Talk:Metric space/Archive 2

Split metric space into metric (mathematics) and metric space

I split metric space into metric (mathematics) and metric space. Reasons were

• metric space article has become to long
• norm (mathematics) and normed vector space are separate articles too
• different metrics can be used to define the same metric space (depending of course on how one defines sameness)

MathMartin 12:28, 8 Apr 2005 (UTC)

These used to be separate and then got merged. Now they are separate again. I'm not going to complain too loud, but someone might. One could argue that norm (mathematics) and normed vector space should be merged. A topology is defined on the topological space page. -- Fropuff 16:19, 2005 Apr 8 (UTC)

To be honest I am not completely sure my split was a good thing. I have looked in the history but could not find the merge. Do you know the exact date ?

Norm (mathematics) and normed vector space should not be merged. A Frechet space can be defined using a countable collections of semi-norm (which should be merged with norm (mathematics)) so in this case it certainly makes sense to only talk about the norm and not the normed vector space.

My point is if your are talking about a topological vector space (a mixed structure) it is clearer to say the space is endowed with a norm and a metric than to say the space is a metric space and a normed vector space. For example it is clearer to say the space has a topology induced by a translation invariant metric than to say the space is a metric space with a translation invariant metric.

I do not care strongly about this split but I would like to get some other opinions before I merge the pages again.MathMartin 17:27, 8 Apr 2005 (UTC)

I think a split is good, as soon as a "critical mass" is reached, provided that sufficiently tight links are maintained, in order to avoid duplication of material, especially examples.

Concerning the concrete case, I think it is good to have the page separated into "norm" which could allow (more algebraic) discussions about norms (as functions) (definitions, constructions (product spaces, inner products,...)...) and their properties (equations, inequalities,...), while "normed space" could focus more on "global properties", remarkable subsets, and examples of such spaces. — MFH: Talk 19:36, 2 May 2005 (UTC)

I just learned about gage spaces, uniform spaces where the topology is defined by a family of pseudometrics. So I think the split in metric (mathematics) and metric space was reasonable.

Just added suggestions for principles (and implicitly clean-up) to the discussion in http://en.wikipedia.org/wiki/Talk:Metric_(mathematics)#Split_metric_space_into_metric_.28mathematics.29_and_metric_space PJTraill (talk) 19:36, 11 March 2012 (UTC)

Isometry

Isn't an isometry bijective? It is defined this way in Wikipedia. If so, there can be some simplification here. --JahJah 08:55, 21 September 2005 (UTC)

In my experience usually yes. However the usage is not completely standard. Looking at the books on my shelf gives the following:

• Steven Willard, General Topology (1970) defines an isometry as an order-preserving bijective (1-1 and onto) map. Defining two metrics to be isometric if there is an isometry between them.
• Eduard Čhec, Point Sets (1969) defines an isometry as an order-preserving injection (1-1), but says two metric spaces are isometric if there is a surjective (onto) isometry between them.
• Steen and Steenbach, Counterexamples in Topology, doesn't use the term, but defines isometric the same as the others.

I expect that Willard's and our definition at Isometry is the more standard (and modern?) especially since everyone defines isometric in the same way. If no one objects I would suggest changing this article to conform to the definition at Isometry. Paul August ☎ 15:47, 21 September 2005 (UTC)

I've checked a few references, and there is a certain amount of disagreement: however, for consistency I suggest changing this article to reflect Isometry. --JahJah 08:58, 22 September 2005 (UTC)

I would tend to agree. But in any case we should mention the varied usage at Isometry. Paul August ☎ 16:39, 22 September 2005 (UTC)

I'd call the "into" notion an "isometric embedding", though where clear from context it could slip to "isometry". BTW there's something a little redundant-sounding in the definition at isometry -- a "distance-preserving isomorphism"? What would be a non-distance-preserving isomorphism? A metric space doesn't have any structure that can't be recovered from the distance function. --Trovatore 17:00, 22 September 2005 (UTC)

I will raise this question at Wikipedia talk:WikiProject Mathematics. Paul August ☎ 16:32, 22 September 2005 (UTC)

Is this really a conflict? For an isometry in Euclidean geometry all of the plane must be in the image and pre-image; but in more general contexts we may want to consider, say, a ball within Euclidean 3-space. We need some way to talk about the common metric, and isometry seems a plausible word. It's quixotic for Wikipedia to try to enforce consistency when mathematics itself does not. Context is everything, so long as each is careful with its definitions. The problem is, Wikipedia is context free, so inconsistency must occur if Wikipedia is to be complete. (Hmm; that sounds eerily familiar.) Personally, I'm inclined towards Trovatore's distinction between isometry and isometric [map], insisting that isometric spaces be related by an isometric map that is further required to be bijective (and hence an isometry). --KSmrq^T 18:06, 22 September 2005 (UTC)

By the way, PlanetMath's Isometry and MathWorld's isometry also require an isometry to be a bijection. JahJah, would you please say which references you found which do not? Paul August ☎ 16:47, 22 September 2005 (UTC)

Munkres, Topology (a standard undergraduate/beginning graduate reference) defines isometry as distance-preserving but not onto. However, this is only in an exercise: the book concentrates on equivalence of metrics. --JahJah 07:38, 24 September 2005 (UTC)

The word "iso" also shows up in "isomorphism" where it is a bijective morphism. According to webster, the word comes from Greek "isos" meaning "equal". Ultimately I guess we can make it a convention that in this encyclopedia "isometry" will mean bijective distance-preserving map. This will be our ISO standard! Oleg Alexandrov 22:36, 22 September 2005 (UTC)

I prefer the names isometric map and isometric embedding for the 1-1 case, and isometry for the bijective case. While we are on the subject of conventions, I am not content with the usage of the term compact on WP for spaces which are not Hausdorff. The term quasi-compact was created for this reason (compact not-necessarily Hausdorff spaces); compact is reserved for Hausdorff spaces only. This is the standard in algebraic geometry as far as I can tell, and I feel that it is an important distinction. Whatever we decide for isometries should also end up here: Wikipedia:WikiProject_Mathematics/Conventions. - Gauge 00:29, 24 September 2005 (UTC)

I'm not a topologist, but that's not what I recall either as definition or usage. I have certainly seen the phrase compact Hausdorff space often enough to doubt it is redundant. What nationality are your textbooks? Septentrionalis 18:22, 24 September 2005 (UTC)

Bourbaki requires compact spaces to be Hausdorff, but that is not standard. For example, Steen and Seebach, and Willard don't require Hausdorff. Paul August ☎ 04:34, 25 September 2005 (UTC)

Homeomorphism and continuous functions

A metric structure on a given set induces a unique topological structure. This means that the notions of continuous function and homeomorphisms should be defined on the metric structure without any reference to topology. I think someone should rewrite that part of the article accordingly. Tomo 10:40, 30 October 2005 (UTC)

Is this an error?

Right now the article reads:

A metric space M is called bounded if there exists some number r, such that d(x,y) ≤ r for all x and y in M. The smallest possible such r is called the diameter of M.'

Is it the 'smallest' or 'largest'? Just trying to understand better. Plowboylifestyle 20:20, 29 December 2005 (UTC)

There is no mistake. The idea is that if you can squeeze the whole metric space into a ball of radius 4, you could even easier have it into a ball of radius 200, which is larger. Then, what you care about, the the smallest ball which still contains the whole space, and that's called the diameter. I hope that answers your question. Oleg Alexandrov (talk) 20:27, 29 December 2005 (UTC)

That's the problem with Wikipedia. When you don't understand something at first, you think it might be a typo. The math pages are pretty good though. Plowboylifestyle 02:52, 30 December 2005 (UTC)

british rail metric not a metric?

Does the British rail metric satisfy d(x,y)=0 iff x=y? I can only see this property happening only for the origin. I don't think it is a metric, but it *is* a norm on the cartesian product of the normed space in question:

||(x,y)|| = |x| + |y|

and hence can be turned into a metric on the cartesian product space by

d((x,y), (z,w)) = |x - z| + |y - w|

Any thoughts?

Seems to have been fixed since this question was asked. 82.42.16.20 00:47, 9 March 2006 (UTC)

In France (and French speaking countries) this is called "métrique SNCF" - would this merit being added on the main page ? — MFH:Talk 04:59, 10 March 2006 (UTC)

Superfluous condition?

Is the first condition really a definition condition for a metric?

It seems the non-negativity is not a condition, but rather a property of a distance function. Take any two points, A and B, and apply the triangle inequality to the A–A distance:

d(A, A) ≤ d(A, B) + d(B, A)

By the identity condition on the left side and the symmetry on the right side we get:

0 ≤ 2 × d(A, B)

Divide by two, and here is the result: d(A, B) ≥ 0.

Of course the non-negativity condition is necessary in quasimetric space, which does not guarantee d(x,y)=d(y,x), or in semimetric space, which does not guarantee the triangle inequality. But in the metric space it seems superfluous.
--CiaPan 22:44, 3 July 2006 (UTC)

Well, you're right I guess, but most textbooks etc. on metric spaces do in fact state the first condition as a condition rather than a property. So maybe we should follow convention and leave it the way it is? --HellFire 13:49, 18 July 2006 (UTC)

I would agree. I recall some author (a better mathematician than me) writing something to the effect (in this or a similar situation), that although it is possible to create a simpler set of axioms / conditions, the benefit is minimal, and the standard set have the advantage of greater clarity. Madmath789 21:04, 18 July 2006 (UTC)

I'm no great mathematician, but I think that there is a great benefit in separating conditions and properties: If you are trying to prove that a function is a distance function, the proof should have exactly as many sections as there are conditions; if you are trying to use a distance function to write a proof, then you might benefit by reviewing both the conditinos and properties. I understand moving some properties into the conditions to give a reader a more intuitive feel of something, but my personal feeling is that the logical structure of Conditions and Properties is worth more -- definitely though, I'm all for putting positive definiteness as the first derived property. (Jenny Harrison says something similar in Talk:Norm (mathematics).) If the inessential conditions are left in, I suggest we add a note saying that they are not actually conditions, but can be derived from the other conditions, that way you have the clarity that you (madmath, hellfire) are after but accuracy as well. MisterSheik 19:20, 21 July 2006 (UTC)

The positity is not inessential. I would suggest you cite a book or two where the positivity axiom is left out. As far as I am aware, it is always in. Oleg Alexandrov (talk) 06:53, 22 July 2006 (UTC)

I have added a note to the effect that (1) is not necessarily always part of the definition, together with a reference to a text that mentions it merely as a property. Hammerite 21:46, 24 July 2006 (UTC)

Thanks. I shortened it a bit, but agree that this needs be said. Oleg Alexandrov (talk) 03:05, 25 July 2006 (UTC)

I am deleting the "more compact but equivalent" definition from the definition section, as it seems to be wrong. Consider the following example metric d(x,y) = y-x. This clearly fulfills d(x,x) = 0 and d(x,z) <= d(x,y)+ d(y,z). However, it is not a metric according to the "less compact" definition. --Reineke80 (talk) 00:39, 30 October 2010 (UTC)

'our intuitive', 'interesting', POV

Current article says:

The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space.

What group of people does the pronoun "our" encompass? Is it known for certain that everyone a priori conceives of space in 3DE? It's arguable that some people don't -- the text seems patronizing. Safer to say "The most familiar space might be 3-dimensional Euclidean space", or even use a qualifier like "most familiar space to western society".

(BTW, I recall an Irving Adler popularization asserting Kant could not imagine any other space but 3DE, and thus built some of his metaphysics on it.)

And:

...by using a different metric we can construct interesting non-Euclidean geometries...

"Interesting" shows POV. It's a fact that various great and lesser mathematicians and scientists have been and are interested in such geometries, and that these geometries have been useful. It's an opinion that the geometries are interesting, because interest is not some universally agreed upon property. Maybe the passage's author meant "mathematically interesting", but that adjective is vague as it could fairly apply to any category of math.--AC 04:35, 1 September 2007 (UTC)

I think "interesting" is more a byword for that which presents (or has presented) fruitful opportunities for mathematical investagation not found in every geometry. It's not necessarily merely a case of "I like" or "mathematicians like". --Mark H Wilkinson ^{(t, c)} 06:44, 1 September 2007 (UTC)

So this byword means "having many ramifications"? First draft recast: "..by using a different metric we can construct non-Euclidean geometries with many ramifications..." Sounds a bit high tone, perhaps something better will come to mind, but at least it is more neutral.

Or perhaps you're saying that since math bywords exist, they must be tolerated (and learned) to read WP articles. Seems like the same argument could be made for all slang and jargon. No, the point of general encyclopedia is save time by minimizing the amount of new syntax and vocabulary needed. If I need a quick definition of rap music terms like 'shizzle' or 'biotch', a definition written in more rap terms is of little help. Same with math -- nothing against the utility or beauty of such language taken a whole, but readers want "just the facts", not to learn secret handshakes or to join the tribe, and not because they don't like or respect the tribe, but because they haven't enough time. --AC 08:42, 1 September 2007 (UTC)

Oh, no, I'm not trying to say that the obscurities of mathematics communication ought to be foisted on a wider audience without explanation. I was offering my understanding of its usage in order to help inform a rephrasing. --Mark H Wilkinson ^{(t, c)} 08:52, 1 September 2007 (UTC)

OK, and thanks for the clarification -- sometimes I'm too suspicious. 2nd draft attempt: "..by using different metrics a wide and productive range of non-Euclidean geometries have been discovered..." --AC 06:07, 5 September 2007 (UTC)

Content syncronizing

Please see Talk:Metric (mathematics)#Content syncronizing. `'Míkka 17:52, 8 October 2007 (UTC)

Non-commutative metric space?

One of the requirements of a metric space is that d(x,y) = d(y,x). Are there such things as a non-symmetric (non-commutative) metric spaces where this is not the case? — Loadmaster 21:27, 11 October 2007 (UTC)

Yes, they're called quasimetric spaces. --Zundark 21:31, 11 October 2007 (UTC)

computer memory metric?

Does this edit make sense? I'v never heard of such metric. :( --CiaPan 07:32, 7 November 2007 (UTC)

I don't think it makes sense (e.g., the time will depend on the state of the CPU cache). In any case, there are no Google hits for the term, so it must be pretty obscure, probably original research. I've removed it. --Zundark 08:57, 7 November 2007 (UTC)

Hyperbolic example

In listing examples of metric spaces we elucidate the concept by drawing on a student's previous experience. It may be that a student of metric spaces has not been introduced to hyperbolic geometry. In that case, it is only reasonable to point to a model of the hyperbolic structure that is asserted to exist. For this reason I have edited the example to guide the student to links for such models. To claim baldly that a hyperbolic structure is a metric space can only discourage an inquistor trying to build a new concept in mind. The notion is a bit subtle since we are dealing with a mathematical model of a mathematical structure. Nevertheless, pointing to the model in the reference makes plain the path to assimilating an important instance of this metric space concept encountered in topological study.Rgdboer (talk) 22:08, 8 February 2008 (UTC)

Once again an editor has pruned the statement, "Any model of the hyperbolic plane is a metric space" to "The hyperbolic plane is a metric space." In the lives of Lobachevski and Bolyai there was a "hyperbolic plane" but only faintly a metric. It took the algebraic models of the hyperbolic plane to define a metric, usually a logarithm of cross-ratio. So all the models of the hyperbolic plane are isomorphic metric spaces, but to grasp a distance function some particular model must be in hand. Thus I recommend statement of the example in terms of a model; fortunately the editor left the anchored link to the model section of the hyperbolic geometry article.Rgdboer (talk) 21:55, 24 November 2008 (UTC)

Corrected an error

I found the following material on this page:

---

The following construction is useful to remember:

If $(M_{1},d_{1}),\ldots ,(M_{n},d_{n})$ are metric spaces, and N is any norm on Rⁿ, then

${\Big (}M_{1}\times \ldots \times M_{n},N(d_{1},\ldots ,d_{n}){\Big )}$ is a metric space, where the normed product metric is defined by

$N(d_{1},...,d_{n}){\Big (}(x_{1},\ldots ,x_{n}),(y_{1},\ldots ,y_{n}){\Big )}=N{\Big (}d_{1}(x_{1},y_{1}),\ldots ,d_{n}(x_{n},y_{n}){\Big )}$ ,

and the induced topology agrees with the product topology.

---

Unfortunately, this is not actually true, as was pointed out on Ben Green's Metric and Topological Spaces, Example Sheet 1, Easter 2008. Specifically, let $\scriptstyle \langle .\rangle$ be the inner product on R² given by the formula

$\langle x,y\rangle =x\cdot {\begin{bmatrix}3&-2\\-2&3\end{bmatrix}}y.$

Note that the given matrix is symmetric and positive definite, so indeed defines an inner product; let N be the associated norm. Note that $\scriptstyle N((4,1))={\sqrt {35}}$, $\scriptstyle N((2,2))={\sqrt {8}}$, and $\scriptstyle N((2,1))={\sqrt {7}}$, so that if d₁ and d₂ are both just the usual norms on R, then

$N(d_{1},d_{2}){\Big (}(0,0),(4,1){\Big )}={\sqrt {35}}>{\sqrt {8}}+{\sqrt {7}}=N(d_{1},d_{2}){\Big (}(0,0),(2,2){\Big )}+N(d_{1},d_{2}){\Big (}(2,2),(4,1){\Big )},$

in violation of the triangle inequality.

I have fixed this error by requiring that the norm in question be the usual Eudlidean norm, and mentioning that there are other norms for which the normed product metric as defined is really a metric. - skeptical scientist (talk) 23:48, 10 July 2008 (UTC)

d_2(f(x), f(y)) ≥ d_1(x, y)

Does anyone know how a function of this form might be called? It's essentially a generalization of isometry. (So, the map is injective and has closed range, assuming completeness.) It's similar to Lipschitz-ness, but not really the same. -- Taku (talk) 11:31, 17 April 2009 (UTC)

M₁, M₂ -> X, Y

Me again. I would like to propose to change M₁ and M₂ to X, Y. To the best of my knowledge, X, Y are much more commonly used than M₁ and M₂. I'm putting the proposal instead of boldly making the change because this may be discussed before. Any objection? -- Taku (talk) 23:28, 19 April 2009 (UTC)

Metrizability

The phrase

A topological space which can arise in this way from a metric space is called a metrizable space; see the article on metrization theorems for further details.

is deleted by Lovysinghal saying "Def of metrizable spaces isn't exactly the one given". Why? What was wrong with the definition? I restore it. Boris Tsirelson (talk) 16:05, 9 October 2010 (UTC)

My apologies User:Tsirel! I misread and misinterpreted the written sentence as "... which arises in this way ..." instead of "which can arise ..." Lovy Singhal (talk) 07:19, 30 October 2010 (UTC)

Diameter

The picture given to represent diameter seems to misrepresent the definition of diameter; by looking at the picture, one is lead to believe that the diameter of a set $A$ in a metric space $M$ is twice the radius of the smallest ball that contains the set (i.e., $\inf\{2r\in \mathbf {R} :A\subset {\overline {B}}(x,r){\text{ for some }}x\in A\}$). This is incompatible with the definition stated both within the article and in the diameter article; consider an equilateral triangle (or any regular $(2n+1)$-gon, $n\geq 1$). The proper diameter is the length of a side $s$, but the full triangle is contained in a closed ball of radius $r={\frac {2s{\sqrt {3}}}{3}}>s$ centered at the circumcenter $O$, and it is clear that this is best (if the triangle is labeled $ABC$, any point in $ABO$ is further from $C$ than $O$ is).

Should we scrounge up a more pedagogically useful diagram? kielejocain (talk) 02:18, 1 March 2013 (UTC)

Identity of Indiscernibles

Why does the condition $d(x,y)=0\,$ iff $x=y\,$ link to the page on the identity of indiscernibles? The principle of identity of indiscernibles says that two objects have all the same properties iff they are equal, but this condition is only for a single property.

On the metric page this property is also called the coincidence property. I would suggest changing the name "identity of indiscernibles" to "coincidence property". — Preceding unsigned comment added by Catrincm (talk • contribs) 18:03, 17 September 2014 (UTC)

I agree that the "identity of indiscernibles" is irrelevant here. But nevertheless this comment is not true. The statement that $d(x,y)=0\,$ iff $x=y\,$ has nothing to do with x and y each having some property the same. It is related only to x and y together having one property as a pair of points, namely zero distance.Daqu (talk) 20:16, 20 June 2015 (UTC)

Utterly untrue and confusing statement in introductory section

The introductory section ends with this passage:

"In the most general definition of a metric space, the distance between set elements can be negative."

Wrong. A metric space has only nonnegative real numbers as its distances.

I don't know the correct generalization that covers negative distances, but it is not a metric space.

This also disagrees with the definition of metric space given just a little bit lower in the article!

Then change it lower in the text. Verdana♥Bøld 17:56, 22 June 2015 (UTC)

Which makes it very confusing for someone trying to learn the concept.

It's already confusing for people who don't understand it — like you.

A space with a (3,1) metric signature has one dimension with negative length relative to the other three, by definition.

Einstein's spacetime metric: d² = x² + y² +z² - ct²

This is the metric for 4D distance, which is called the "interval" between two points.

If you increase x and want the distance to remain unchanged, you have to subtract some y or z, or you can ADD an amount to t. This is exactly the situation for every point on the null cone. Each of those points has zero distance from the center, but positive and negative distances from each other. That's right; 4D distance in a space with a (3,1) signature is intransitive.

Call it what you want; I don't care. Call it a "fake" or "pseudo" metric if you prefer. But Einstein and every book on SR says that "the metric for spacetime is [the above equation]." Paypal me five bucks and I'll look one up for you. Verdana♥Bøld 17:56, 22 June 2015 (UTC)

The next sentence reads:

"Spaces like these are important in the theory of relativity."

This may well be true, but that does not change the definition of metric space!)

Define it however you want; that doesn't change the topological mathematics that rules the physics. Verdana♥Bøld 17:56, 22 June 2015 (UTC)

Daqu (talk) 20:08, 20 June 2015 (UTC)

That part was added by @Verdana Bold: in this edit on 19 April 2014, and it seems untrue to me, too. I think one could say

There are some generalizations of a metric space definition, which allow the distance between set elements be negative. (...)

but I really doubt it is in the article's scope. Even if it is, it should rather be mentioned in Metric space#Generalizations of metric spaces section, not in lead. --CiaPan (talk) 08:13, 22 June 2015 (UTC)

Take it out of the lede then; I don't care. But you can't then say that distances are always positive. Verdana♥Bøld 17:56, 22 June 2015 (UTC)

And by the way, what is "important in the theory of relativity" is probably Minkowski space, a case of Pseudo-Euclidean space; there, in some sense, squared distance may be negative, not the distance itself.

[Wrong AGAIN, Albert!|http://www.youtube.com/watch?v=mg8_cKxJZJY]

The negative squared distance will always be negative if the distance in time is > the distance in space. That's because time is just another dimension, like x, y, and z. The ONLY difference between them is that distances in time are negative real distances. A yardstick pointing in the "time direction" would be -36 inches long. If you prefer, you can say that the time distance is a positive imaginary distance instead of a negative real distance, but that's unnecessarily, uhh... complex.

But I do not understand you. A negative distance means a positive squared distance, right? A negative squared distance means an imaginary distance, right? (And an imaginary number cannot be "positive" or "negative", since the choice between +i and -i is only a convention; their properties are identical.)

See Pseudo-Euclidean space#Distance. Boris Tsirelson (talk) 09:30, 22 June 2015 (UTC)

Well, I've deleted the confusing statement. It is enough that "Metric signature" is mentioned in "See also". Boris Tsirelson (talk) 17:11, 22 June 2015 (UTC)

Questions to User:Verdana Bold

During 3 days you did not address my question above. Is it not visible enough? Here I repeat it:

A negative distance means a positive squared distance, right? A negative squared distance means an imaginary distance, right?

And another question. No doubt that pseudo-Euclidean space is important in relativity theory, and no doubt that this space is endowed with something that physicists routinely call "metric". Here is the question: do they call it "metric space"? I suspect that the term "metric space" is always (or almost always?) used according to the "mathematical" definition. That is, terminologically, "your generalization" is a metric (on a space), but not a "metric space".

Waiting 3 more days for your reaction. Boris Tsirelson (talk) 14:53, 25 June 2015 (UTC)

More on the terminology. Search for "indefinite metric" (on WP) gives many pages, but I never see "metric space" in that context. I see "indefinite inner product" in "Indefinite inner product space" and "Minkowski space"; "indefinite metric" in "Suraj N. Gupta", Topological quantum field theory, and Four-vector; "indefinite metric tensor" in "Generalizations of the derivative"; "indefinite or mixed signature" in "Metric signature". But "indefinite metric space" would be a neologism, right?

See also Metric tensor#Signature of a metric and Pseudo-Riemannian manifold#Lorentzian manifold; there, such objects are never called "metric spaces".

Also, Google search for "indefinite metric space" gives some occurrences of "indefinite-metric space" and in rare cases "indefinite metric space". Most authors hesitate to use the latter term (while others probably do not care about the difference between

• A man-eating shark is a shark that eats humans.
• A man eating shark is a man who is eating shark meat.

see Hyphen#Varied_meanings). The former term indicates that this space is not quite a "metric space".

Thus, such remark as "In the most general definition of a metric space, the distance between set elements can be negative. Spaces like these are important in the theory of relativity." (but corrected: the squared distance can be negative) fits better in "Metric (mathematics)#Important cases of generalized metrics" or, if you want to emphasize it more, in the lead of that article, but not here. Boris Tsirelson (talk) 16:29, 25 June 2015 (UTC)

___________

Before talking about math, I want to say that actually, I spent nearly three hours writing (what I think was) a response both jocular and eloquent. [this one took 2½ hours). But when I tried to post it, your post caused an edit conflict, and the screen that goes with edit conflicts is confusing and by then, I was sick of the whole thing. I did NOT want to spend yet more time on it, as I was already mad at myself for ignoring my homework for the entire evening. So I said "the hell with it", never intending to come back here. But I saw the "new message" line, was beguiled, and came anyway.

Another reason I didn't reply is that I learned never to argue with "grownups," particularly on Wikipedia. If someone takes you to arbcom, just abandon your wiki-name (cut your own wiki-throat) and create a new account. I'll do that at the drop of a hat if someone who disagrees with me "owns" an article I worked on or has an admin friend. But if you get banned, you can't edit again unless you get a new ISP. Three of my friends were treated shockingly unfairly and permanently banned for insisting on including material from a seminal article in a important academic journal. The "owner" just didn't like the information in it.

Now then...

A negative squared distance means an imaginary distance, right?

Yes, and you're living in a huge one.

I don't know if you've seen the metric of the manifold called "spacetime", but it's the Pythagorean theorem with an extra term (for time), and that term is SUBTRACTED from the sum of the other three squares. Per Einstein:

d² = x² + y² +z² - ct²

Perhaps easier to understand:

d² = x² + y² +z² + (-ct²)

That also preserves the integrity of the pythag theorem. And BTW, "c" is only there so we can refer to time as a spatial distance in the same units as x, y, and z. (e.g., light years instead of years).

"ct" is expressed in positive units (like 4 light-years). The "negative" comes from the minus sign to the left of the squared value.

But I do not understand you.

That's okay; nobody else does, either.

A negative distance means a positive squared distance, right?

Yes yes; so what?

A negative squared distance means an imaginary distance, right?

The Force is strong in this one!

And an imaginary number cannot be "positive" or "negative"

Yeah, only complex numbers can be. Thank you for educating me.

the choice between +i and -i is only a convention; their properties are identical.)

Yes, yes; they're complex conjugates. What's your point? Unfortunately, you leave it out, so I'm not sure what it is. It might help to look at the metric again:

d² = x² + y² +z² + (-ct²)

We observe that both ct and ct² are positive. They're just like the x, y, and z terms. The Jedi mind trick happens when you have to SUBTRACT that ordinary distance from the squares of the other three distances. It is then that the time term becomes a negative real distance.

I hear you ask, "Why must you subtract the time distance from the sum of the others?"

This is usually just asserted in special relativity books instead of being derived, but Einstein said so, and his derivation gives true insight into the nature of space and time. If you add ct² instead of subtract it, all kinds of other equations become internally inconsistent (i.e., they contradict themselves).

The rest of your post is about the meaning of words. But semiotics is both boring and irrelevant—particularly in math and physics.

Again, I'll trade you references on all this for 10 bux paypal. Then I can buy beer from the guys in the dorm, which I won't be able to buy at the store for many years. But it's either that or give public BJs.

Verdana♥Bøld 06:02, 26 June 2015 (UTC)

Well, I am sorry for the inconvenience caused by the edit conflict, but I do not feel guilty. and in fact, I also had such inconveniences, this is the life in WP. Also, I never appeal to arbcom (check it if you doubt), has no admin friends (well, this is hard to check, just believe me), and I will not edit-war with you (or anyone); if we cannot agree then I rather wait for more participants, in order to get a consensus.

You do not need to explain me the meaning and structure of the indefinite metric of the space-time; I understand it well enough (the last 50 years). But I cannot agree when you write "The rest of your post is about the meaning of words. But semiotics is both boring and irrelevant—particularly in math and physics." Note that this discussion was not started by me. Other persons were disturbed with your formulation. And I understand, why. You may neglect "the meaning of words" if you like, but probably not when contributing to an encyclopedia. Otherwise you are at risk of being not understood.

As for me, all math you wrote (above, not in the article) is OK, and still, the problem (posed by others and me too) is not resolved. Boris Tsirelson (talk) 07:35, 26 June 2015 (UTC)

See also Wikipedia talk:WikiProject Mathematics#Another terminology problem: what is a metric? Boris Tsirelson (talk) 17:41, 27 June 2015 (UTC)

It's very simple

Wow, what a lot of verbiage over such a cut-and-dried thing. Semi-Riemannian manifolds (non-positve-definite ones) are very important. But they are not metric spaces. Done. --Trovatore (talk) 00:08, 28 June 2015 (UTC)

A "metric space" is something standard in mathematics that can be found in any number of elementary textbooks. It is clear that pseudo-Riemannian manifolds of indefinite signature (including the Lorentzian manifolds of relativity theory) are not "metric spaces" in the sense that reliable sources mean by the term "metric space". An attempt to insist that they are, contradicting a host of reliable sources, is not appropriate for an encyclopedia. Also, I think Trovatore means pseudo-Riemannian manifolds rather than semi-Riemannian manifolds. Semi-Riemannian manifolds actually are metric spaces, but do not possess what would normally be regarded as a metric tensor. These are different, but related concepts, that only agree in the case of Riemannian manifolds. Sławomir Biały (talk) 00:19, 28 June 2015 (UTC)

Maybe you have a different usage of "semi-Riemannian" in mind? The link you gave points to pseudo-Riemannian manifold, which claims that the two terms are synonyms. I took differential geometry from O'Neill, whose book is called "Semi-Riemannian Geometry", and I don't have it handy but I'm pretty sure it used the terminology as I have used it here. --Trovatore (talk) 00:38, 28 June 2015 (UTC)

Sorry, I was thinking of sub-Riemannian manifolds. Sławomir Biały (talk) 15:11, 28 June 2015 (UTC)

Okay, look, both of you:

You may neglect "the meaning of words" if you like, but probably not when contributing to an encyclopedia.

I don't give a fchh about this article anymore. As I've said several times, I abandon articles when other people get excited and hostile about telling me I'm wrong [whether I'm right or wrong]. Let the fanatics own articles. Because of the corrupt, fu cked up management I've seen (twice), I don't really care very much about this (WP) project anymore, either. I just like making things better (like grammar).

The reason I'm still here is that i've only been reading this stuff for 2 years and i wanted to see others' opinions about whether it is legitimate to view imaginary distance as negative real distance. Because in the spacetime interval metric (pyth theorm in 4D), it blatantly ACTS like neg distance.

Before y'all got all worked up about whether a pseudometric is called a metric, it seemed like nobody has any real problem with seeing imaginary distance this way.

If you still want to fight about what names to call stuff, fine 4 U. But do it w/o me because I don't care.

------

What I'm really about is that if imaginary distance is negative real distance, then it is absolutely, unquestionably trivial to calculate h-zero (rate that the universe expands) using only its age and c. No gravity enters into it. The answer thus predicted is the correct expansion rate to within the accuracy with which we can measure the expansion rate.

Nobody has ever noticed this before because no one ever thinks of time as negative spatial distance and because people automatically dismiss it if a 15-y/o (at the time this occurred to me) kid came up with it. Verdana♥Bøld 02:04, 28 June 2015 (UTC) — Preceding unsigned comment added by Verdana Bold (talk • contribs)

Hi Verdana if that is your name :-).

Yeah, my mom named me after a Windows font. My sister is Helvetica oblique. Verdana♥Bøld 08:05, 28 June 2015 (UTC)

It is not about proving you wrong,

Say, that's too bad! If I had any, I'd pay BIG MONEY for my model to be proven wrong. I hate being wrong. It's like a horrible pus-boil you don't know you have. If I have one on me, I want to know about it immediately. --Verdana♥Bøld 08:05, 28 June 2015 (UTC)

it's not about your age, and it's not about owning anything.  It is, to some extent, about what to call things, because our readers rely on us to use standard terminology that corresponds to what is found in the standard literature.  It doesn't matter whether it's the best possible terminology; it just needs to be standard, so that when users come across other works that discuss metric spaces, they will be able to map the concepts found in those works to the ones we discuss. 

Accck! You people just won't stop lecturing me about terminology, as if I'm arguing with you about it!

The closest thing i said was that names don't matter to me; that I only want to understand stuff, not what names people give the stuff. I never said word one about terminology on WP. --Verdana♥Bøld 08:05, 28 June 2015 (UTC)

Original insight is a great thing, but not in an encyclopedia.

!!! You think I want to add my model to WP? NO! Astrophysical Journal, sure. But only if it's not crackpot, which is my greatest fear. Verdana♥Bøld 08:05, 28 June 2015 (UTC)

I am not sure exactly what you mean by imaginary distance acting like negative real distance

I mean that for any manifold of metric signature (3,1), you increase the total distance by adding to any of the 3 (spatial) vectors, or by SUBTRACTING from the 1 imaginary-length vector (in this case, time). This is the definition of a (3,1) signature. Verdana♥Bøld 08:05, 28 June 2015 (UTC)

but it sounds intriguing,

Yeah. Good. I'll get you a front-row seat for my speech at the Nobel ceremony.Verdana♥Bøld 08:05, 28 June 2015 (UTC)

and I'm not going to say off the top of my head that it's not true.

Say, that's 2 bad! Come back if you can ever prove it's not true. That's what I seek.

 The place to ask is at the mathematics reference desk, WP:RD/Math.  --Trovatore (talk) 02:22, 28 June 2015 (UTC)

Now, I'm gonna try reeeal hard not to come back to this page.Verdana♥Bøld 08:05, 28 June 2015 (UTC)

Happy Nobel ceremony. Boris Tsirelson (talk) 08:49, 28 June 2015 (UTC)