Talk:Minkowski space

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Rant on merging[edit]

ARRGH Please someone explain how I was browsing through a series on "timelike hotopy" and clicked on a link to get the exact definition of "timelike" and was brought to this page of technical jargon. For the love of god, don't make wikipedia inaccessible by merging every single article that is remotely similar. Eventually there'll just be one god damn article and it won't say anything useful, no matter how much of the irrelevant stuff we have to search through. —Preceding unsigned comment added by 61.68.184.249 (talk) 17:40, 22 December 2006

I fully support this rant. Time-like and Space-like must not redirect here. They should either redirect to Spacetime#Space-time_intervals or have their own article which would combine explanations from Spacetime#Space-time_intervals and from Speed of light. --206.169.169.1 21:11, 20 June 2007 (UTC)

Agreed there should be an unmerge. I volunteer to work on it when I get some time, but any contributor should start. It is against nature for Wikipedia's introduction on time-like intervals to be so off-putting. One of the most important things which must change is using approachable variables such as Δt and Δx rather than the Minkowski vectors. For example:

A time-like interval is a description of the space-time distance between two events. In such calculations, space and time are related by the speed of light. In a time-like interval, the squared interval of the time component ()of separation measured between the two events is greater in magnitude than that of the spatial distance component, that is: .
In this case of time-like intervals, the calculated proper time () is used to represent the separation of the events. Heathhunnicutt (talk) 08:01, 8 January 2008 (UTC)

Typos in "Causal structure"?[edit]

The article currently says:

  • is timelike if and only if
  • is spacelike if and only if
  • is null (lightlike) if and only if

Isn't this wrong? The middle part of each math statement looks wrong to me; it gets the sign of the timelike component wrong. If it were really correct, the result would be that no vectors would be timelike.

I think what is intended is the inner product:

But I'm not certain enough to make the edit. --Jorend 17:35, 5 January 2007 (UTC)

By definition .
The pont is that . I think that everything is OK here.
XCelam 21:09, 5 January 2007 (UTC)
I would propose to start with choosing a signature. As far as I can see for the signs chosen, the following phrase should start the second sentence: "Given the (+,-,-,-) signature ". 92.54.93.79 18:15, 30 May 2009 (UTC)

I also believe that the article should stick to one signature. In the Causal Structure section, the time/space-like inequalities seem to be defined using the (-+++) signature instead of (+---) which was used earlier in the article. I'm just been introduced to the subject, so whoever feels comfortable please make the appropriate changes. Mppf (talk) 22:02, 30 January 2011 (UTC)

As far as I can see, the article only refers to (+---) as a secondary possibility in some cases where (-+++) has already been used. Is there some exception to this which I have not noticed? Please be specific as to the section and paragraph, and give a quotation. JRSpriggs (talk) 09:42, 31 January 2011 (UTC)

Minkowski's nationality[edit]

I wouldn't be sure if it is important but this article calls Minkowski a "German mathematician" while by clicking at the guy's surname you can easily learn that he's a "Lithuanian mathematician". The information should be either coherent or omitted, IMO. What is even funnier, it then reads that he was born "to a family of German, Polish, and Jewish descent" and in fact his family sounds rather Polish (it would be also quite a good Jewish or German surname, still it wouldn't as Lithuanian nowadays since they add those "-is" and "-as" suffices to all surnames or so it looks like; I've seen a plaque to Dzordzas Busas, the president of USA in Vilnius).

Pigeon-holing every person into categories of that kind is a nasty habit of WP. Since it isn't relevent to this article I removed it. But I recommend that you edit the main article on the person Minkowski, to make the opening more factually correct and preferably to remove the horrible implication from the lead that his nationality and ethnicity are the absolute most-important two defining facts for readers to first know about him. Cesiumfrog (talk) 23:50, 22 November 2010 (UTC)

Math markup[edit]

The equations in this article are written in HTML instead of the TeX math markup used for equations in most of Wikipedia. A number of the math symbols, such as the 'element of' symbol and some of the brackets, don't display in my IE7 browser, so I'll bet this page doesn't display correctly for a significant number of viewers. I think the equations should be rewritten in math markup. --ChetvornoTALK 11:14, 13 January 2009 (UTC)

With this new dimension[edit]

With this new dimension, it can now be more likely for time travel to become a reality, the time dimension and wormholes. Albertgenii12 (talk) 20:38, 9 March 2009 (UTC)

imaginary length?[edit]

The inner product of a timelike vector with itself is negative. Does this mean that the length of a timelike vector is imaginary? I think this point needs to be clarified a little in the discussion of timelike and spacelike vectors.

Etoombs (talk) 03:23, 14 April 2010 (UTC)

You cannot define length from the Minkowski 'norm'. It is not a norm. It is a pseudo-norm, i.e. it looks like one but isnt.94.66.66.21 (talk) 11:25, 20 October 2010 (UTC)

It's worse than that. The article currently states the Minkowski norm ||v|| of a vector v, defined as ||v||2 = η(v,v), need not be positive (and mentions some misnomers relative to pure mathematics). This makes no sense. If read literally this definition means that the Minkowski norm is double-valued (since any value for the norm itself can be multiplied by minus one and will still satisfy the definition) and is sometimes not even real but imaginary (since by that definition the norm is still proportional to the square root of the metric product that isn't positive definite). Is it possible that the intention was to identify the Minkowski norm with ||v||2 (and NOT with ||v|| itself), in which case the norm would be real (not complex and not more than single-valued) and simply not always non-negative (as per the clause that follows the definition) however there would still be other differences (e.g., for spacelike vectors the norm itself would have units of length-squared rather than just length, and the symbol for the norm would have to always incorporate the superscript in order to remind to take its square root before applying formula derived for standard norms). Which is it? Cesiumfrog (talk) 23:42, 22 November 2010 (UTC)
Usually, the "norm" as it is defined in relativity is , which is just the proper length. I have changed the article to reflect this more standard usage. I think we should avoid using "norm" also to refer to , even if it is typical to do so in informal treatments. Sławomir Biały (talk) 14:26, 7 December 2010 (UTC)
It might be worth adding that in his 1908 "Space and Time" lecture Minkowski never referred to a norm. The "Minkowski norm" is a later invention. Neither did he talk of orthogonality but used the term "normal" instead. He was much more careful than his modern commentators.JFB80 (talk) 18:48, 5 October 2011 (UTC)

Minkowski norm[edit]

Minkowski norm redirects here. But that seems to be something different. At least, there seems to be a totally different notion of Minkowski norm, which is related to Finsler spaces.--Trigamma (talk) 22:07, 7 May 2010 (UTC)

ict picture and rotation[edit]

How should we present the elaboration of the x0 = ict picture? It is mentioned briefly in one paragraph, but I think it's worth presenting fully because of its beauty and the transformation-as-(ordinary)-rotation picture. What do you think? CecilWard (talk) 02:55, 23 December 2011 (UTC)

You may put it into a separate section near the end of the article. JRSpriggs (talk) 04:15, 24 December 2011 (UTC)
I don't think that's a good idea. As it is so obsolete, giving it a section of its own would i.m.o. give wp:undue weight to it, and it is already prominently mentioned in the history section. - DVdm (talk) 10:17, 24 December 2011 (UTC)
I wrote the historical note and think the complex Minkowski representation important. It is in several respects much clearer than the standard affine space view and deserves a separate page to itself. It is one form of the hyperbolic theory of special relativity. The standard view is horribly muddled in this article which should be rewritten in simpler form without tensors.JFB80 (talk) 16:14, 24 December 2011 (UTC)
I don't see how it would help: in one sense it's mathematically no different from more modern ways of doing things which being modern have much more theory and sourcing behind them, as it produces the same results so must involve the same calculations. But it's also potentially confusing: complex rotations, as described at e.g. Rotation (mathematics)#Complex numbers, are usually Euclidian. These are non-Euclidian and so far from ordinary. That can be explained away but again it ends up duplicating mathematics that is already there.--JohnBlackburnewordsdeeds 17:25, 24 December 2011 (UTC)
I disagree and as I said before think the complex space method deserves a special page. (a) The method used in this article is just Minkowski's 2nd method of the 1908 'Space and Time' lecture expressed in terms of 'Minkowski norm' (not used by Minkowski and unsatisfactory) with a hint of tensor notation. I'm not clear on what could be these 'more modern ways with more theory and sourcing' you talk about. I am not of course saying that they shouldn't be also described. It's not one or the other.
(b) The method you prefer does not produce the same results as the complex space method. You say the complex space method produces a non-Euclidean result. So it should – perfectly correct. I did remark that it is a form of the hyperbolic space theory of special relativity. Your final remark is: 'they can explained away but again it ends up duplicating mathematics that is already there'. I don't believe it. How and where?JFB80 (talk) 19:57, 27 December 2011 (UTC)

So, an article on Minkowski space![edit]

Alright...so here are my thoughts. Hopefully they're helpful!

First off, as has been pointed out, this article is much too technical much too soon, jumping into terms, axioms, and derivations without even definitions. Reading this now, knowing what a Minkowski space is to a reasonable extent, I can understand the material, and it seems that it has definitely been put forward correctly. However, it most certainly has not been put forward introductorily! If I didn't know what a Minkowski space was, what context its terms were in, or what kind of elements it had, I suspect I would be rather lost. Now, obviously the article can't be self-contained, but it can be much more reader-friendly, even through simply defining terms and using more well-known objects to define things at first (and later telling us that such a structure has a name). I'm referring, of course, to the sudden technical punch that begins the discussion on structure—'a nondegenerate symmetric bilinear form with signature (-,+,+,+)' or similar. While fine if you're familiar with the terms, this is quite unnecessarily intimidating to one without such previous familiarity. These terms do help to pick Minkowski space out of a broader, more general class of spaces, but that is not a helpful initial definition—rather, we should build up Minkowski space from more-likely-to-be-familiar and more accessible vector-related notions. Also, I think putting Minkowski space in a mathematical relativistic context early on (perhaps after the initial definitions) is important—after all, that's why this specific space has an entire page! Also, I've noticed that this page doesn't focus much on defining the particular term "Minkowski metric"—even though the term is a misnomer, as said in the article, it is quite common, and one looking for a good definition of "Monkowski metric" on Wikipedia, having been redirected to this page, would have to be halfway through the Structure section to notice it, and even there it's rather hidden as a secondary name for "Minkowski inner product" (and only for the fact that it's a misnomer). As the article shows, this space is interesting precisely because of its inner product/metric—however, it's not clear at all what's so interesting about this "metric" from the article, even though it's properly defined. This article should focus on defining the Minkowski "inner product"/"metric" in context and from more basic structures, and on exploring its relevant ramifications and interpretations in physics in (reasonably) commonly accessible terms. For instance, the section on Lorentz transformations doesn't explain how these are physically relevant or what they represent to the extent an article so important to relativity should. So, if it's alright with you who have been working on this page, I'll set about organizing and expanding this page in the immediate future—note that I'll maintain at least all of the information already put forth (just organized differently). (And note that I'm also responding positively to the "Rant on merging" section which is in this Talk page.) Just wanted to bring this up on the Talk page before I changed the page! (Of course, if I don't get a response, I'll just start—it can always be undone if someone finds an objection, after all.) Anyway! I hope what I plan to do will help the page!

Trmwiki (talk) 07:51, 28 August 2012 (UTC)

I agree. Glad to see someone who believes like I do, that we've got to stop those complaints by the public that "the only people who understand WP articles are the ones who write them"! I'm an engineer so my level of comfort is the Lorentz transformations, but the higher math is a little unfamiliar. From my perspective, what I'd most like to see explained for the general reader is the crucial difference between "distance" in Euclidean space and "interval" in Minkowski space, which is now expressed by that cryptic phrase: "(-,+,+,+) signature" Most people get the idea of a spacetime created by adding a time dimension to the 3 space dimensions (although that could also use a little explaining, maybe bringing in the idea of a worldline). But the article doesn't explain anything about how the Minkowski metric creates the light cone structure at an event, dividing spacetime into future and past causally connected regions and noncausally connected region. It discusses it in mathematical terms in "Causal structure" but not lay terms, except for the good diagram. Also as you say the Lorentz transformations, and why the speed of light is a universal speed limit. BTW, my feeling is most of the existing article is good and should be kept, just additional sections could be added giving nontechnical explanations. Cheers! --ChetvornoTALK 10:41, 28 August 2012 (UTC)
I would like to a comment because in my opinion the article certainly needs to be rewritten. Principally I think an article about Minkowski space should pay some attention to what Minkowski actually said which the main part of the article does not do. Minkowski talked about two different space-time representations (as I tried briefly to explain in the historical remarks) .His first was complex Minkowski space, using (x, y, z, ict) and pseudo-Euclidean ideas (orthogonality, distance etc) His second was affine Minkowski space using (x, y, z, t) with affine geometry (oblique axes not preserving angles). Almost all the WP article is written about someone else's creation called Minkowski space which mixes and muddles the two using some of the ideas of affine Minkowski space together with a Euclidean style pseudo-metric (plus some fancy notation to go with it) Pseudo-metric only works in complex Minkowski space. Affine space was essential to Minkowski's 2nd presentation (the 'Space and Time' one usual nowadays) because he showed how the Lorentz transformation could be understood in terms of a skewing of axes. So he never referred to orthogonality but always to conjugate directions. And the space time diagram and the ideas of the light cone structure i.e. 'time-like', 'space-like' were presented by Minkowski in an affine way too even though the diagram in the article shows them looking Euclidean which their definition doesn’t depend on. So why not at least take a look at Minkowski's 'Space and Time' lecture on http://en.wikisource.org/wiki/Space_and_Time ?JFB80 (talk) 04:54, 5 September 2012 (UTC)
I would like to see more graphical representations of concepts. first figure looks like a good start than just using TEX data equations which are not vetted and not informative , this is a not a reference book! Juror1 (talk) 18:49, 23 February 2017 (UTC)

Incorrect though Popular Attribution: "Einstein's theory of special relativity"[edit]

Although interested in the relativity theory and history of it I am not an expert. However, based on my research the statement "Einstein's theory of special relativity" is vastly misleading. I point to the Articles on the Poincare group, the Lorentz Transformation and Minkowski Spacetime. From what I can determine Einstein played zero role (other than popularization) in the development of the theory of special relativity. Far more prominent was Poincare who developed the theory to a level that Einstein never even managed to copy. In fact Poincare did state the principle of relativity before Einstein and he developed it in terms of a beautiful and far more general mathematical theory involving groups. Poincare did acknowledge Lorentz for the famous Lorentz transformations central to the theory. I therefore suggest Poincare-Lorentz special theory of relativity with mention of the work by Minkowski. Little or no credit goes to Einstein. Apparently Einstein may have played some role in the development of general relativity but there again Hilbert was involved. However, Einstein did successfully predict the advancement of the perihelion of Mercury, however, this is general and not special relativity. — Preceding unsigned comment added by Berrtus (talkcontribs) 08:33, 17 April 2013 (UTC)

The world seems to disagree with that point of view. Check Google scholar and Google books. - DVdm (talk) 08:56, 17 April 2013 (UTC)

Admittedly, you are correct. The world disagrees with the point of view that I put forth. But luckily this is not an issue of popular agreement. On this issue we can ascertain the facts. From what I have been able to determine Poincare came up with a far more general mathematical description of relativity theory than Einstein did before Einstein published his results. Further Einstein did not give proper attribution to Poincare although Einstein had read Poincare's results. However, I must also say that Einstein did put forth the relativity postulates more forcefully, although even he did not totally abandon the aether. Most likely the theory was a collaborative effort. But I see Lorentz- Poincare - Minkowski as the men who truly developed this theory especially in the mathematical details and generalizations. Einstein was more like the Carl Sagan of special relativity. §— Preceding unsigned comment added by Berrtus (talkcontribs) 09:52, 18 April 2013 (UTC)

(ec)
Please sign your talk page messages with four tildes (~~~~). Thanks.
We don't have to attribute the genuine authors of the theory. We have to reflect what the world says. This is just an encyclopedia, not a textbook, or a forum where we can put things straight, or straighter, or curved along the results of our personal research or someone else's fringe viewpoints. And this talk page is where we are supposed to discuss the content and format of the article, not a place where we make challenges — see also wp:TPG - DVdm (talk) 10:02, 18 April 2013 (UTC)
Einstein realized that this new symmetry between space and time applied to everything, not merely to electromagnetic phenomena. He discarded the old 3+1 way of thinking entirely; and noticed the importance of the fact that simultaneity is relative to the state of motion of the observer rather than an absolute relationship.
He was also the first to appreciate the fact that the equality of gravitational mass and inertia (i.e. the equivalence principle) is the defining property of gravity and that it implies that spacetime is curved. JRSpriggs (talk) 10:00, 18 April 2013 (UTC)

"We don't have to attribute the genuine authors of a theory." I disagree. At least if we have substantial evidence of who they are. "We have to reflect what the world says." I disagree especially if we have substantial evidence to the contrary, or if we do we should mention it. Going along with a known false status-quo is simply not acceptable. I think it is unfair to personalize this or to say that it is a fringe viewpoint. Those are just personal attacks. I disagree that proper attribution of a theory is an inappropriate topic for the talk page. Someone might rightly simply change the page to say the Lorentz - Poincare - Minkowski theory of special relativity, but I did not do that. So none of what you said gets to the actual issue. I see your comments as mostly personal and off issue.

As to the comment that Einstein was the first to apply his theory to everything. Please correct me if I have this wrong but was it not Poincare that applied this to Maxwells equations? And on gravitational mass that is general relativity. please note my comments are on special relativity. But thanks for the on issue comments! — Preceding unsigned comment added by Berrtus (talkcontribs) 10:29, 18 April 2013 (UTC)

Please note it was Poincare who developed the synchronization procedure for clocks (simultaneity) Berrtus (talk) 10:40, 18 April 2013 (UTC)

We don't have to correct you if you have this wrong. This is an article talk page where we discuss the article, not a chat room where we discuss a tangent of the subject. See wp:talk page guidelines. - DVdm (talk) 10:46, 18 April 2013 (UTC)
Berrtus, if you have reliable sources that Poincare, Lorentz, and Minkowski are more responsible for Special Relativity than Einstein, then the issue can be discussed. Otherwise it's WP:OR. --ChetvornoTALK 15:14, 18 April 2013 (UTC)
And not just some rant from the incorrigible plagiarist please — read about our wp:UNDUE policy. - DVdm (talk) 16:22, 18 April 2013 (UTC)

Berrtus' opinion that Lorentz and Poincaré deserve more credit for the theory of relativity than does Einstein is not at all an original viewpoint. See Relativity priority dispute. Red Act (talk) 07:52, 27 May 2013 (UTC)

lorentz transformation's importance[edit]

it seems to me a lot of people are incorrectly stating that Minkowski's Raum Und Zeit paper showed that Lorentz' transformation was invariant.

This is not true. DVDm has forced me to post here, but it's quite clear in the link I posted in the edit (page 293).

Minkowski's geometric theory of spacetime was far more general than what lorentz was doing. He states that there is an equivalence, but that does not mean Lorentz' transformation is invariant.

It seems to me that Minkowski was making a small reference to the lorentz transformation, and everyone else decided this meant he was providing additional strength to Lorentz' hypothesis. this is incorrect.

Minkowski's paper simply was showing the geometric properties necessary to establish a full axiomatic system for a geometric theory of gravitation.

The lorentz transformation does not have any *mathematically* valid interpretations of geometry. Minkowski's paper is providing a *geometric* basis for a relativistic theory, complete with the conditions (like the sum of the squares of the measures should total to 1).— Preceding unsigned comment added by 174.3.213.121 (talkcontribs) 22:42, 22 November 2014‎ (UTC)

Please sign your talk page messages with four tildes (~~~~). Thanks.
Nowhere it was written "that Lorentz' transformation was invariant.". The article said that Maxwell's equations are invariant under a Lorentz' transformation. I don't think that our interpretation of the text that you quoted really matters here—see wp:NOR. - DVdm (talk) 09:56, 23 November 2014 (UTC)
you shouldn't be citing minkowski directly after an interpretation. all references to original work should be cited in place for clarity--i.e., to "bookend" any reference to previous work.— Preceding unsigned comment added by 174.3.213.121 (talkcontribs) 05:12, 24 November 2014‎ (UTC)
What? What do you mean? What is "bookend"? JRSpriggs (talk) 09:47, 24 November 2014 (UTC)

Naming: norm squared (or separation squared?)[edit]

I made the article precise where it talks of the square of the (Minkowski) norm. However, arguably the quantity v2 is actually more useful, as I think JRSpriggs's edit seems to be suggesting (I'd agree that the square of the Minkowski norm doesn't merit being mentioned). Can we find a suitable name for this ubiquitous quantity – or more correctly, of the scalar value v,v? —Quondum 22:33, 26 February 2015 (UTC)

The article says "The Minkowski norm of a vector v is defined by ", so saying that the norm squared is is obviously wrong, because the latter can be negative, so I was puzzled by JRSpriggs' reversal of my correction. So it is because he does not realize/agree that norm squared means the square of the norm?. - Patrick (talk) 00:29, 27 February 2015 (UTC)
If v2 is supposed to be a variable it should be written as such, just a letter. If v is the variable it should be defined explicitly, "v2 = .." has two solutions. - Patrick (talk) 01:32, 27 February 2015 (UTC)
v is the four-vector we're dealing with, not something that we're solving for. JRSpriggs's comment "there is no basis for talking about absolute values here; the absolute value of this has no physical meaning or significance" is valid: the square of the Minkowski norm has no particular utility, other than when its square root (i.e. the Minkowski norm itself) is needed. The problem was the mismatch between the text and the formula, and the revert restored the mismatch. So I put it into a correct, if somewhat useless state. The ideal edit would restore the formula to without the absolute value, but would match the words to the formula. The quantity v2 ≝ ⟨v,v is the primary invariant scalar associated with any vector v, and is thus something that we should mention, just like we find ds2 = gμνdxμdxν occurring everywhere. The sign of the square is significant: it distinguishes spacelike from timelike vectors. My problem is that I do not know what the name for this is: not absolute value, not magnitude, not norm. Penrose, for example, simply seems to refer to "the quantity v2", thus avoiding a name for it. —Quondum 04:34, 27 February 2015 (UTC)
Earlier I encountered in the article: "the norm-squared of a vector v is " and corrected that into a correct statement about the square of the previously defined norm , but it seems now that what was meant was v2 ≝ ⟨v,v⟩ = , and no statement about the square of the norm was intended, "norm-squared" was apparently just a poor formulation. I removed it, added the definition of v^2, and restored the original formula. - Patrick (talk) 09:22, 27 February 2015 (UTC)
Much better. Though now I'm realizing that the notation v2 is inappropriate in this context (it is used, and correctly at that, in contexts such as geometric algebra, where the metric tensor is implicit in the product of two vectors, as well as in vector contexts where the "inner product" between vectors is denoted as some operator, e.g. u · v). When you came to this article, it already had these inappropriate/inaccurate references ("norm-squared of a vector" and v2), and your stumbling over them has highlighted this. You have already removed the first problem; I've now removed the second. —Quondum 15:36, 27 February 2015 (UTC)

Splitting off Minkowski geometry[edit]

Following on a discussion at Talk:Hyperbolic geometry#Lorentzian ≠ hyperbolic To me this article seems to be about a couple of different (and only loosly related) subjects:

And I think it does neither subject any good to be combined in one article and I would suggest therefore splitting it up.

To be honnest on the talk page of hyperbolic geometry this idea was rejected, so now I am trying it here, (this is also the better place to discuss it) WillemienH (talk) 09:29, 7 July 2015 (UTC)

I think what I mean with minkowski geometry is more the geometry of the minkowski plane ( a kind of Benz plane) not that I understand it all (I just stumbled upon it) but the pictures seem similar. WillemienH (talk) 21:08, 7 July 2015 (UTC)

Very quiet here :) , maybe my suggestion was to radical, but still the article is to complex, could we change the structure a bit?

my suggestions

  • split the geometry of the physics (maybe we also could use a section on 3 dimensional minkowski space (I am thinking about a 3 d space with a signature (+ + -) , just to bridge the gap between minkowski plane 2- dimensional planeand this page 4 dimensional spacetime.
  • more consistency the minskowski metric really seems to use the wrong signature
now: should this not be yes just to keep to the right signature.

But even after this the article is still to complicated.WillemienH (talk) 13:02, 19 July 2015 (UTC)

On the first point, I don't know what you mean by "geometry of the physics". There is little geometry in the article, unless you promote pretty much every structure in the article to fall into the confines of geometry (which is an entirely acceptable POV). Concepts as straight lines, angles, length etc are, in the article, undefined. This should perhaps change with additions to the new empty subsection called "Geometry". Edit: To be clearer, both the former and this version tackles Minkowski space in an almost entirely algebraic fashion. I believe the foundation should be algebraic because it follows the most easily from the postulates. We also have an article on Pseudo-Euclidean space. Minkowski geometry is a special case of that, and perhaps you should look there for material to split off.
On the second point, there is no such thing as the "correct signature". It is made abundantly clear in the article. The article takes on a neutral POV (hence the ±) with respect to the signature, except in the very last section where a choice is made explicit for preservation of space.
On your third point, please tell me exactly what you find too complicated. Verdicts as "article too complicated" are useless. In the process, I suggest that you bring up the latest version side by side with the version just prior to my edits (that I gather from my talk page that you fancied more). Then you can compare section by section, and tell me which section/paragraph/sentence has not improved. Note that some statements from the old version are totally gone. I found those too unclear to keep. The intention is to reformulate some of these and restore them, some in the geometry section. Please keep in mind that things should be presented as simply as possible, but no simpler than that. YohanN7 (talk) 14:05, 19 July 2015 (UTC)
For example: I find what is said about the quadratic form η(v, v) and just very confusing.
it says , but also
are η(v, v) and the same thing?
Also how to combine " " and " Two vectors v and w are said to be orthogonal if η(v, w) = 0. " (so Two vectors are said to be orthogonal iif one of them = 0?) and later " they are to be mutually orthogonal vectors { e0, e1, e2, e3 } such that"
somewhere I am missing something or getting confused.
You do explain there are two different signatures (very well) but just use only one signature in the article (it is a bit like driving on the right and driving on the left they are both safe, but only if everybody stays at the same side) WillemienH (talk) 05:55, 20 July 2015 (UTC)
I took the liberty of indenting your reply. It is our way of keeping track of who is responding to what. I'll work backwards through your points.
  • As I tried to demonstrate above, the article is neutral with respect to the signature. It is therefore you see the ± in places. With a particular choice, the ± wouldn't be there. I also added a note in the article to that effect in response to your first post. The sole exception is the last section, to save space. A choice is there made explicit. To get the other choice, flip sign (−→+, +→ −) in (the matrix representation of) η. I honestly don't know how to make this more clear.
  • The non-degeneracy condition says in words that no vector is orthogonal to every vector. Each vector can be orthogonal to some vectors, but not all unless it is itself the null vector. The way this is formulated is standard. Maybe the explanation in words should go in?
  • It is mentioned (in the "intro" section of "Mathematical structure") that the metric tensor, the bilinear form, the Minkowski inner product are all the same object, and that it is good to keep that in mind. You can also add the quadratic form to that list since the polarization identity allows you to pass back and forth between a bilinear form and a quadratic form. But that would be an unusual way of promoting a quadratic form to a full blown tensor. The matrix [η] also refers to the same object. It, however, is basis-dependent. This does not mean different appearance in cartesian coordinates for different inertial frames, but it means different appearance in, say, spherical coordinates. I am emphasizing this because it is the major key to all of special relativity. The invariance of the interval (now first thing of substance in lead) forces coordinate transformations between inertial systems to take the form of Lorentz transformation. These are exactly the ones that preserves the interval, or , equivalently the bilinear (or quadratic) form with the matrix η being what it is in these coordinates.
I see a spot or two where I can use a better notation and a better formulation. For instance, η(v, v) and η are not really exactly the same thing. One of them is the metric tensor/bilinear form/Minkowski inner product/quadratic form when given two equal arguments, i. e. a number that it prodces, while the other is the metric tensor/bilinear form/Minkowski inner product/quadratic form, i. e. a function that will produce a number when fed arguments. (Standard notational abuse in physics, but not math.) I'll make a small edit (in due time).
You have not mentioned the involvement of tangent spaces in the article. I thought that this may be the major obstacle for understanding for most, since the topic of manifolds, from which this concept stems is almost always introduced later than introductory special relativity. The rationale for introducing tangent spaces (which I kept from the previous version) at all here is that it facilitates comparison with general relativity. Another rationale is that the article is not only aimed at the beginning relativist/geometer/whatever. It is (in an ideal state) supposed to be a reference for people familiar with the subject, but perhaps being rusty needing to look up some definition (this article is very much not there yet). Do you think that tangent spaces are treated in a comprehensible fashion? YohanN7 (talk) 09:40, 20 July 2015 (UTC)

Thanks for your explanations , I do think I miss interpreted the non-degenerate condition. I guess I still needs to learn more. I did put an hatnote refering more to Pseudo-Euclidean space. Maybe better to copy bits to there (and start subsections on 2 dimensional and 3 dimensional Pseudo-Euclidean spaces). Maybe I am getting more and more confused: There seem to be two unrelated types of hyperbolic plane (the "2 dimensional plane in hyperbolic geometry" and the " hyperbolic plane as Isotropic quadratic form") two or even more unrelated types of Minkowski plane (the Benz plane one, the affine geometry one, and maybe even more see http://math.stackexchange.com/q/1352447/88985 ) on introspection this article seems to be about the four "dimensional" Isotropic quadratic form and I am just getting confused.

I don't like the always indent one more indenting structure, my ideas is to keep it less indented, just the op (original poster) has no indention, the first reply-er gets one intention, the second reply-er gets 2 indentions, and reply ers keep their indention level as they go along, it is just a much more efficient use of screenspace . (but I guess you don't agree with this) Thanks for your eplanation, ps I am still looking forward to your reply at Talk:Hyperbolic geometry#GA nomination and archivingthat is all for now.WillemienH (talk) 16:27, 21 July 2015 (UTC)

Someone with better knowledge of geometry should chip in. My knowledge is limited. But it does include the fact that "Minkowski geometry" is somewhat of a misnomer. The Minkowski metric is a pseudo-Riemannian metric (standard terminology). But it is not Riemann that is pseudo, it is the geometry. Only when you restrict to certain subsets (hyperboloids of dimension one less than spacetime) you get a space with a true geometry (including lengths, angles and all). See surfaces of transitivity of spacetime. YohanN7 (talk) 18:34, 21 July 2015 (UTC)
You can interpret that as the hyperboloid (and hence the other model spaces) being isometrically embedded in spacetime when given the metric induced by the Minkowski metric. YohanN7 (talk) 18:59, 21 July 2015 (UTC)

Instead of Minkowski space why not use 4D Euclidean space with meters=i*c*seconds?[edit]

Einstein shows in Appendix 2 of "Relativity" that Minkowski space is formally equal to 4D Euclidean space if you make the simple substitution meters=i*c*seconds where i*c, not c, is the exact and universal conversion factor between meters and seconds that needs to be incorporated into Planck units to make them a lot simpler by removing several of the "units" altogether, not just making the constants "1". A lot of contradicting talk over "speed" in special relativity examples crop up from not doing this because not using this more strictly makes people think c is a constant and "speed" is a variable (speed is unitless if you take meters and seconds being equal seriously, so it can't be a physical measurement) when the opposite follows Occam's razor more closely. If you let c be defined for a particular reference frame, then length, meters, mass, and entropy do not change for any reference frame. Wouldn't 1 physical quantity varying instead of 4 be nicer? Lorentz transformations would no longer be needed in theory if the experimenters insert the Lorentz adjustment when they measure in meters, mass, and I think entropy. Seconds could remain the same. Ywaz (talk) 13:09, 9 August 2015 (UTC)

I found your reference to Relativity here. Making the analogy to 4-D Euclidean space is helpful,(and in-source) but your point about making the speed of light frame-dependent is not in the source. (WP:OR)
What frame is a photon tied to if its speed is dependent on a frame? The speed of the source object? Destination? Something else? The speed of light in a frame would change depending on the observation frame as well. A reliable source might bring some more specificity on the math here. Forbes72 (talk) 04:16, 2 September 2015 (UTC)
Why give Einstein credit when he is merely restating what Minkowski had previously done? Minkowski in 1907 had taken units with c=1 which is equivalent to what you are proposing isnt it? The attempt to interpret Minkowski space as Euclidean is flawed because it just is not Euclidean. JFB80 (talk) 04:39, 5 September 2015 (UTC)
ict is out of fashion. It used to be popular for introductory texts, but rarely for the advanced texts, and never in general relativity. It deserves mention though, as an existing variation. YohanN7 (talk) 12:57, 29 September 2015 (UTC)

Minkowski manifold[edit]

Should Minkowski manifold redirect here? — Preceding unsigned comment added by 70.247.173.205 (talk) 21:00, 28 February 2016 (UTC)

After Googling it, I do not see enough usage of that phrase to justify having any redirect. JRSpriggs (talk) 02:37, 29 February 2016 (UTC)
A Manifold is a special type of a Topological space. All manifolds are spaces, but not the other way around. Spirit Ethanol (talk) 18:21, 2 March 2016 (UTC)

Nonsense ?[edit]

Again I have reverted ([1]) this unsourced nonsense:

The signature (+,-,-,-) which has been adopted here has several physical implications. Principally it is positive for physically observable events and for the difference between two such events. The attempt to keep a Euclidean analogy by using the opposite signature fails completely.

- DVdm (talk) 17:24, 1 September 2016 (UTC)|

We could perhaps add (the essence of) the following:
  • The groups O(p, q) and O(q, p) are isomorphic
  • Interchanging O(p, q) ↔ O(q, p) amounts to flipping sign in some definitions. For instance, an interval is "timelike iff ds2 < 0" in one convention becomes "timelike iff ds2 > 0" in the other. Physics, i.e. in this case the essence of timelike does not change.
  • Examples of notable figures adopting one or the other notation. I have in mind Einstein (mostly minus) and Weinberg (mostly plus). None of them crackpots, and well aware that one could equally well adopt the other convention. Don't know if it is true, but there is a story about Einstein getting the question; is there physics in the choice?. E replied "yes" while trying to keep a straight face.
That takes care of the first part in the quoted edit summary. The last part can be taken care of by noting that the x = (x1, x2, x3, ict) convention makes the effective Lorentz group a subgroup of O(4, ℤ). This does make some formulas appear to be Euclidean, and is made precisely to hide the indefiniteness of the metric. It used to be popular, at least at the introductory level. This convention too has no physical implications. The middle part in the quoted edit summary is nonsense (in any convention). YohanN7 (talk) 09:21, 2 September 2016 (UTC)
No problem with that. But please, whatever you add, make sure you have a proper source at hand Face-smile.svg. See, for instance this: [1]

References

  1. ^ Wald, Robert M. (2010). General Relativity. University of Chicago Press. p. 11. ISBN 978-0-226-87037-3.  Extract of page 11
- DVdm (talk) 09:29, 2 September 2016 (UTC)
I don't plan on adding anything at the moment. You or someone else might? YohanN7 (talk) 09:39, 2 September 2016 (UTC)
"Nonsense" is your personal opinion. Clearly you have not understood my remarks so you call them nonsense. I don't understand your group remarks either. Should I therefore call them nonsense? They certainty do not add any clarity. Groups should not come into it; they just make things unnecessarily abstruse and befuddle the simple picture. Also, it should not be necessary to quote sources for simple, easily verifiable, mathematics. Should we source 1+1=2? It was a welcome change when Yohani recently switched over from +++- to +--- in one section. But he didn't finish the job which I tried to do adding a few comments as well. For a long time the Minkowski article has had a bias towards the choice of signature +++-. Choosing this signature is common but it makes little sense having few useable properties and further does not at all retain the character of Euclidean geometry as was shown in detail by Robb in his book (The Geometry of Space and Time 1936 Cambridge) The signature +--- (Landau-Lifshitz convention) on the contrary makes a lot of sense: it is positive for time-like events and intervals which are the physically realizable ones since they correspond to motions with velocities less than that of light [sic: yes it is not nonsense, just simple algebra - no groups or fancy things!]. These are governed by a partially ordering relation > meaning one event is reachable from another at a velocity less than that of light. Further the pseudo-norm has an important inequality, the reversed Cauchy inequality, from which can easily be deduced a corresponding reversed triangle inequality. To verify these properties is just simple algebra which to add here would need a lot of Wiki markup and would take me some time to do. The only source reference really necessary is to the reversed Cauchy inequality which I can easily give. One reference, which I believe to be the original one, is Aczel Ya: Some general methods in the theory of functional equation in one variable, Usp Mat Nauk 11 1956 3-68. As for Yohani's comment about Minkowski's ict and the representation as a 4-dimensional sphere it is important to notice that this is not an ordinary Euclidean sphere but a sphere with imaginary radius. Changing sign it becomes a sphere with Landau-Lifshitz radius equal to cτ, τ being Minkowski's proper time. JFB80 (talk) 18:32, 2 September 2016 (UTC)
Re "Also, it should not be necessary to quote sources for simple, easily verifiable, mathematics": This is not just a routine calculation as in wp:CALC, and this is not about mathematics. Your quoted remark is about physical meanings of mathematical equations. So for statements like "Principally it is positive for physically observable events and for the difference between two such events," we need a source. I doubt that you can find such a source because the statement is just wrong. Please correct me if I'm wrong, but I assume that your "physically observable events" are events within the past light cone, so an observer at the origin can "physically observe" them by seeing them. Now just take timelike events with coordinates (t,x,y,z) = (-2,-1,0,0) and (-2,1,0,0) which are both "physically observable" and which indeed have positive squared interval. The "difference of these events" clearly has not, as they are spacelike related. - DVdm (talk) 19:05, 2 September 2016 (UTC)
I see that what you understand as 'observable' are events the past light cone and I am meaning events in the future light cone or more generally accessibility of one event from another by a light signal. But in either case we are talking about time-like intervals which have positive +--- signature. So simple. My remarks were only intended to explain that it is the +--- signature which is actually the useful one. JFB80 (talk) 20:45, 2 September 2016 (UTC)
And their "difference" can have negative +--- signature. So simple too Face-smile.svg. And note that some autors clearly find -+++ signature useful. See? That is why Wikipedia has this wp:policy about wp:verifiablity.- DVdm (talk) 21:28, 2 September 2016 (UTC)
You are able to produce your counter-example because you are using your own definition. So I will quote from the article itself (which incidentally is not sourced but you did not worry about that - your insistence on the rules appears to be selective)
(1) Chronological and causality relations: x chronologically precedes y if y − x is future-directed timelike. This relation has the transitive property and so can be written x < y; x causally precedes y if y − x is future-directed null or future-directed timelike. It gives a partial ordering of space-time and so can be written x ≤ y. (2) Reversed triangle inequality: If v and w are both future-directed timelike four-vectors, then in the (+ - - -) sign convention for norm, |v+w|>=|v|+|w|.
Do you have a counter-example to disprove these statements? JFB80 (talk) 04:12, 3 September 2016 (UTC)
Yes, wp:OTHERSTUFFEXISTS and wp:OTHERCRAPEXISTS. By all means, when you encounter unsourced problematic text, feel free to remove it, or, if it has been sitting there for quite some time, tag it with {{citation needed}}, or the article or section as {{unsourced}}. - DVdm (talk) 08:24, 3 September 2016 (UTC)
Do you accept what is said in these article quotations or not? If not please explain your objection. Do you consider it also "nonsense"? JFB80 (talk) 21:17, 3 September 2016 (UTC)
This talk page section is about my not accepting your "Principally it is positive for physically observable events and for the difference between two such events." If you insist on having this in the article, you will need a source—see wp:BURDEN. - DVdm (talk) 09:00, 4 September 2016 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── @JFB80, since you don't understand groups, let's skip the Lorentz group. The end points of a meter stick at any given time in a particular Lorentz frame in which the meter stick is at rest are two events that are very much measurable. You simply record their spatial locations (xi) (you can use the meter stick itself for the purpose) and note the common time. Now take their difference and compute the interval,

.

This is positive with the (−, +, +, +) metric and negative with the (+, − ,− , −) metric. Right?

Is this a "physical implication" of any of the two conventions? No, putting in such an interpretation is nonsense – in any convention. What happens is that you get a sign flip in the definition of proper length in terms of the interval, illustrating a point (one of the bullets) I made in my first post.

By the way, this example illustrates that restriction of the (−, +, +, +) metric to space yields the Euclidean metric. (But this is not important.)

Moreover, your edit made the rest of the article inconsistent. The rest relies on the choice of the (−, +, +, +) metric for definiteness and preservation of (article) space. Spelling out both choices is just a waste of space since they are trivially related. If you feel religious about the (+, −, −, −) signature, by all means change, but do so consistently throughout and please, please don't add anything about physical implications. There aren't any. YohanN7 (talk) 09:32, 5 September 2016 (UTC)

Yohani I am glad you wrote, I wasn't making much progress against the cries of nonsense and the barrage of Wiki Law! Yes I do know about groups but often they are quoted in a way which is not enlightening. I am saying two related things. (1) Minkowski's terms 'space-like' and 'time-like' tend to give the wrong impression. It is time-like events which have a physical reality and space-like events are the unrealizable ones. Take your example of measuring a length. A simultaneous measurement of the ends gives positive (-+++) signature i.e. space-like. But you can't observe both ends simultaneously - it is not physically realizable in relativity. The two ends have to be visited to observe them by some means moving less than or equal to light speed. The observations you make are time-like with positive (+---) signature. (2) The bilinear form and pseudo-metric with (-+++) signature have no useful mathematical properties and lead to a collapse of familiar geometrical properties. But with (+---) there is a nice theory relating partial ordering, reversed Cauchy inequality and reversed triangle inequality as in the WP article excerpt I quoted. This is incidentally very much related to the hyperbolic theory of Special Relativity. That's all - quite simple and verifiable without source references (which are usually quite unobtainable anyway). JFB80 (talk) 14:44, 6 September 2016 (UTC)
I don't think that Minkowski's terms 'space-like' and 'time-like' tend to give the wrong impression. On the contrary. Unless "unrealizable" has a special meaning, there is nothing "unrealizable" about space-like events. Your phrase "A simultaneous measurement of the ends gives positive (-+++) signature" again sounds like nonsense. A length is positive. Its square is positive, and minus its square is negative. Nothing more, nothing less. Note that without source references none of this will be taken in an article, specially when "unobtainable anyway". Also note that we are close to disrupting this talk page. We are supposed to discuss the article here, not the subject - see wp:talk page guidelines. This talk page is not a forum (see warning at the top) and no place to vent our thoughts about the subject. So unless you have a proposal to make a concrete edit to the article, backed by a proper reliable source, this discussion should probably be closed. - DVdm (talk) 15:29, 6 September 2016 (UTC)
I agree. It seems to me the vast majority of GR books describe the choice of metric signature as a sign convention. I'd suggest that term be used in the article, rather than the more ambiguous phrases "not standardized" or "left open". By definition, a choice of sign convention has no "physical implications" --ChetvornoTALK 16:07, 6 September 2016 (UTC)
Probably a good idea. I went ahead and did this, which has the benefit of adding two wikilinks. Feel free to hone. - DVdm (talk) 16:26, 6 September 2016 (UTC)
Chetvorno: 'A choice of sign convention has no physical implication'. Isn't c²dt² - (dx²+dy²+dz²)>0 (time-like) the condition for velocity to be less than that of light? (Einstein's 2nd postulate). Positive (+---), negative (-+++) If you don't think this suggests a sign convention then whatever would? JFB80 (talk) 19:47, 6 September 2016 (UTC)
This is off-topic. See wp:talk page guidelines. - DVdm (talk) 21:04, 6 September 2016 (UTC)

────────────────────────────────────────────────────────────────────────────────────────────────────@JFB80. Finally you say something that is precise enough to be responded to properly, because it is not again of the sort not even wrong. This time it is finally clear. What you say is simply false or plainly wrong. This is an improvement over nonsense, a term that you probably don't like and a term that becomes unavoidable (of sorts) when you or anyone else is less than crystal clear. This is good.

  1. Minkowski's concepts of timelike and spacelike are unambiguous and independent of coordinate systems and metric conventions. Period. If you cannot see this, I can't be of help.
  2. You have the wrong impression of "physical reality". Not "wrong" in the sense that "I do not agree", but rather wrong period. I'll point out one misconception below.

You need to get acquainted with the concept of Lorentz observer. I'll not spell out the details, but a Lorentz observer is someone who has access to a complete record of spacetime events in his Lorentz frame. The concept requires "observers" at arbitrary points in space of the Lorentz frame in question, at rest relative to the coordinate system. These observers have synchronized clocks. They report to a main office. The Lorentz observer can read all their reports. Lorentz observers provably exist in principle (can be realized), see for example *Sard, R. D. (1970). Relativistic Mechanics - Special Relativity and Classical Particle Dynamics. New York: W. A. Benjamin. ISBN 978-0805384918.  In particular, they can be realized in practice when it comes to noting the end points of a meter stick at an instant.

How accepted do you think special relativity would be if we could not measure the length of things at rest in our world?

Thus when you talk about undefined concepts such as "physically realizable events". not having understood the meaning of "Lorentz observer", then you will in the first place either be misunderstood or tossed off as being a crackpot. Invariably. Period. If you have a reference defining these things of yours, just burn it.

Finally, the "reversed Cauchy inequality and reversed triangle inequality" have analogues in the (, +, +, +)) metric. The inequalities in this case reverse direction (to the standard direction) with the price being paid absolute signs. Thus the signatures each have imperfect versions of theorems that hold in true inner product spaces. Also, the section about causal ordering does not refer to the signature of the metric. It is an thereof independent concept. Nothing new under the sun.

I urge you to think twice or even ten times deeply before you proceed to respond. I don't wish to spend much more time on this and had in fact promised myself not to respond further because experience suggests that it is a total waste of time. I will respond, but only if I can see that you have read up on the basic concepts and use a language common to us both, free of your own definitions. YohanN7 (talk) 06:57, 7 September 2016 (UTC)

As this is not directly (or even indirectly) about the article anymore, would you mind continuing this on your user talk page(s)? Perhaps you could move this to User talk:JFB80. Thx. - DVdm (talk) 08:05, 7 September 2016 (UTC)
JFB80 has changed the article (getting reverted) and argued here why the sign convention is, according to him, obvious. Responding to those arguments is very much about the article. YohanN7 (talk) 06:58, 8 September 2016 (UTC)
Agreed. But the article is properly sourced and so are the responses, while his arguments are not. If we don't cut this short, anyone can go to any article talk page and start an endless argument about why they don't agree with something in it. I know, it can be a thin line. Cheers. - DVdm (talk) 07:26, 8 September 2016 (UTC)
You have a point. If old arguments about "measurable events" (the big misconception) are simply repeated, we should cut it short. New arguments and references belong here. There are rationales for choosing particular sign conventions, usually to avoid ugly minus signs. This is usual for particle physics where the (+, −, −, −) convention is common, but not universal even within particle physics. An arguments for the (−, +, +, +) convention is that it naturally extends the standard metric on space, (3), and metrics are traditionally associated with distance measurements rather than time differences. (The best references usually do not motivate their sign convention with a single word though. It is taken as part of the notation, and that's it.)
Such arguments can go into the article, but the article should not take sides. YohanN7 (talk) 07:45, 8 September 2016 (UTC)
I agree with DVdm. Wikipedia editors are not allowed to include their own conclusions from sources in articles, that is WP:SYNTHESIS. @JFB80: all your arguments above are irrelevant, and are inappropriate for the Talk page. If you want to put the phrase "The attempt to keep a Euclidean analogy by using the signature (+,-,-,-) fails completely", or any other phrase preferring one signature over the other in the article, you have to find a source that says that explicitly. And not just one, but show that it is the general opinion of relativity experts that your favorite signature is better. And then find some source that explains why half the GR texts out there use the "wrong" signature. So far, your arguments are just WP:OR.
@YohanN7: I'd like to see a section discussing the rationales of using the different signatures (making clear that the only difference is notational convenience and not "physical implications") but again, it must be sourced. --ChetvornoTALK 08:44, 8 September 2016 (UTC)
@YohanN7: Thank you for your remarks. I wish you would allow me to discuss them with you. May I merely correct misrepresentation? (1) I never said anything to contradict what you say about Minkowski's definitions space-like, time-like. (2) I never said the length of a rod could not be found. It just has to be done indirectly with physically realizable observations (something like a Lorentz observer). @Chetvorno: re. 'attempt to keep Euclidean analogy ...' I did source it with Robb 1936 who discusses it at length. Please read the earlier part of the comments JFB80 (talk) 15:46, 8 September 2016 (UTC)
LATER: Robb's book is on the internet. See Wiki article Alfred Robb for link to 'Geometry of Time and Space', Introduction, pp.1-5. That's a source reference which can actually be read! JFB80 (talk) 18:59, 8 September 2016 (UTC)
I glanced through pages 1–5. Nowhere does he define ημν, and even less is there anything remotely like

The signature (+,-,-,-) which has been adopted here has several physical implications. Principally it is positive for physically observable events and for the difference between two such events. The attempt to keep a Euclidean analogy by using the opposite signature fails completely.

stated. It looks more like he is doing something along the lines
where the last step (not given by Robb) seems to be customary in string theory. Note the minus sign in front. Note too that the (−, +, +, +) metric is in effect in string theory. See Zwiebch, A First Course in String Theory for this variant. It is all a matter of definition. As long as you are consistent, things are fine. Whether Robb defines the metric tensor later (in any which way), I don't know. It is irrelevant to this article (but relevant to his book). YohanN7 (talk) 13:21, 9 September 2016 (UTC)
Robb was quoted for 'The attempt to keep a Euclidean analogy' etc seems irrelevant. In you can change ds to cdτ where τ is Minkowski proper time and then what you need are Weierstrass coordinates. You will notice that changing c to ic transforms to a Euclidean sphere.JFB80 (talk) 04:52, 10 September 2016 (UTC)
Changing t to it transforms (-+++) to (++++). See Special_relativity#Comparison_between_flat_Euclidean_space_and_Minkowski_space:

The actual form of ds above depends on the metric and on the choices for the X0 coordinate. To make the time coordinate look like the space coordinates, it can be treated as imaginary: X0 = ict (this is called a Wick rotation). According to Misner, Thorne and Wheeler (1971, §2.3), ultimately the deeper understanding of both special and general relativity will come from the study of the Minkowski metric (described below) and to take X0 = ct, rather than a "disguised" Euclidean metric using ict as the time coordinate.

Perhaps that is what you have in mind? - DVdm (talk) 09:11, 10 September 2016 (UTC)
Are saying he does not choose a metric signature because it is pointless because it will not look like the Euclidean metric? All this seems to be about choice of definition of the interval. This is not the same thing as choosing a metric.
The metric (with either signature) is used extremely frequently in the literature most relevant to the concept of Minkowski space, and in just about every calculation. It is definitely time to close this discussion. YohanN7 (talk) 09:29, 10 September 2016 (UTC)

Another problem[edit]

I'm not too fond of this passage:

Vectors v = (ct, x, y, z) = (ct, r) are classified according to the sign of c2t2 - r2. A vector is timelike if c2t2 > r2, spacelike if c2t2 < r2, and null or lightlike if c2t2 = r2.

It is unusual to speak events as being timelike, etc. Is would not be a fully Lorentz invariant concept (i.e. not Poincaré invariant). It should be emphasized more that we are talking about intervals here, i.e. the interval between v and the origin (0, 0, 0, 0). This is a fully Lorentz invariant concept. For some 4-vectors, e.g the momentum 4-vector, this works automatically since there is an implicit derivative. But the event (0, 2.5 million light-years, 0, 0) referring to an event in the Andromeda galaxy 2000 years ago (according to us) is according to us spacelike, but the Andromedans might disagree.

What makes this confusion possible is the canonical identification of tangent spaces with the spacetime manifold itself. This does not work in general relativity. YohanN7 (talk) 07:17, 8 September 2016 (UTC)

How do they formulate this in the literature? Why not pick one and add an inline source? - DVdm (talk) 07:26, 8 September 2016 (UTC)
I'll check out L&L, Goldstein and Kleppner&Kolenkow when I get at them. If you get at it before me, it is well, because it will take a me few days. YohanN7 (talk) 07:55, 8 September 2016 (UTC)
Google Books is our friend:
  • Landau & Lifshitz mention intervals of events being timelike/spacelike: [2]
  • Kleppner & Kolenkow mention events: [3]
  • Goldstein is not available: [4]
  • Hazewinkel talks about vectors, just like we do in the article: [5]
I think it depends on whether the approach is more physical or more mathematical, events leaning to the former, vectors to the latter. Our article is somewhere half-way in between I guess Face-smile.svg... - DVdm (talk) 08:26, 8 September 2016 (UTC)
Since the article is about Minkowski space and it was Minkowski who gave these names timelike, spacelike, why not start from what he said - or at least make mention of it? JFB80 (talk) 02:03, 25 September 2016 (UTC)
That's exactly what the article does. Sea also the history section Minkowski space#Minkowski space. In the original [6] Minkowski used "zeitartigen Vektoren" and "raumartigen Vektoren". In the translation [7] they use "time-like vectors" and "space-like vectors". We probably dropped the dashes and use "timelike" and "spacelike" as they are more common in the literature. - DVdm (talk) 09:17, 25 September 2016 (UTC)
I am not talking of German/English translation. You checked out what was said by Landau & Lifshitz, Kleppner & Kolenkow, etc but didn't think to checkout Minkowski himself as though that was of no importance.JFB80 (talk) 08:21, 26 September 2016 (UTC)
I agree with your edit. In my previous reply above I checked Minkowski himself, even if Minkowski is not the most important source, as our articles should reflect the current literature. Compare that with Maxwell's original writings and our article on Maxwell's equations, or with Einstein's original writings and our article on special relativity and mass-energy equivalence. The original wordings can of course be mentioned in an historical context. - DVdm (talk) 09:00, 26 September 2016 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── Perhaps this [8] is acceptable. There is absolutely nothing timelike, spacelike, or lightlike about events per se. But when one refers to events as vectors, then an origin is implied (we then have a coordinate system on the spacetime manifold). YohanN7 (talk) 08:27, 29 September 2016 (UTC)

Concerns about the poor description of Minkowski's ideas[edit]

Concerning current literature, I think that Minkowski's paper is wonderful compared with the confused version given in this article which is largely unsourced and apparently 'own research' (but that doesn't trouble anyone!). For example Minkowski did not define a norm (there isn't one), he talked about conjugate vectors. But he then introduced the proper time relation which in effect gives the +--- norm (as per my 'unsourced nonsense') and relates directly to the Schwarzchild solution of General Relativity. JFB80 (talk) 17:07, 29 September 2016 (UTC)
Minkowski's paper is referenced.
Then be specific: What is largely unsourced and apparently 'own research'? Give a list. No general hubbub, give special short examples, and we'll fix them if they are OR or need sourcing.
But he then introduced the proper time relation which in effect gives the +--- norm (as per my 'unsourced nonsense') and relates directly to the Schwarzchild solution of General Relativity. Face-smile.svg
You can totally forget bout introducing archaic terminology (conjugate vectors) YohanN7 (talk) 08:49, 30 September 2016 (UTC)
Of course Minkowski's paper is referenced but I would like to see its ideas talked about in this article, not just 'time-like' and 'space-like' (which are incidentally unsourced to Minkowski). You may regard his ideas as old-fashioned compared with your own advanced ideas but, even so, they deserve mention in an article supposedly on his ideas, not just yours and those of other advanced thinkers like you. As I said 'Minkowski norm' (or 'Minkowski metric' as it has now become) is a misnomer because Minkowski did not define it and he did well because it just isn't a metric. So Minkowski metric needs proper sourcing to the person who first introduced this dodgy idea. Preferably it needs deletion with just a note about its attempted use. And who first used the two forms -+++ and +---? Generally speaking all the pseudo-metric generalities and tensor talk is just pure mathematician's padding (= own research) with little relevance to the real use of Minkowski space in physics. For appropriate sourcing you may also like to search in the sections: pseudo-metric generalities, standard basis, causal structure, chronological and causality relations, reversed triangle inequality, flat versus curved space.
Also you make fun of my attempt to say something important about the relation of Minkowski's version with general relativity'. In my 'unsourced nonsense' above I quoted Minkowski's basic relation c²dτ²=c²dt²-(dx²+dy²+dz²)(>0) only to have it immediately deleted (censored) by DVdm as 'off-topic'. May I suggest you actually look at the article on Schwarschild metric (#The Schwarzschild Metric) and compare it with the censored Minkowski equation? I notice that the article does say, to its credit, that the GR metric reduces to the Minkowski form in certain circumstances so maybe you would not also be laughing at that although you would continue in your conviction that the -+++ form is essential for a relation with GR.
But I feel rather bored to continue this particular discussion as I try to make such a simple and important points without result and more hours have now been wasted trying to explain to someone determined to misunderstand me. JFB80 (talk) 22:18, 30 September 2016 (UTC)
You add too many points for one post. This is very frustrating. I'll reply to one of them and comment on one other and point out a misconception of yours. Please address one point at the time in the future.
  • Reply:The use of terminology is explained in "nb2" (a footnote). There simply is no standard terminology for "inner product thingie", "norm thingie" and "metric thingie". But the article has to talk about these thingies, like it or not. They are important because they are used. The thingies are thus given names for local use in this article. You might find it instructive to read this Physics Forum link. In it you'll find references to "real" literature. The link illuminates very well what is going on. Namely, the name of the things varies, and the metric signature is the signature the author in question happens to use.
  • Comment: You talk about my "conviction that the -+++ form is essential for a relation with GR". What? Where? Really, you are the one (probably the only person in the world) that thinks there is one and only one choice of signature. I have said before that you can switch to +−−− in the few places where a definite choice is made (for space saving purposes). But do so consistently, and add no drivel about physical consequences that aren't there – or that one or the other choice is "superior".
  • Misconception of yours: You claim "c²dτ²=c²dt²-(dx²+dy²+dz²)(>0)". No, there is an error. I leave it to you to find it yourself. It will teach you something. It is probably the source of this whole affair.
Please stick to one point at the time. We can sort them all out. YohanN7 (talk) 10:16, 1 October 2016 (UTC)
I did have a look at Schwarzschild metric. It says
In Schwarzschild coordinates, with signature (1, −1, −1, −1) the line element for the Schwarzschild metric has the form...
This means to me that instead using signature (−1, 1, 1, 1) it will look slightly different. So? Obviously you interpret this very, very differently. YohanN7 (talk) 10:35, 1 October 2016 (UTC)
Thank you for your reply and I will be answering. It needs thought. JFB80 (talk) 13:40, 1 October 2016 (UTC)
  • Minkowski norm I have looked at Physics Forum. They are just saying various conventions are in use which we already know. I also looked at Giulini - terribly clever and sophisticated stuff but he misses the point as they all do. I just follow Minkowski, who no-one thinks of quoting, and think he got it right. The equation c²dτ²=c²dt²-(dx²+dy²+dz²)(>0) was given by him and I still believe it 100% correct. I wish you would reveal the secret of what you think is wrong with it. JFB80 (talk) 18:40, 1 October 2016 (UTC)
Jrheller1 has pointed out that the +--- convention is consistent with the known relation while the -+++ convention is not. Would you be calling this 'drivel' ? JFB80 (talk) 11:33, 2 October 2016 (UTC)
I made a correction at your talk page. The -+++ metric is actually written which is consistent with the correct value of the Lorentz factor But this means that the infinitesimal amount of spacetime distance, is imaginary. With the +--- metric, the infinitesimal amount of spacetime distance is real. Personally, I prefer the +--- metric for this reason. Jrheller1 (talk) 16:54, 2 October 2016 (UTC)
YohanN7, you claim there is an error in This is definitely the correct form of the +--- metric, and it is definitely greater than 0 for any material object since nothing can travel faster than light. So what is the error? Jrheller1 (talk) 17:04, 2 October 2016 (UTC)
The error is the last part. Not all intervals are timelike. Some are lightlike and some are spacelike. The equation
is holds independently of metric convention (and is actually a definition of proper time), but isn't defined for all intervals. It simply has nothing to do with the (1, −1, −1, −1)/(−1, 1, 1, 1) discussion. Then that the Lorentz factor would be dependent on metric convention is, ...., well, ... . The definition of timelike, etc has nothing to do with sign conventions in the metric. YohanN7 (talk) 09:25, 3 October 2016 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── People, does someone have a specific proposal to change something, backed by a relevant source, to the article? If not, we are discussing our personal preferences and views of some aspect of the subject of the article, which is not allowed per the wp:talk page guidelines. If no one has a specific proposal, this discussion should be closed here. It can of course be continued on one of our user talk pages, like for instance User talk:JFB80#Spacetime metric sign convention. Thanks. - DVdm (talk) 18:58, 2 October 2016 (UTC)

I agree. This is not the place to discuss what "we think" about the Minkowski norm, that is not relevant to the article (WP:TALK#OBJECTIVE). Unless someone has a specific proposal for new wording in the article, this discussion should be closed per WP:TALK#USE.--ChetvornoTALK 08:42, 3 October 2016 (UTC)
I disagree because the confusion shown in this discussion is reflected in the confusion in the article. I notice that DVdm very conveniently stepped with Wiki-law avoiding an answer being given to the crucial question of Jrheller1 just as he did when I previously asked the same question. As a specific proposal I suggest the concept Minkowski metric should be properly historically sourced making it clear that Minkowski himself did not use it. JFB80 (talk) 03:08, 3 October 2016 (UTC)
We might be making progress. Ok, please make a specific suggestion here below by providing the exact source, and what you'd like to change to the article. - DVdm (talk) 07:10, 3 October 2016 (UTC)
Chetvorno, please note that I am actually defending the article from personal preferences. It is others that want personal preferences in, making believe there is "physics" in them, like here. It is pretty silly to suggest that either of the likes of e.g. Weinberg, Landau and Lifshitz (they use different metric signatures) "miss" the physics because they supposedly misunderstand high school school math. YohanN7 (talk) 11:06, 3 October 2016 (UTC)

The tone~and Changing posts replied to...[edit]

My tone of discussion isn't the best in my latest post. Well, there are reasons. That is, there were reasons. After my last reply, JFB80 went back and changed his post from downright offensive to more neutral words. This would suffice for someone else than me to take it to the higher powers. YohanN7 (talk) 09:04, 3 October 2016 (UTC)

Indeed, that should not have happened. JFB80, please have a careful look at WP:REDACT and restore what you originally wrote. You can comment here. Thanks. - DVdm (talk) 09:12, 3 October 2016 (UTC)
DVdm The attitude of YohanN7 using such descriptions as 'gibberish' and 'drivel' had finally put me in a bad mood when I wrote the comment you refer to. But I soon wanted to change it using more moderate words. Perhaps you could suggest to YohanN7 to do the same. As for redacting I must ask you to do it as you are expert in such matters and I am not. JFB80 (talk) 10:24, 3 October 2016 (UTC)
Ok, I have restored the original. I propose we keep cool and let this rest now, and concentrate on your upcoming proposal here below. - DVdm (talk) 10:36, 3 October 2016 (UTC)
JFB80, the term "gibberish" (and also "nonsense") has rightly been used, not only by me, about your edits. "Drivel" was, as I recall, in response to the post that you subsequently changed. YohanN7 (talk) 11:11, 3 October 2016 (UTC)
DVdm Thank you for making these changes although I still fail to understand why I did not have the right to modify what I had said. Now I am now tired of going on and on trying to explain the obvious only to be ignored or misinterpreted (e.g. as initially with 'physical realizability'). So I will let the matter rest. I can give some thought to historical origins as suggested which hopefully is non-controversial. JFB80 (talk) 19:40, 4 October 2016 (UTC)
It is very important for you to understand why you can't go back and change what you wrote when people have replied. You know why you changed, right. You changed because it will make you appear in a better light. No doubt you also figured out that, as a nice side benefit, it will also make ME look a little worse. Right? Don't lie to yourself now. The bad tone on my part (and possible on part of others) initially came after you couldn't take getting reverted and promptly reverted back, here and elsewhere. You have only yourself to blame. I stand behind my bad tone towards you so far, but I am perfectly willing to change it (in the future, not the past) if you present fresh arguments in the future. (No, you being tired of explaining "the obvious" to us doesn't count. Take a Wiki-break!)
Oh, the importance for you lies in that you sooner or later will be blocked/banned if you don't get this near trivial point. But I can guarantee that I'll never bring this or similar matters to the ANI, but others will sooner or later – I can guarantee that too.YohanN7 (talk) 10:43, 5 October 2016 (UTC)
Actually you are wrong and I could explain but the talk page is not for settling personal disputes. Could you please transfer to a user page. JFB80 (talk) 06:33, 6 October 2016 (UTC)

Terminology[edit]

Is the treatment of the terminology in the article of the terms

  • Minkowski inner product
  • Minkowski metric
    Resolved
  • Minkowski norm

acceptable? They have long been here in one or another form. They aren't standard, but there is no standard to use. YohanN7 (talk) 14:47, 5 October 2016 (UTC)

Geometry[edit]

I removed the empty section "Geometry".

Should we have one? I think (but I am just guessing here, not my field) it is the case that the restriction of the Minkowski metric to one sheet of two-sided hyperboloids in fact yields a true Riemannian metric. An idea is to pull this metric back via a parametrization and explicitly demonstrate the result. Perhaps then from there calculate the (presumably hyperbolic) distance function (a true metric). Suggestions? YohanN7 (talk) 14:53, 5 October 2016 (UTC)

This would work. I found a reference, the Lee book recently added to the ref list. YohanN7 (talk) 09:59, 6 October 2016 (UTC)
 Done


Rest list:

  • Demonstration that H1(n)
    R
    is a homogeneous and isotropic (in the Riemann sense) space.
  • Calculation of geodesics This is relatively easy in the present hyperboloid model. They project down to the circles of the Poincaré ball model confirming the result.
  • Calculation of the metric (as opposed to the Riemann metric).
  • Calculation of the curvature = −1/R2.

To be honest, hyperbolic geometry is probably of less interest in special relativity than what special relativity is in hyperbolic geometry. Reason: The geodesics of H1(n)
R
are not geodesics of flat spacetime (in the same way great circles on the sphere aren't geodesics of 3). That said, I think it may be of value to the reader to see curved things embedded in flat spacetime like is now done. YohanN7 (talk) 14:21, 11 October 2016 (UTC)

Grammar bad[edit]

Under "History", the following phrase appears, "In 1905, and later published in 1906...". The grammar might be improved.

Wrong[edit]

The effort of 14:53 G.M.T. on 20/10/2016 seems to be wrong. — Preceding unsigned comment added by 92.26.14.17 (talk) 10:09, 21 October 2016 (UTC)

Please sign all your talk page messages with four tildes (~~~~). Thanks.
 Yes, corrected now. Thanks. - DVdm (talk) 10:20, 21 October 2016 (UTC)
Indeed, yes.
I made another edit aimed at clarifying the issue for beginners. It is not easy to spell out what is actually meant, and someone else may do it better. There is inherent notational abuse in tit. Some notation like φ(t, x, y, z) = (φμ(t, x, y, z)) = (x, y, z, it) ≡ (X, Y, X, T) would have been preferable, since then T = it and Tt explicitly. But nobody ever writes it that way. And we can't say the nonsensical tt "because the t on the left isn't necessarily the t on the right". YohanN7 (talk) 11:19, 21 October 2016 (UTC)
We should really have a subsection of the history section on the (time-varying) popularity of the ict convention. There is also historical terminology to be explained: The "east cost metric" and the "west coast metric". This could (should) be non-technical and hopefully entertaining to read. YohanN7 (talk) 12:30, 21 October 2016 (UTC)

Orthogonality[edit]

In the section Pseudo-Euclidean metrics the following appears:

The links refer to positive definite quadratic forms and are not appropriate here. The material further down in the article on space-like and time-like vectors needs to precede this discussion. For space-like vectors orthogonality is right, but hyperbolic orthogonality applies between a time-like and space-like vector.Rgdboer (talk) 01:59, 9 December 2016 (UTC)

I don't see how the links refer to positive definite quadratic forms only. As far as I can see they point to articles in which the non-Euclidean case is mentioned too. See, for instance Orthogonality#Mathematics_and_physics where hyperbolic orthogonality is indeed mentioned. LIkewise, see the section Orthonormal_basis#General_unit_vectors. So I think the links are indeed appropriate. - DVdm (talk) 08:55, 9 December 2016 (UTC)
In Minkowski space there is only one type of orthogonality, namely that with respect to the bilinar form, precisely as defined in the article, and for that matter by Minkowski.
It is standard as far as I know to refer to this as orthogonality without further qualification thus abusing terminology. The only reasonable alternative as I see it would to use the term pseudo-orthogonality, but this is simply not used. The article orthogonality instead should be updated with this extended use. Minkowski used the term normal, but that never caught on.
I am sure the complicated geometric interpretation in hyperbolic orthogonality would be interesting to some, but it will merely confuse most people (like Minkowski diagrams do).
To be very frank, I have a distinct feeling that it is the author of that article that has invented the term. I apologize in case I am wrong. I can't find it in reference 1 (it uses the term perpendicular), reference 2 is not available. Reference 3 (Minkowski) uses - as mentioned - normality. I can't figure out how to get to page 441 in reference 4 (only how to flip pages one-by-one). Reference 5 makes no mention. I don't particularly mind, but it should at least be addressed whether reference to hyperbolic orthogonality should remain in the relativity template and elsewhere. YohanN7 (talk) 09:59, 9 December 2016 (UTC)
"Hyperbolic orthogonality" is the same thing as orthogonality defined here. It is a distinction without any difference. JRSpriggs (talk) 03:23, 10 December 2016 (UTC)

Thank you YohanN7 for calling attention to the lack of naming reference. One has been supplied with online link. The author uses "inner product" including the indefinite case, not restricted as our inner product; we must use bilinear form. The author also develops hyperbolic numbers which clarifies the usage with orthogonality. In the interests of making Minkowski space a better article, the above comments were suggested to clarify the special nature of the temporal axis in orthogonality.Rgdboer (talk) 02:11, 12 December 2016 (UTC)

I put in a link after the definition. YohanN7 (talk) 07:36, 12 December 2016 (UTC)

"time itself"[edit]

The section on four-dimensional spacetime includes a useful note comparing time when thought of in a four-dimensional field as compared to "time itself" as when measured by a clock. My edit was reverted, but its intent was to eliminate this identifying "time itself" to be the experience of time as when measured by a clock (aka, sequentially/linearly). Newton might have considered time to be a sequence, but if Einstein proved anything, it is that Newton's understanding of linear time was insufficient and therefore could not be considered 'time itself.' Rather, by showing that physical phenomena like gravity were best modeled in Minkowski Space, Einstein was arguing that it is this 4-d conceptualization of time that should be thought of as 'time itself' and that the sequential perception of time as perceived by humans is merely a local perception of "time itself" (4-d spacetime) as interpreted by the mind of a great ape. My objection is only with that label; it is very useful to highlight how fundamentally different the two mathematical models of time are. Schray (talk) 20:18, 23 January 2017 (UTC)

Confusion between inner product and norm[edit]

The statement The Minkowski inner product is defined as to yield the spacetime interval between two events when given their coordinate difference vector as argument. is wrong. The interval between x and y is sqrt(|<x-y,x-y>|), the norm, not the inner product. Shmuel (Seymour J.) Metz Username:Chatul (talk) 20:58, 14 February 2017 (UTC)

I have reverted your edit and added a source. Please read the article, and see Inner product space#Generalizations. - DVdm (talk) 21:14, 14 February 2017 (UTC)
Chatul I support you except that the definition applies only to time-like vectors. The article is about Minkowski space so we are not interested in abstract generalizations. However DVdm always insists so what can we do? JFB80 (talk) 04:34, 15 February 2017 (UTC)
You'll find all variations in the literature. My feeling is that the term "interval" usually refers to rather than . It doesn't matter though, since your edit should really have allowed for imaginary intervals (of the type, like L&L does in a spot or two) to avoid the ambiguous "... inner product is defined as to yield the positive or negative square ..."
Let me quote an unreliable soure, namely this article in Minkowski space#Minkowski metric:
It is a fact of life that one has to check out the definitions first thing when one consults the relativity literature.
It is unsourced OR (by me) of which I am proud. In fact, it should have a place in the lead (and a place in the mind of some editors around here). YohanN7 (talk) 09:59, 15 February 2017 (UTC)
The fact is that nowhere in the literature is the subject thought out clearly and a norm in the usual sense does not appear to be possible. Minkowski was wise enough not to define a 'Minkowski norm'. JFB80 (talk) 20:43, 16 February 2017 (UTC)
In all of the established literature is the subject thought out clearly. Indeed, the Minkowski metric isn't a metric in the usual sense. But nor is a Riemannian metric a metric. Even the metric in general relativity (always referred to as the metric) isn't a metric in the ordinary sense (it is closer to a Minkowski metric as defined here). And a Sierpinski sponge isn't a really a sponge.
It is also common practice for non-crackpot scientists to refrain from naming things after themselves. But the bilinear form producing all these things is indeed in Minkowski's paper (it is just not called a bilinear form (and much less, the Minkowski bilinear form), the terminology and style of writing was painstaking 100 years ago, few people understand it). Otherwise Minkowki would not have been famous. Did you notice that nowhere does Minkowski mention Minkowski space?
Terminology differs between sources, but the meaning never does. You know all this but insist on keeping trolling. At least, you should know this after trolling the talk page for years. YohanN7 (talk) 09:43, 17 February 2017 (UTC)

Incorrect definition of the Minkowski inner product?[edit]

Under Standard basis we have

Shouldn't it be

? — Preceding unsigned comment added by 147.91.66.6 (talk) 08:43, 3 March 2017 (UTC)

No, since
where "lowering of an index with the metric was used". See the following section Minkowski space#Raising and lowering of indices where it says: . - DVdm (talk) 09:58, 3 March 2017 (UTC)

Split proposal for section Geometry[edit]

It has been proposed that the section Geometry be split out.

  • Neutral: I wrote that section. Before I wrote it, I probed (scroll a little, link works poorly) whether we should have the section, but no-one expressed an opinion. On the one hand, there has been requests for a geometry section, but on the other hand, those requests might have anticipated something less technical. On the third hand, with the section, there's an excellent spot to exhibit the connection between Minkowski space, De Sitter space, and Anti-de Sitter space. YohanN7 (talk) 08:49, 5 May 2017 (UTC)