Talk:Pseudo-Euclidean space

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Relativistic version of the Pythagorean theorem

Invariant mass mentions this, and IMHO it should be explained here. Incnis Mrsi (talk) 13:30, 26 January 2013 (UTC)[reply]

Terminology

Standard terminology makes our articles more supportive of learning. Currently the article starts with a quadratic form. The article should show that this induces a symmetric bilinear form on the pseudo-Euclidean space. The term "dot product" is not appropriate since it assumes a positive definite space not found here. The article also introduces "angle", mentioning that it may be an "imaginary angle". The appropriate concept is hyperbolic angle which occurs in planes where the quadratic form goes both positive and negative.Rgdboer (talk) 20:09, 16 April 2013 (UTC)[reply]

The article shows a symmetric bilinear form

⟨ , ⟩

associated with

q

. It is possibly incorrect that I call it the dot product, but which term is appropriate: is scalar product better? Feel free to rewrite Pseudo-Euclidean space #Angle with “hyperbolic angle” and anything else necessary – it is actually my original research. Incnis Mrsi (talk) 11:08, 19 May 2013 (UTC)[reply]

“In affine space there is no metric”

https://wiki.riteme.site/w/index.php?title=Pseudo-Euclidean_space&diff=555917232

Similar assertions could be:

“In vector spaces there is no norm”
“In manifolds there is no metric tensor”
“In Euclidean spaces there is no orientation”
“In groups there is no topology”
“In fields there is no order”
⋮

All these are true in the sense that “this additional structure is not canonical”, but all these are false in the sense that “this additional structure cannot be compatible with the primary structure”. Thanks, Quondum. Incnis Mrsi (talk) 12:54, 20 May 2013 (UTC)[reply]

Sure thing. This logic is of course clear enough. On the other hand, mathematics is littered with terminology that can be confusing and must be disambiguated by context and subdiscipline (this was a huge adaptation for me on WP, where articles span a myriad mathematical disciplines). Pertinent here is that it is not unusual for terms to be used alternately for the vector space or the affine space: Minkowski space is an example specialization of the pseudo-Euclidean case. To make the distinction clear in all these cases seems to me to be appropriate. — Quondum 13:50, 20 May 2013 (UTC)[reply]

It seems to be you are making a fuss about nothing. For example the definition of vector space contains no reference to norm. But we often can define a norm and then it becomes a normed vector space which has its own precise general definition. Similarly the definition of affine space contains no reference to norm. If you can define one fine but then rename it accordingly. But you should not change established concepts arbitrarily and still call the modification by the same name (as has been done in the affine space article which redefines an affine space as being without origin)JFB80 (talk) 19:58, 22 May 2013 (UTC)JFB80 (talk) 20:09, 22 May 2013 (UTC)[reply]

We are not redefining concepts, which seems to be the point that you are missing. A normed vector space is still a vector space. No matter how much struct you add to a definition (thus making it more specific), an object of a subclass still adheres exactly to the defining axioms of the containing class. You made a claim that directly contradicts this. You cannot, for example, claim that a vector space has no norm – or in this case, that an affine space has no metric. Adding the structure of a metric to an affine space does not stop it being an affine space. — Quondum 21:34, 22 May 2013 (UTC)[reply]

Thank you, you may be right, I’m thinking about it. By saying an affine space has no metric I was of course meaning by definition and common use. I referred to a similar remark (not mine): ”an affine space has no metric structure” in the WP article “affine geometry (projective view)”. Of course you don’t have a metric, only a pseudo metric so the usual axioms for metric don’t apply.JFB80 (talk) 19:45, 25 May 2013 (UTC)[reply]

(Later) I have been browsing Weyl’s “Space, Time, Matter” which makes affine geometry very understandable by addition of successive localized vectors. Looked at this way the introduction of a metric appears rather obvious though, apparently, not done. On the other hand, looked at in the way you suggest (point-line) I don’t see how you can introduce a metric (or pseudo-metric).JFB80 (talk) 15:29, 26 May 2013 (UTC)[reply]

Rather good now, but make yet a little effort towards understanding the article, please. Nobody claims that pseudo-Euclidean geometry is (some case of) affine geometry. The article states that a pseudo-Euclidean space can be understood as an affine space: it has the corresponding structure, but along another structure. In other words: any pseudo-Euclidean morphism is an affine transformation, but not every affine transformation preserves the magnitude form on vectors. Incnis Mrsi (talk) 15:45, 26 May 2013 (UTC)[reply]

Markup

The following markup for the bilinear form does not show on my screen:

⟨ x, y ⟩ = 0

.

This template is not familiar to me. Standard math markup in traditional wikitext does appear.Rgdboer (talk) 20:33, 20 May 2013 (UTC)[reply]

I'm guessing that you see everything but without the angle brackets. If so, your browser simply does not support the angle bracket Unicode characters with the fonts you have selected. You might want to select a font such as Cambria or Cambria Math as your browser's serif font. — Quondum 00:09, 21 May 2013 (UTC)[reply]

The angle brackets show up with usual math markup, so the text has been changed. Suggestion that the problem is in my browser is bogus. If I see a faulty character, it is due to our markup. Please refrain from troublesome edits. The follow from the subsequent paragraph is also faulty:

N = ⟨ ν ⟩

.

Here you are using the angle brackets for another use besides the bilinear form, perhaps introducing confusion.Rgdboer (talk) 21:56, 21 May 2013 (UTC)[reply]

First part: if you see a mojibake, whitespace or boxes in ⟨ ⟩, then templates {{langle}} {{rangle}} are not rendered properly in your environment. Implement Quondum’s suggestion or try to address the problem to WP:Village pump (technical), but if you do not see brackets, then it is a problem of your rendering of these templates, not a “faulty markup” in the article. Using HTML/Unicode for mathematical formulae is a well-established practice.

Second part: Yes, the same brackets, but without a comma and, not least, with a hyperlink. Are you actually confused about

(a + b)\cdot c

which means “the sum of

a

and

b

multiplied to

c

” and

(a + b, c)

which means “an ordered pair, the first component of which is the sum of

a

and

b

and the second component is

c

”? Incnis Mrsi (talk) 10:12, 22 May 2013 (UTC)[reply]

It might be helpful to get clear on what is actually being seen by Rgdboer. Right now we are guessing, none of us knows exactly what the other is seeing. Here is a translation of the two indicated lines into direct unicode (without template use or bolding).

⟨x, y⟩ = 0

N = ⟨ ν ⟩

Replacing the angle brackets with similar (but more angled) ASCII characters, these become

<x, y> = 0

N = < ν >

Perhaps Rgdboer could describe the way in which the original template rendering differs, so that we may diagnose the problem? — Quondum 11:39, 22 May 2013 (UTC)[reply]

To clarify, I see white boxes with the template markup, but angle brackets are rendered with math markup, for example $\langle x,y\rangle$ . It may be that other readers also note such failure to render.Rgdboer (talk) 19:25, 22 May 2013 (UTC)[reply]

See a followup at WP: Village pump (technical) #.texhtml revisited. Incnis Mrsi (talk) 20:08, 22 May 2013 (UTC)[reply]

@Rgdboer: what kind of operating system, browser and respective versions of those do you have ? It's important context for these kind of font problems. —TheDJ (talk • contribs) 20:16, 22 May 2013 (UTC)[reply]

While this context information will no doubt be helpful, what Rgdboer is describing is not uncommon – I have encountered it with diverse editors. It appears to be a font support problem (the local loaded font does not support a particular character, and so a white box or similar is diplayed). In this instance, the Unicode character code is identical but the display is different, so it is definitely a font problem. The correct fix would not be to insist that everyone experiencing this problem should adjust their local settings or download fonts. Incnis's diagnosis and suggestion at the followup that he linked makes sense: fixing this would be valuable for maths articles generally, since using a large variety of Unicode characters is unavoidable in this context. MathJax evidently has the problem solved; it makes sense to make use of the solution. — Quondum 21:57, 22 May 2013 (UTC)[reply]

I vehemently disagree. The whole purpose of the math tag is to workaround browser bugs or lack of implemented features in browsers. When NOT using the math tag, you are expressly opening yourself to the state of the client platform. As such you shouldn't use that method in cases where that might lead to exposing these browser limitations. So I would argue that you SHOULD be using the math tag, because it is the whole purpose of the math tag to prevent issues like these. —TheDJ (talk • contribs) 06:34, 23 May 2013 (UTC)[reply]

<math> for stand-alone formulae is arguably the preferred format, but it is known to make several inconveniences in a text line. You think that if 1% of readers develop a problem with ⟨ ⟩ as {{langle}} {{rangle}}, then these brackets should be remade in <math>. It is not obvious, because \textstyle <math> in a text line implies barely legible PNG rectangles without transparent background, which sometimes disrupt text formatting due to inconvenient height, for those 80% of readers who use the PNG (LaTeX) renderer. Incnis Mrsi (talk) 08:05, 23 May 2013 (UTC)[reply]

But at least it behaves reliably (although ugly) on the client side, and your solution doesn't. —TheDJ (talk • contribs) 13:08, 23 May 2013 (UTC)[reply]

Now I can only theorize that it will eventually be fixed. If it will not be, then the purpose of {{langle}} and {{rangle}} is defeated. Incnis Mrsi (talk) 18:30, 23 May 2013 (UTC)[reply]

It is reassuring to see that this problem is being addressed in the appropriate forum, the tech room of Village Pump. The discussion there is in a direction in which I am not able to contribute. Thank you both, Incnis and Quondum, for your attention and contributions.Rgdboer (talk) 23:18, 22 May 2013 (UTC)[reply]

Edit of June 3

The edit[1] was at least a double degradation.

First, “split-complex number” is unambiguous (if not well-known), but “hyperbolic plane (quadratic forms)” is something obscure. Moreover, the former linked article has topical content and images, but the latter redirect is confusing and does not lead the reader to a target related namely to $n = 2, k = 1$ case.

Second, it was an egregious mistake to treat hyperbolic rotation and squeeze mapping as synonyms. Any hyperbolic rotation is a squeeze mapping, but not versa. If currently the former redirects to the latter, then it is a problem of the article to explain that hyperbolic rotations are a special family of squeeze mappings which are rotations with respect to certain quadratic form. In the second instance this Rgdboer’s edit replaced a precise and standard term with a vague and less intuitively appealing one.

Also, plane (geometry) was a confusing/misleading link, because a “plane” means a two-dimensional subspace in whatever space (and over whatever field), but split-complex numbers form a (real) two-dimensional space which is represented parametrically, not as a subspace as “plane (geometry)” may suggest. Incnis Mrsi (talk) 04:10, 4 June 2013 (UTC)[reply]

It is a question of mathematical structure. Pseudo-Euclidean spaces are characterized by their quadratic forms. In the example section the hyperbolic plane is an appropriate example and not that obscure if you see the list of references for isotropic quadratic form where it redirects. As a key concept in the study of quadratic forms, the hyperbolic plane is a basic example. On the other hand, split complex numbers form an algebra over a field so the structure is somewhat richer than the context of this article. Your other arguments are similarly flawed, but for discussion let us deal with comparison of hyperbolic plane with split-complex number.Rgdboer (talk) 21:15, 4 June 2013 (UTC)[reply]

Of course, split-complex numbers are themselves a richer structure. But it is a direct example, concretely over real numbers, for which pseudo-Euclidean-properties are discussed in details. And your “hyperbolic plane” first, is an ambiguous term which requires a parenthetical disambiguation, and second, is a marginally-topical subsection of another article, which discusses quadratic forms over arbitrary fields. What you try to do is to dump the reader into a large article, force him/her to find a relevant title, then to stuff his/her brain with unnecessarily abstract notations and generalization for the gain of a stub subsection without a single image. What do you try to say about “hyperbolic plane” can be done entirely inside Pseudo-Euclidean space #Examples. Incnis Mrsi (talk) 04:21, 5 June 2013 (UTC)[reply]

The structure of hyperbolic plane (quadratic forms) is at the appropriate level for pseudo-Euclidean space. The negative descriptions "dump", "force", and "stuff" of the linking are unfair.Rgdboer (talk) 22:15, 5 June 2013 (UTC)[reply]

You do not like my negative attitude, but it is not baseless. Instead of continue this pettyfogging, you could explain that your “hyperbolic plane” is an important example of a pseudo-Euclidean subspace. Just in this article, but in a separate section. It would be really useful, along with explanation that each positive-dimensional linear/affine subspace is either Euclidean, pseudo-Euclidean (of which aforementioned h.p. is the simplest), or degenerate. Of course, example of “hyperbolic plane subspace” would implicitly present it as an example of a pseudo-Euclidean space on its own right. Incnis Mrsi (talk) 07:39, 6 June 2013 (UTC)[reply]

Poincaré

The following reference concerning non-Euclidean geometry was removed:

Poincaré, Science and Hypothesis 1906 referred to in the book B.A. Rosenfeld, A History of Non-Euclidean Geometry Springer 1988 (English translation) p. 266.

Using Google to search Science and Hypothesis turns up nothing for "Pseudo-Euclidean". Confusion between this article and that topic is understandable, but editors must distinguish them. — Rgdboer (talk) 22:35, 31 July 2017 (UTC)[reply]

The correct source, "On the dynamics of the electron", published by Circolo Matematico di Palermo is now cited, as well as Rosenfeld's mention. But Rosefeld does not define "pseudo-Euclidean space", and introduces it as "what we now call" it. His 16 page index skips Pseudo-Euclidean space. On page 372 he again uses Pseudo-Euclidean space to discuss an article by Felix Klein on the Lorentz group and Cayley-Klein metrics. — Rgdboer (talk) 23:52, 1 August 2017 (UTC)[reply]

Relation to Minkowski space and concepts in general relativity

Minkowski spacetime is given as an example of a pseudo-Euclidean space. It might be useful to physicists studying general relativity if the relation between the quadratic form q and the line element ds were shown here. As a physicist, I don't understand how q could be anything but a differential, so I found the article confusing from the start. (Having arrived at this article from a link under "Riemannian geometry" I was not expecting pure math with no notation familiar to physicists.) 71.32.47.231 (talk) 14:16, 31 August 2018 (UTC) Kathleen A. Rosser[reply]

Reference to differentials and Riemannian geometry are part of differential geometry. For a person (physicist or not) to become familiar with the manifolds of differential geometry, some familiarity with flat spaces like Euclid’s is needed. The position of this article is in linear algebra augmented with quadratic forms. Pseudo-Euclidean spaces are tangent spaces to pseudo-Riemannian manifolds, a topic in physical cosmology. So the link was right to bring you here as this bit from linear algebra supports the study of theoretical physics. But costs and revenue offer another interpretation of pseudo-Euclidean space. — Rgdboer (talk) 01:51, 7 January 2020 (UTC)[reply]

New Image

File:41114 2019 23 Fig2 HTML.webp shows "light con" in an space with signature (2,2). Please check if it might be included here or elswhere. Thanks.--Ernsts (talk) 19:02, 12 June 2022 (UTC)[reply]

The notion of "signature" is missing

I quote from Minkowski space.

Minkowski space is thus a pseudo-Euclidean space with total dimension $n = 4$ and signature $(3, 1)$ or $(1, 3)$ .

But pseudo-Euclidean space doesn't have a definition of "signature". Could somebody fix this issue? 2A02:AB88:2A82:CD80:E0A8:F70:B373:170E (talk) 06:59, 12 July 2024 (UTC)[reply]

OK, a link signature (quadratic form) has been inserted. Adjustment also at Signature (disambiguation) to include that usage. — Rgdboer (talk) 23:09, 12 July 2024 (UTC)[reply]