Talk:Pseudo-Riemannian manifold

Don't understand

I don't understand one bit of this stuff! —Preceding unsigned comment added by Graham P. (talk • contribs) 22:46, 20 February 2005

Holy crap

No kidding. Articles generally should start out with a simple explanation. This starts out with no explanation and is therefore a terrible article. The only person capable of understanding the article will already know the content. Meaning that someone who simply wants to begin understanding it has nowhere to go except away from Wikipedia. My lowest rating. 1 of 10 stars. —Preceding unsigned comment added by 192.103.41.201 (talk) 23:36, 15 January 2006

Comment on Context Tag

I added the context tag. I had followed the Lorentzian manifold link (redirected to here) on the General relativity page hoping the get a better understanding of the subject. However, this article is difficult to follow and could use improvement. Judging by the comments above I'm not the only one. Ryan Roos 22:40, 1 December 2006 (UTC)

Do you have a specific question you wish to ask? If you haven't studied Riemannina geometry, you won't magically grok this article no matter how much "context" is added to it. FWIW, this article is nt well written, but it says nothing that is actually complicated, this is pretty introductory material. linas 23:41, 1 December 2006 (UTC)

I suspect most of the complaints regarding this page are coming from people who arrive here from a general relativity article looking for a definition of a Lorentzian manifold when they don't even know what a manifold is, much less a Riemannian one. Calabi-Yau manifold gets similar complaints. Perhaps its necessary to fork off Lorentzian manifold to its own page and start the discussion at a much lower level. -- Fropuff 00:27, 2 December 2006 (UTC)

linas and Fropuff thank you for your thoughtful replies. I found the discussion on the Talk:Calabi-Yau manifold of the technical tag that had been used there helped to resolve some of my problems with the article. The main basis of my concern was result my lack of knowledge of manifolds. (On a side note linas I enjoyed following several of the links on your user page) Ryan Roos 02:04, 3 December 2006 (UTC)

WikiProject class rating

This article was automatically assessed because at least one WikiProject had rated the article as stub, and the rating on other projects was brought up to Stub class. BetacommandBot 10:01, 10 November 2007 (UTC)

Lorentzian Manifold

I'd like to separate out an article about Lorentzian manifolds. This could emphasise their use in physics and in general relativity. It would link to pseudo-Riemannian manifold. What do people think? SJ 17.00 5 December 2007. —Preceding comment was added at 17:13, 5 December 2007 (UTC)

Oxymoron

A manifold must be locally Euclidean. A Pseudo-Riemannian metric cannot be used to form a family of neighborhoods of a point. We are in the business of clear communication in an encyclopedia. Occational misnomers are an unavoidable part of human communication when people race ahead of their understanding using inadequate vocabulary. In Minkowski space we have a few of these like Minkowski metric which is not a real metric (mathematics). The metric tensor article has been nuanced to avoid claiming to establish a metric. So far, so good. But this article is where general relativity is given too much license with the use of an oxymoron. Certainly it is a term in common use (though no references are yet produced), but our article needs to help readers. Learning should not be a continuous torture watching abuse of intellectual designations. The article must be clear on how the space is a manifold, not rely on presumption.Rgdboer (talk) 22:56, 5 July 2009 (UTC)

• The page is correct: a pseudo-Riemannian manifold is a differentiable manifold with an additional structure, the pseudo-Riemannian metric. The manifold structure must be imposed prior to the choice of pseudometric. One might then wander if the topology is induced by the pseudo-Riemannian metric, and it is. For example, one can make the following construction in Minkowski space: Take the point about which you wish to construct a point, x. Construct two distinct null curves from the point. Pick two distinct points along the curves, y_1, y_2. The Lorentzian separation between x and and each of y_1 and y_2 is zero. The sets B(y_i,\epsilon), which consist of the points which have a separation less than \epsilon from y_i are open. Take the intersection of B(y_1,\epsilon_1) and B(y_2,\epsilon_2). This intersection is an open set containing x. Given any point z, there are values of \epsilon_1 and \epsilon_2 such that z is not in the intersection. The topology constructed this way is the standard one. 129.215.255.13 (talk) 20:24, 19 August 2009 (UTC)

In section two, one finds the definition that spells out the manifold hypothesis; section one apparently is necessary to lay the groundwork for the whole structure. Thank you computer 129.215.255.133 at Edinburgh University for pointing out the stipulation. As for your discussion of a topology generated by the pseudo-Riemannian metric, I'm not sure what you consider standard topology of spacetime.Rgdboer (talk) 20:47, 19 August 2009 (UTC)

Hi, you said on that page that you didn't understand what I meant by the standard topology. I'm not sure what the point of confusion is. If I start with things like second-countable and a base, would that make sense to you? —Preceding unsigned comment added by 129.215.255.13 (talk) 17:28, 20 August 2009 (UTC)

When you say, "Given any point z", you must mean "Given any event z with a non-zero separation from x". I note that such a topology does not separate events on the light-cone.

There is a significant alternative spacetime topology defined by Erik Christopher Zeeman in the journal Topology 6:161–170 (1966). The article is called "The topology of Minkowski space". Topology of spacetime continues to be an active investigative area; often writers just assume some Euclidean coordinates are around to make good. For pseudo-Euclidean metrics one often finds them built of a Euclidean metric with some minus signs thrown in, so the Euclidean metric is in the background. Zeeman's topology focuses on the important invariants in spacetime, so it contributes to an understanding of how the science of open sets can be applied in modern physics.

May I suggest, contributor in Edinburgh, that you log-on as a WP:User and gain the benefits including a User Talk, such as mine where you have been corresponding. I will respond here, since our focus is on the Pseudo-Riemannian manifold.Rgdboer (talk) 20:04, 20 August 2009 (UTC)

Hi, It's me again. Yes, registered user and talk page might make sense, but I prefer being anonymous. Sorry. You wrote: "When you say, 'Given any point z', you must mean 'Given any event z with a non-zero separation from x'." What I meant was take any event z which is distinct from x. There is no restriction on the separation between x and z. I will use the terms point (assumed to be in spacetime) and event interchangeably. Allow me to continue working in Minkowski space with the standard coordinates. I'm going to change the names of the points to P and Q so that I can use (t,x,y,z) as coordinates, and denote the separation between P1=(t1,x1,y1,z1) and P2=(t2,x2,y2,z2) by d(P1,P2) = -(t1-t2)^2 +(x1+x2)^2 +(y1-y2)^2 +(z1-z2)^2. With out loss of generality, I may assume that P is at the origin (Otherwise, I can translate the coordinates.) I may assume that Q lies in the t-x plane (Otherwise, I may rotate the axes until this is the case.). You were concerned about points on the light cone, so, to handle this case, let's assume the coordinates of P are (0,0,0,0) and for Q they are (a,a,0,0) for some positive number a. (P is at the origin. To get Q to have positive coordinates, I may apply reflections. This gives that the t and x coordinates are equal.)

To prove my claim it is sufficient that I construct an open set, defined by the separation, which contains P but not Q. The key point here is to construct the set about a different point. (I called such new points y_1 or y_2 previously. Now, I will call it R.) Let me introduce a new point R with coordinates (a,-a,0,0). Consider the new point R and the separation from P and Q. The separation from P is zero, but the separation from Q is 4a^2. Construct the set of points such that the separation from R is less than or equal to a^2, i.e. if Minkowskian separation is denoted d(M,N) for points M and N, the set is {S : d(S,R)<a^2}. This is an open set in the topology induced by the separation. This set contains P (since d(P,R)=0<a^2), but it does not contain Q (since d(Q,R)=4a^2 > a^2). Does that make sense? Does that convince you?

I'm not sure what the Zeeman paper says, but I'll try to look at it if I can. 86.177.228.248 (talk) 23:40, 3 September 2009 (UTC)

As stated above, now I see that you adequately define the concept since the manifold condition is specified in the second section. I retract the "oxymoron" label. There really is no convincing necessary.Rgdboer (talk) 21:25, 4 September 2009 (UTC)

This discussion comes to the point: For General Relativity we need the concept of a pseudo-Riemannian manifold. A reference to Euclidean spaces is inappropriate, since there is no Aether. So we need to define a manifold as something which is locally a (flat) real four-dimensional vector space (without every reference to a Euclidean structure thereupon). The structure, we must deal with, is the pseudo-Riemmannian one (i.e. the Minkowski-product locally). Any topology must be derived from there. Zeeman's topology fulfills this condition, but it is not known (to me) whether it is the only one. Perhaps there are other ones, equivalent or not in what sense whatsoever. Such a "Minkowkian"-topology should work inside, on and outside the null cone, giving an idea of neighborhood. Who has solved the problem, without which General Relativity is not well-based? — Preceding unsigned comment added by 141.89.80.203 (talk) 15:32, 4 October 2011 (UTC)

Special characters or symbols

There are a number of special characters or symbols that do not display correctly, or at all, in four different browsers I have tried:

• Internet Explorer shows some as " " [box symbol for non-display characters] and some as a " " [a blank space]
• Firefox and Safari show all as blank spaces, so it is not clear something is missing
• Opera shows all as empty boxes, so it is clear something is not showing

I have tried looking at the source code, editing the page within wikipedia, and copying the characters/symbols to a word processor. I cannot figure our what the writer intended. This requires the author of the text to fix this problem if it is desired to communicate completely with the reader. Colin.campbell.27 (talk) 21:08, 27 March 2011 (UTC)

copyvio?

This is odd. If I do a google search on '"locally non-decreasing" sylvester signature', http://www.google.com/#hl=en&sclient=psy-ab&q=%22locally+non-decreasing%22+sylvester+signature&oq=%22locally+non-decreasing%22+sylvester+signature&gs_l=hp.3...720.10192.0.10353.48.44.2.0.0.0.167.3998.30j14.44.0.les%3B..0.0...1c.1.W7Ns9Kjyc-0&pbx=1&bav=on.2,or.r_gc.r_pw.r_qf.&fp=a7217b4391070716&biw=1280&bih=905 , I get a hit from google books for a book by Sussman and Wisdom, which contains text from this article. When I saw this, I assumed the article was plagiarized from the book. But when I click through on the google books link, I get a link that claims to be to a *different* book, by Kleppner and Kolenkow. And even this is wrong. I have the book by K&K, and what's in google books in not that book -- it's a compilation of wikipedia articles. Very mysterious.--75.83.64.6 (talk) 00:00, 2 October 2012 (UTC)

I took 3/4 of an undergraduate physics major and the introduction made my head hurt!

Seriously, I'm a biologist now, but when I write WP articles, I pretend that my readers are morons--seriously, it's hard to teach people if they can't understand what you're saying. Furthermore, drawing on experience explaining evolution to undergrads (an experience roughly analogous to skateboarding into the same giant stack of rusty nails and CMUs three hundred times in a row), I can safely say that the simpler language you use to explain a complex topic (even if it takes 300x longer to explain it that way than you could with the proper jargon) the more people (even people conversant in said jargon) will understand the points which you are trying to make. Seriously, explaining a topic in plain English does not reflect poorly on your knowledge of a given topic. Quite the opposite, it shows that you have an intuitive understanding of it.--99.112.106.168 (talk) 18:27, 15 September 2013 (UTC)

Seriously, I just noticed that I used the interjection "seriously" three times in a row. I hope that underscores just how serious I am!--99.112.106.168 (talk) 18:28, 15 September 2013 (UTC)

topological obstructions...

...to a manifold for admitting a Lorentzian metric. There is a large body of literature on this. Will be nice to add a word or two about it. - Subh83 (talk | contribs) 07:01, 10 November 2013 (UTC)

Meanwhile…

While the talk page is cluttered with pettifogging, the article states that a flat space with the

$g=dx_{1}^{2}+\cdots +dx_{p}^{2}-dx_{p+1}^{2}-\cdots -dx_{p+q}^{2}$

metric is called… the Minkowski space. Can I believe that nobody of you knows the correct name of this concept for general p and q? Incnis Mrsi (talk) 19:12, 19 February 2014 (UTC)

It does not seem to say that; perhaps you misread? My guess is that pseudo-Euclidean space is reserved for the indefinite cases, and is thus not general. Flat space? —Quondum 21:53, 19 February 2014 (UTC)

First use?

Does anyone know what was the first use of pseudo-Riemannian manifold? David edwards — Preceding unsigned comment added by David edwards (talk • contribs) 15:47, 2 June 2018 (UTC)

My revert of a good-faith edit aimed at addressing WP:REDIR#ASTONISH

I reverted an edit (Since "Lorentz metric" redirects here, I added some information that is appropriate per WP:REDIR#ASTONISH. This is based on my understanding of my textbook, if I'm wrong, please try to correct any mistakes while leaving the content intact) with the promise of elaborating here.

This is an article that deals with mathematical aspects of a metric tensor on a pseudo-Riemannian manifold, which need not be flat, even in the Lorentzian case, and example being general relativity. As such, defining even the separation between two points entails defining a curve along which to integrate, and this usually is taken to be a geodesic, the definition of which in turn requires a lot of theory based on the metric tensor. So in a sense, describing a Lorentzian metric tensor in terms of distances is inherently circular. In the flat case (a Minkowski space), a lot of short-cuts are possible and the circularity can be avoided, but that intuition does not work to define distances directly in this context. A second major point is that the ideas of a "time separation" and a "space distance" are ill-defined in this context, and even in the Minkowski space case need specification in terms of a selected frame of reference. Without mentioning this (in the Minkowski case, where it applies), one risks confusing the reader with Galilean intuitions of absolute time and space. Defining such frames in the Lorentzian case are not possible in general. One is left with building the concept of distances along curves from the concepts already stated, and one cannot start from the concept of distance or time separation: one cannot really appeal the intuitions of special relativity except in the limit of shrinking a region to a point.

A more productive approach might be to resolve the WP:REDIR#ASTONISH problem by looking at where the redirect was used. Note that the redirect itself seems to be appropriate; a Lorentzian metric (which I presume "Lorentz metric" might be used in place of) is more general than a Minkowski metric: allowing non-flat space in addition to an arbitrary number of spatial dimensions; there should be no violation of the no-astonishment principle with this redirect. —Quondum 12:06, 19 August 2020 (UTC)

This is not the correct explanation. The metric is defined on the tangent space (of the spacetime manifold), where it is linear as in linear algebra. No appeal to geodesics is needed to define the metric. Quite the opposite: *after* defining the metric (on the tangent space), you can ask what the shortest distance between two points on a manifold. Be sure to read and grok manifold. The shortest distance is gotten via variational principles, solved by the Hamilton-Jacobi equations. These are partial differential equations, and when you integrate them, only then do you get the geodesics. There aren't any circular definitions here. 67.198.37.16 (talk) 08:40, 18 November 2023 (UTC)

Lorentzian Metric ...again (1)

Currently (Aug 8, 2022), in the LM section of this article "... signature of the metric is (1, (-1)(n-1 occurrences) or (equivalently, (-1, 1(n-1 occurrences))" is a hot, stinking mess. My first suggestion is that the FULL signature (p,q, 0) be mentioned. (I assume that's what (p,q) means, as per convention). I also note that if p, q must be integers, that calling them numbers is just plain sloppy; no reason for that. Anyway, my suggested correction is "...signature of the metric is (1, (-1)(n-1)) ( or equivalently, (-1, 1(n-1)). (see Sign Convention)" but also include an explanation of n AND to explain how -1 is allowable when p and q are to be NON-NEGATIVE. So, that's like four things. 1. Easiest: clean up the parenthesis - "or (blah blah)" is wrong, should be "(or blah, blah)" 2.Most obvious: remove "occurrences" from formula, it is NO WHERE defined. 3. Almost as obvious: Before using n, DEFINE (or explain) it. 4.Lastly:YOU MUST EXPLAIN why negative (integers?) are allowable as a "special case" when non-negative integers are required in the more general case.174.131.48.89 (talk) 17:24, 8 August 2022 (UTC)

The dimension of space is always a (non-negative) integer, like "1D space and 2D space and 3D space" and its unfair to ask that an advanced article to stop and explain this first, before doing anything else. You will have an easier time of it all if you get a good book on special relativity and another on differential geometry and get through the basics, first. Also, Feynman "Lectures in Physics" are great. 67.198.37.16 (talk) 08:53, 18 November 2023 (UTC)