article is presumably directed towards high school kids, or maybe fresh undergraduates. For any one of them knowing the zoo, there's a hundred knowing real and perhaps complex numbers. Keep those in mind first. YohanN7 (talk) 08:38, 5 July 2017 (UTC)

My contributions have most definitely been directed towards high school kids. I've asked several outside reviewers to look over Sections 1-4 and 9. As guidelines, I've indicated:

• 1:Introduction (middle-to-high school)
• 2:Basic mathematics of spacetime (high school)
• 3:Beyond the basics (high school to first semester college physics/calculus)
• 4:Introduction to curved spacetime (high school)
• 9:Section summaries (middle-to-high school)
• 5–8:Legacy chapters, not written by me (anything goes, but I've jettisoned several obscurantist sections)
• 0:Lead paragraphs, written by Greg L (middle-to-high school)

My latest section, "Introduction to curved spacetime", has so far received only a very limited review by reviewer #3. I am awaiting a review on this section by reviewer #1 (a Norwegian college professor) and by reviewer #4 (a Polish mathematician). If you could check this section for inaccuracies ahead of my response from these two, I'd much appreciate it.

Although the outside reviewers haven't (yet) caught me making outright blunders, the outside reviewers have caught me making unclear and/or ambiguous statements, suggested better rewordings, etc.

Discussing what you term "zoo animals" is completely outside of the scope of anything that I have written.

Re "jettisoned obscurantist sections": My impression of the legacy sections that I've completely discarded is that they represented copy-and-paste from advanced textbooks by non-expert college kids who didn't really understand the material they were copying, and who reworded the original material only just enough to avoid copyvio. Some of the statements that I removed were ridiculously false but had survived for years unnoticed because nobody had actually bothered to read the paragraphs in which they were embedded. (For example, "It is the intermingling of electric and magnetic manifestations, described by Maxwell's equations, that give spacetime its structure.") In general, though, a typical jettisoned section comprised a bunch of true statements strung together but not forming a coherent progression of thought.

Stigmatella aurantiaca (talk) 09:33, 5 July 2017 (UTC)

Simplification needed!

This article badly needs simplification. The spacetime article should not be a place for a tutorial on special (nor general) relativity-- the reader should be directed to the special relativity (or general relativity) article for that. This article should be about spacetime: that is, how it is that relativity unifies space and time.

Delete the relativity tutorial: focus on the topic. Geoffrey.landis (talk) 15:18, 7 July 2017 (UTC)

@Geoffrey.landis: You have a valid point of view, as do I. The fact is, that the articles Special relativity and Theory of relativity provide almost no historical context, while History of special relativity provides far too much historical context at a level of detail that few would want to tackle.

I intended the Introduction section to be readable by middle-to-high-school students to get a decent overall feel for what the subject is all about and to provide an inkling of the motivations that lead to the development of the theory. I think you will agree that neither Special relativity, History of special relativity, nor Theory of relativity provide adequate historical context at a level that a young student can comprehend.

Unlike the featured articles General relativity and Introduction to general relativity, Special relativity and Theory of relativity suffer from too much churn by well-meaning editors with different points of view, resulting in a almost unreadable mishmashes.

Re the current article, a good compromise should leave neither of the disputants totally happy.

Over the weekend, I'll try working out a compromise between your viewpoint and my viewpoint. I imagine there will be some teeth-gnashing on both your part and my part, but if the measured pounds-per-square-inch of your jaw-grinding comes out to approximately the same as my jaw-grinding, we will have achieved something.

Stigmatella aurantiaca (talk) 17:57, 7 July 2017 (UTC)

I appreciate your thought. I would like you to keep in mind, however, the question "who is reading this article"? People who are looking for a historical survey of special relativity will not, in general, find this article. The people who do find this article will more likely be people asking "what is this concept 'spacetime' that seems to be important to physicists?"

So, addressing your intent "and to provide an inkling of the motivations that lead to the development of the theory" should be focussed on "the motivations that caused people to develop the notion of spacetime," and not "the motivations that caused people to develop the theory of relativity."

Addressing "the motivations that caused people to develop the theory of relativity", on the other hand, is important-- but should be addressed in the articles Special relativity, History of special relativity, or Theory of relativity , where the people looking for this material will find it. Geoffrey.landis (talk) 18:36, 7 July 2017 (UTC)

My only problem with that, is the quality of the articles Special relativity, Theory of relativity, and History of special relativity. The first two sort of suck, in my opinion, and really don't address motivations and historical context at all. And even though I've read a lot of Miller, Stachel, Renn, Janssen etc., I find History of special relativity tough going. Very dry, like unseasoned jerky. Stigmatella aurantiaca (talk) 18:50, 7 July 2017 (UTC)

Sure, but you won't make those articles any better by putting material into this article. They will still "sort of suck" regardless of what is put here. Geoffrey.landis (talk) 19:19, 7 July 2017 (UTC)

Trying to improve Special relativity and the special relativity sections of Theory of relativity is like trying to write in sand. The GR sections of Theory of relativity are relatively stable, but every high school kid thinks they know enough to write about SR, so there is a lot of non-productive churn. Compare those articles with their versions as of three years ago, and the SR portions are completely different, and not improved. That's why I put as much historical stuff here as I did.

On the other hand, most of what you've done so far is pretty decent. But the link to History of special relativity??? Have you actually ever tried to read that article??? Stigmatella aurantiaca (talk) 20:00, 7 July 2017 (UTC)

But historical stuff added here doesn't help actual readers, because readers who are looking for historical stuff won't be looking here. Readers here will be the ones looking for information on spacetime-- not on the history of relativity.

On the other hand, some good material on the history of developing the concept of spacetime in the theory of relativity-- the development of the Minkowski geometrical formulation over Einstein's original algebraic formulation-- would be very useful. Geoffrey.landis (talk) 20:08, 7 July 2017 (UTC)

The lede had was the product of lots of collaboration by multiple editors over a long period of time. I find that Geoffrey’s deletion of so much material is overly bold and without consensus so I reverted it. If you, Geoffrey, think the article is too complex and needs material removed rather than tweaked, please give the courtesy to discuss some of your plans keeping in mind that with several active editors, consensus should prevail before wholesale deletion of content. Greg L (talk) 20:50, 7 July 2017 (UTC)

P.S. I completely share your sentiment, Geoffrey, that the article needs simplification. Stigmatella and I are solidly behind you on that score. Precisely “how” to accomplish that requires some discussion given the complex subject matter and the extensive amount of material we have to deal with here. And please bear in mind that Stigmatella created the vast majority of illustrations here, which requires a huge investment of time and creative energy and adds greatly to the article. He adds value to this proposition… and so do you as we finally seem to have a knowledgeable editor dedicated to the cause of simplification. So let’s all collaborate. Greg L (talk) 20:59, 7 July 2017 (UTC)

Thank you. "Be bold" is one of the foundations of Wikipedia. I'm sorry that you don't like the changes, but "two editors liked this material" is not a reason to keep material, nor is "a lot of editors collaborated on a previous version" a reason to revert a change. The reason to revert a change-- the only reason to revert a change-- is because the article is better with the material included than with the material deleted. It shouldn't matter how many editors "collaborated" on a previous version. The only question that is relevant is: is the change better, or worse?

If you wish to revert the material again, please explain why you are making the reversion, this time with an explanation that states why the article is better with the material included. Geoffrey.landis (talk) 21:01, 7 July 2017 (UTC)

Unlike the anonymous IP editor who gave us so much grief, Geoffrey actually knows his stuff. His point of view disagrees with mine and I will partially revert some of his changes, but some of his bold changes strike me as valid. For instance, moving the section on measurement vs appearance to Special relativity was a valid move. He just did it in a very sloppy fashion and I had to fix up the Special relativity article because it didn't have a reflist group = note. Wikipedia is a collaborative effort, and not everything will go our way. Geoffrey should realize that not everything will go his way either. As I noted before, a good compromise leaves NEITHER party totally happy. Stigmatella aurantiaca (talk) 21:05, 7 July 2017 (UTC)

Responding to Geoffrey, Wikipedia is collaborative writing environment. That two editors are in disagreement with you is actually an important concept for everyone to abide by to avoid editwarring. The material in the lede is often all that many visitors ever read and should be sufficient to solidly ground the reader in what spacetime is and how we got to the current understanding of the concept. Greg L (talk) 21:06, 7 July 2017 (UTC)

Do two editors disagree with me? I have yet to hear a clear statement that even one editor disagrees about the cut material. You have not yet actually stated you disagree with the edit-- you have merely stated that several editors worked on the previous version (which is not a statement that the article is better with the cut material restored.). Stigmatella aurantiaca and I have been discussing the changes (in the long thread above) but so far I'm awaiting his changes-- I haven't yet seen a clear statement from him that the material deleted in fact belongs in the lede.

Let me ask you this directly: Do you, in fact, think that the material I cut should be in the lede? If so: why should it be in the lede? If a reader is looking for information specifically on spacetime, is the cut material the important thing that you think they are looking for? Geoffrey.landis (talk) 21:22, 7 July 2017 (UTC)

@Geoffrey.landis: Geoffrey, you need to cool down and take a deep breath. I explained why the lede is better above, but you apparently are too agitated to read what others are writing and and jump straight to angrily pounding on your keyboard. That gets us nowhere. And please don’t revise others’ posts by adding posts and unsigned pieces of posts right in the middle of them.

@Stigmatella aurantiaca: Stigmatella, I prefer the extra material for the reason stated above (The material in the lede is often all that many visitors ever read and should be sufficient to solidly ground the reader in what spacetime is and how we got to the current understanding of the concept). Do you agree? Greg L (talk) 21:30, 7 July 2017 (UTC)

Responding to Stigmatella. I’m happy to hear that Geoffrey is far from a crank and knows his stuff. And I am extra-pleased that we finally have someone who not only knows his stuff, but is highly motivated to simplify the article. But “consensus rules” on Wikipedia, even though one editor firmly believes his views are superior. With 3+ of us active here, we should always be able to achieve a consensus on tough decisions. And, just to be sure we are all on the same wavelength here is a link to WP:Consensus, which governs all plain ol’ disagreements over editorial content, which is all we’re dealing with here. Greg L (talk) 21:19, 7 July 2017 (UTC)

Motion: I motion that we three take a 20-hour break from editing the article and also get a good night’s sleep before making anymore comments here on this talk page while upset. We have infinite amounts of digital whitespace to fill with arguments and a lifetime to improve the article. Greg L (talk) 21:35, 7 July 2017 (UTC)

(edit conflict with Greg. Edits composed before the 20 hour break suggestion)

Responding to Geoffrey. There can be legitimate differences of opinion on what a reader expects to find when he/she looks up an article on "Spacetime". Probably over 95% of the time, the person is interested in spacetime as it relates to special relativity or general relativity. But there are the occasional readers who would be interested in, say, the 2+1 model spacetimes studied by Carlip et al. as an approach towards quantum gravity, or maybe the 4+2 spacetimes with multiple time dimensions studied by Bars et al., or Kaluza-Klein theory, or spacetime as an emergent property of entanglement, or maybe model systems of spacetime arising out of condensed matter physics, etc. etc.

Now, your point of view would probably be that covering these variants of spacetime from the 3+1 spacetime of relativity doesn't belong in an article on "Spacetime". Others may disagree, since all of them are spacetimes.

I just happen to be of the viewpoint that covering some of the history of special relativity that, to the best of my knowledge, is not adequately covered in ANY other article on Wikipedia at a level appropriate to the target readership of this article (which I reiterate, for the "Introduction to" sections is the middle-to-high-school student, while the "Basic mathematics of spacetime" and "Beyond the basics" sections are targeted towards high school to freshman college level) is an appropriate goal. And if I can convince you of the appropriateness of this goal, then much of Greg L's lede must be restored. Stigmatella aurantiaca (talk) 21:48, 7 July 2017 (UTC)