Talk:Special relativity/Archive 14

Tests of special relativity page ?

Is there a page which discusses tests of special relativity (analogous to tests of general relativity)? - guess not. Perhaps there should be, and it would be a good article for at least 2 reasons:

• it would collect together the experiments used to test SR, thus making for better organisation.
• it would be in a more consistent format with the general relativity pages regarding experiments (and there would be some nice overlap too, e.g. Hafele-Keating experiment). There's even a category, Category:Tests of general relativity.

A useful website that lists papers for experiments in SR is the following: http://www2.corepower.com:8080/~relfaq/experiments.html. MP (talk) 14:56, 27 September 2006 (UTC)

Causal Paradox is not adequately explained

It is not enough to show that there exists a frame where the cause happens after the effect. This is explained in detail here on page 11-13.

The problem is that you can't define cause and effect using temporal ordering. You would have to invoke an "arrow of time" but at the fundamental level there exists no such thing.

You have to show that you can create a situation where the effect precedes the cause and that the separation between the two is timelike. Only then do you have a genuine paradox (e.g. the Tachyonic anti-telephone mentioned in the article). Count Iblis 18:17, 29 September 2006 (UTC)

Reasoning for Fact tags placed

I apologize for the delay. We had a "kitten death" in the family and I had to tend to the little ones

The inclusions of the tags were in response to ScienceApologist request made on my talk page from September 26. "Offering helpful criticisms like "please include references for the following facts:" is appreciated." and Kjoonlee's request on this talk page "OK, I've had a look at the article, and there's not a {{citation needed}} or {{fact}} template in sight. Specifically, what sort of cites do we need?" So I responded to those requests. The items I tagged where items that I felt, as a layperson, I would like to be able to verify. I didn't tag the items because I thought they were "false" or "incorrect." I don't doubt the science behind the article or expertise of the editors who watch and maintain this article. But without the benefit of in-line cites, we reach a brickwall when it comes to verifying these items by us non-experts. My concern is for the day when there might not be an expert to watch and maintain this page. What are we to do then? An in-line cite would go a long way towards maintaining the accuracy of this article long after each of you are gone because it allows laypersons to aid in the upkeep. Agne 23:29, 29 September 2006 (UTC)

Most of the points that were tagged follow directly from the postulates of special relativity. In most textbooks they are either derived explicitely or its in some exercise that the student is supposed to do. I.m.o. all that needs to be done is to give a ref. in the beginning of the article to one or more comprehensive textbooks. If you give a ref to all the points that are not imediately clear to a lay pesrson then you get a large number of refs to the same textbooks, which is rather ugly.

In scientific articles where you need to include an important reference to a review article which you are going to use, it is customary to give that reference once at the beginning and then you can freely use it without giving an inline reference at at every sentence. You say, e.g. "we now use the method given in [1]. We define bla bla bla", but you don't write bla [1] bla [1] bla [1] etc. :-).

The way to verify that there is no OR or nonsense is to ask for a review of the article by contacting the wiki physics project members. There is no other way, because even if the refs are included, someone could have changed the contents of the statements. If you are not able to do the maths and see if the statements are correct, then you are unlikely to be able to verify it from a textbook. Textbooks are not just a collection of statements, but they do require the reader to through the math etc. Count Iblis 02:33, 30 September 2006 (UTC)

Reaction

There seems to be no section on the scientific community's reaction to special relativity when it first 'came out,' such as controversy, acceptence etc. Perhaps someone with knowledge on this topic could add it in? --LeakeyJee 04:06, 3 October 2006 (UTC)

Some people embraced it readily. And others rejected it. There was a lot of controversy at first. That is why Einstein got his Nobel prize for the photo-electric effect rather than for relativity. As the experimental evidence has become overwhelming, the balance has shifted in favor of the pro-relativity faction. But people on the other side still exist and are not hard to find. JRSpriggs 10:34, 3 October 2006 (UTC)

I thought that may have been the case, this topic seems relevent enough to have its own section in the article, any one else think so? --LeakeyJee 03:46, 4 October 2006 (UTC)

You could look at Limitations of special relativity. The controversy is really a historical topic which is too big to put into this article (although it would be nice to have a pointer to it). However, an article or section on that controversy would simply attract a large number of cranks and quickly become a worthless article. JRSpriggs 04:32, 4 October 2006 (UTC)

There should be a section on this, with a brief summary, that links to a longer article like Opposition to the theory of relativity that has a full discussion on the subject. However, writing such things takes time. -- SCZenz 04:43, 4 October 2006 (UTC)

A cut-and-dried article on the opposition to SR would be nice, but the last time it was tried the article turned into a crackpot-magnet. Even limitations of special relativity is more of a diatribe than a well sourced and thought-out article. --EMS | Talk 03:11, 7 October 2006 (UTC)

I now started working on a replacement for the "Anti-relativity" article. The new title may be "Opposition to the theory of relativity" but, perhaps better, "Criticism of the theory of relativity" (for how can anyone oppose a theory?). What do you think? It should focus on criticisms by notable "dissidents", instead of on dissidents themselves or on unknown crackpots. I had a look at limitations of special relativity but saw nothing that IMO is worth salvaging. When I get to the point of placing the new article draft, I will immediately (or even upfront) ask for partial protection as a crackpot barrier. Harald88 09:32, 8 October 2006 (UTC)

sequence of references etc.

Now one has to browse through a long list of "see also" and "external links" before finaly reaching the article's references. I propose to correct that: First the references. Harald88 19:28, 10 October 2006 (UTC)

The idea is to put the most easily accessed material first. First, links to other parts of wikipedia. Next, links to other things on the Internet. Next, textbooks. Last, journal articles. JRSpriggs 10:48, 11 October 2006 (UTC)

This way of making the less directly related material more accessible goes at the cost of the accessibility of the articles' notes and references. As the notes and references are part of the article and those links aren't, it effectively rips the article apart. Harald88 19:29, 11 October 2006 (UTC)

While the references are needed to support the article, I do not consider them to be any more part of the article than the other links are. The reality is that most readers will not exert more than a limited amount of effort to follow up on the article. So those materials which they are most likely to use, i.e. the most easily accessed, should be first. JRSpriggs 07:45, 13 October 2006 (UTC)

A Rather....unique GA review has opened concerning this article

The person bringing this dispute claims to have evidence showing that relativity is false or something, and I thought i'd ask editors here about it, because I don't really know how to respond :/. Dispute is at WP:GA/R at the top right now. Homestarmy 16:49, 12 October 2006 (UTC)

A link to “Non-postulated relativity”

Hello everyone,

I did not want to open a discussion about incorporating a single reference (link), but someone deleted this link twice, without providing any explanations. So, I would like to include a reference to the book “Non-postulated relativity” by Lev Lomize [1]. A different version of this book has been already published in the Soviet Union in 1991 (“From high school physics to Relativity”, Prosveshenie, Moscow, 70,000 copies). The publication was supported by theoreticians from the Lebedev Institute of Physics in Moscow. The book was partially rewritten, translated to English, and made available by the author as a PDF file, because Lev Lomize believed that information must be freely available to everyone. That is why I think Wikipedia is an appropriate place to make this reference.

I want to emphasize that author did not try to disprove anything in Special Relativity. To the contrary, he explains why this theory is so valid, although he does that in a highly innovative way. Well, may be his way of presenting Special Relativity is not entirely new, since the similar ideas were shared by Einstein, Janossy, Feinberg and others, as explained in the last chapter of the book by Lev Lomize. I am only sure that his explanation of relativity of simultaneity is completely innovative. Like Janossy, he is talking about limitations of Special Relativity in the end of his book, but every physical theory has certain limitations.

I know that there are numerous books about Special Relativity. So, why include a reference to this one? I think because this book is very special. To explain, I can only cite Lev, with some omissions (Lev passed away recently): “In 1991, the Russian version of this book was published under the name "From High School Physics to Relativity." This book was written after a life time of searching for an explanation of special relativity, which would answer childish questions, such as "What makes the moving rod shorter and the moving clock slow? If the postulates somehow do it, what is the mechanism they use for doing so?" The first hint came out when I was conducting my early theoretical research on the electromagnetic radiation of a bunched beam of charged particles. With the Maxwell equations taken as a starting point for the derivations, I decided to “simplify” the problem by neglecting relativistic “corrections” which were supposed to be made later on. To my extreme surprise, the relativistic “corrections” turned up from the derivations automatically – just by themselves – as though Maxwell had known about relativity. Neither did Newton know about relativity when formulating his universal laws of motion, which are successfully used all over the world in the computations on beam dynamics of relativistic bunches of particles whose speed is very close to that of light. It was Newton who had formally treated the mass in his second law as though it could depend on the velocity. Time and again I had a great pleasure to track down the classical way of explaining relativistic effects without using Einstein’s postulates.

At the first stage of this research, I regarded it as a part of my self-education. But in 1974, I suddenly discovered that I was not alone in my attempts to build a bridge between classical physics and relativity. E.L. Feinberg, one of the best scientific minds in Russia, involved in his research in the field of quantum electrodynamics, spared some of his valuable time to clarify the issue. Supported by V.L. Ginzburg and other theorists from the Lebedev Institute of Physics (Moscow, USSR), in 1974, he published a clear and instructive article devoted to this topic. A thorough inspection of the literature revealed a very interesting history of the issue, starting even from Einstein himself. This inspection convinced me of the need for a new special book that would explain the main relativistic effects in terms of classical physics. “Non-Postulated Relativity” has been written as a fair approximation to it. Only the aberration of light and the longitudinal Doppler effect were not included here in order not to distract the reader from the main ideas. “

Does this sound very strange for some of you? Do you have a lot of questions? Do you disagree? That is precisely why I think this book is so important for understanding Special Relativity. Note that the book is written for undergraduate students, but it takes a lot of effort to read and understand.

Of course, we could argue forever about different scientific topics described in this book. But this is not the point. The point is that reference to this book is in complete compliance with guidelines of Wikipedia. The guidelines say: “The threshold for inclusion in Wikipedia is verifiability, not truth” (so we should not judge who is right and who is wrong), and it says that authors of publishes books are considered as most reliable sources. I know that authors of different books often cotradict each other. But this is fine. Let’s cite many sources to present alternative interpretations, especially if they are interesting and novel. “Divided we stand, united we fall”.

So, if you do not have any serious objections, I would like to include this link. Biophys 18:26, 25 October 2006 (UTC)

Thanks for the reference! At first sight, Lomize's approach has some similarities with that of Bell's "how to teach relativity", and I agree that readers can benefit from such presentations - for example I myself would certainly have benefitted from Bell's presentation if I had not already read similar papers before. But I wonder if there aren't a few significant errors in Lomize's book. Can you be more precise about the original Russian version, You say below that the editor is "Prosveshenie", can you be more precise, maybe with a link to the editor's website?

Apart of that, the phrase you copied refers to topics and not to references. Following your use of “The threshold for inclusion in Wikipedia is verifiability, not truth”, we should certainly include Bell's article as well - and also the zillions of other articles and books about SRT - right? Of course that's not doable, so an editor's pick is needed. Harald88 19:45, 26 October 2006 (UTC)

No. After a cursory examination of the theory the link refers to, I already found serious flaws with it. It will in no way help anyone understand Special Relativity. Tailpig 19:41, 25 October 2006 (UTC)

I am a new person in Wikipedia. So, can someone help me out? I have two technical questions. First, Tailpig is obviously mistaken. There is at least one person who understood Special Relativity better after reading this book. This is me. There are many more. Would you like them to take part in this discussion? Would it help? Those people could tell their names and scientific backgrounds. How many positive opinions do you need to include the reference? So, if a few other people will say "Yes", can I incorporate the reference, or this "No" of a single anonymous person means "No" forever? Second, can I make myself a new Wikipedia article, such as "Non-postulated relativity" by Lev Lomize to explain his ideas? Is that a better way to proceed?Biophys 23:00, 25 October 2006 (UTC)

But are you sure you understand special relativity really better? Or do you now have some twisted idea about special relativity? So, I think you are not qualified to say Tailpig is mistaken. If you'd seen another person get a better grasp of it (and you would be adequately equipped to judge that), then you'd have the right to say that. Errabee 23:21, 25 October 2006 (UTC)

(After edit conflict) Neither of these is a good way to proceed. Non-standard treatments of SR are not going to be a good thing to add to this article, and bringing in people to agree with you won't change the regular editors' opinions on this. See Wikipedia:Sock puppetry#Meatpuppets. Also, creating a separate article is not a good way to proceed; a new approach to physics with no peer reviewed scientific journal publications would not be a good addition to Wikipedia. -- SCZenz 23:24, 25 October 2006 (UTC)

Wikipedia as a tertiary source tries to stay away from primary sources such as this link. See WP:NOR. Wait for Lev Lomize's theory to be properly peer reviewed. There are too many amateur scientific theories to include them in this article. However, Wikipedia is a democracy and so I will not remove the link if the majority of editors agree to include it. Tailpig 23:39, 25 October 2006 (UTC)

I've just been corrected, Wikipedia is indeed NOT a democracy. Tailpig 23:48, 25 October 2006 (UTC)

Tailpig, I also mistook "primary source" to mean what you think, indeed it can lead to much confusion. A review book such as the one here discussed is a secondary source. Personal notes by him in a letter would be an example of "primary source". BTW it is adviced not to base a Wikipedia article on primary sources, but primary sources may certainly be used in articles. Harald88 19:41, 26 October 2006 (UTC)

Whether we see it as a review book (secondary source) or an individual's own original research (primary source) doesn't matter much in this case. There are enough external links to good books in the article. Tailpig 20:32, 26 October 2006 (UTC)

Thank you all for your participation! I only would like to emphasize that this book of Lev is not a primary source! The English vesrion of the book (it includes an additional new historic chapter) indeed has not been peer reviewed. However, this is basically only a translation of the Russian version, which has been reviewed extensively by the very best physicists who worked in the Lebedev Institute of Physics of Soviet Academy of Sciences (FIAN), including Dr. Vitaly Ginzburg (the winner of the Nobel Prize), Dr.B. Bolotovsky, Dr. E.L. Feinberg (a member of Russian Academy of Sciences), Prof. G.Y. Mjakishev (author of physics textbooks, he is well known in Russia) and several others. If it helps, I might be able to obtain some of their reviews on the book and translate them to English. Some of these people asked for the new translated version of Lev's book (this English version was printed in Ann Arbor), and he send signed copies to them. Lev also had a discussion with Academician Andrei Sakharov who was able to understand his ideas very quickly. As far as I know, this work of Lev has not been initially accepted by some of these highly qualified people. So, he had to make a couple of presentations in FIAN to explain and justify his ideas. Finally, everyone was convinced that he is right, and the book was published (the state publishing organization, Prosveshenie, required an approval from the very top of the scientific community). So, if you find this possible, please reconsider. I just thought that putting Lev's work in Wikipedia would be a good tribute to the person who I deeply respect and admire.Biophys 01:45, 26 October 2006 (UTC)

I should add that Lev had close to 100 publications, but most of them were published either long time ago in Soviet journals or as materials with a limited access. He worked in a top secret scientific institution and was allowed to leave this place and move abroad only after his retirement. Two unpiblished papers on his web site are related to the work on the non-postulated relativity, which was his hobby. Biophys 03:12, 26 October 2006 (UTC)

Could you provide a verifiable reference that Dr. Vitaly Ginzburg reviewed this book? Tailpig 02:58, 26 October 2006 (UTC)

Thank you, Tailpig! Yes, I think this may be possible. But will that be enough? Vitaly Ginzburg is a very busy person, so I would not like to bother him without a good reason. What would you need? Lev's friends in Moscow who work in FIAN could ask him to write down a very brief letter about Lev's book, or just say a statement that, yes, he knows this book and approved it for the publication. Would e-mail from one of Ginzburg's associates in FIAN be enough (or may be they can visit Wikipedia and leave a message here?), or you need something like FAX with Ginzburg's signature? Biophys 03:45, 26 October 2006 (UTC)

I'm only asking for a reference out of curiosity. I have doubts that Dr. Ginzburg actually peer reviewed the book. It's not enough to approve it for publication as that does not consist as a peer review. Tailpig 04:23, 26 October 2006 (UTC)

Dear Tailpig, I looked through Wikipedia article about peer review. My understanding is that, yes, the book has been peer reviewed by a large group of highly qualified people. Note that peer review is not necessarily done by persons who are anonymous, but it should be done by persons who are authorites in the field. The book was reviewed by the best authorities in the field in the Sovied Union at the moment of publication, and who are widely recognized as such worldwide. So, I am asking you and others: is it enough if Nobel Prize winner Vitaly Ginzburg will send an e-mail (to the address you tell to me or to someone in FIAN) or go to Wikipedia and state that Lev's book in fact has been peer reviewed and that he is sure that this book by Lev Lomize is correct and great? He was one of the persons who requested the paper copy of the English translation. He has it. So, even if he was not a peer reviewer of the Russian edition (I think he was), are his credentials sufficient to serve as a peer reviewer of the English edition for Wikipedia? Biophys 15:00, 26 October 2006 (UTC)

I personally have published many papers in scientific journals. Of course I know who most of the reviewers were, even though they supposed to be anonymous. Never the reviewers were so outstandig specialists in my field, as those who reviewed Lev's work in his field. The reason is very simple: these people are too busy to review anything except studies that are very interesting and important for them personally. Lev's work was extraordinary interesting and important for them.Biophys 15:00, 26 October 2006 (UTC)

I just found out that an official peer review has indeed been conducted by the state publishing company Prosveshenie ("Education"). Main person who was responsible for this review and final decision was professor G.Y. Miakishev who pulished previously a book "Forces in Nature" (in Russian) and therefore was trusted by the publishing company. If he is alive, it could appropriate to inquire him about the reviewing process. My understanding is that he wanted Dr. Ginzburg, Bolotovskiy and others to serve as reviewers, after looking through Lev's book himself very carefully. I was told that G.Y. Miakishev called to Lev several times during reading the book, ones thinking that he found an error. But it was not there. Finally, Dr. Miakishev was very enthusiastic of the book and wanted best people to read it. Biophys 15:55, 26 October 2006 (UTC)

The article cannot have a link to a book that still uses aether to try to explain Special Relativity. This was proven wrong over a hundred years ago. As for a verifiable reference that Dr. Ginzburg reviewed this book, an e-mail or Wikipedia posting is not verifiable. Since you've dealt with scientific journals and have published many papers, you should know what I mean when I ask for a verifiable source. Are there any reputable scientific journals that reviewed this book? Any verifiable evidence Dr. Ginzburg peer-reviewed this book? Tailpig 16:05, 26 October 2006 (UTC)

Would a FAX with a signature from Drs. Ginzburg or Miakishev be such a verifiable evidence? Then tell me FAX number. How can scientific journals review a book? You either publishing a paper in a journal or prepare a book for publication. In the latter case, the publishing agency is probably responsible for the peer reviewing process, although I have never published a book. Of course, Lev does not say that aether exists in his book!!! Is that the reason you think this book is wrong? He discusses this question in a historical context, and also he uses this to explain everything more clearly for the readers. "Let's imagine for a second that aether does exist. What is then?" Of course, such explanations are appropriate. Dr. Ginzburg and others understood that. To the contrary, Lev says that aether does NOT exist, it "fades away". You are welcome to ask any other scientific questions about this book.Biophys 16:43, 26 October 2006 (UTC)

A FAX is not a verifiable reference. This is not the place to discuss this book. This discussion is over. It lasted long enough and we're still left with a book, on a webpage, with no verifiable scientific backing.

Just for kicks, can you get Dr. Ginzburg to post here "that he is sure that this book by Lev Lomize is correct and great"? Tailpig 17:15, 26 October 2006 (UTC)

Nasty ;-)

Here's another kick from page 266:

"The kind of elasticity we are concerned with is a much more general notion than the ordinary elasticity as used in mechanics. It refers even to such a non-classical object as a single electron. Let us place an electron in the electric field which is so strong that the electron is accelerated up to the speed of light earlier than its mass has time to grow and prevent the electron from overcoming the light barrier. Will this electron reach the speed of light in vacuum? “Surprisingly”, it will. How strong must be that field? An approximate calculation shows that it must be close to the electric field on the “surface” of the positron which “comes in touch” with the electron, both of them regarded as classical spherical objects of the size about their classical radius. Their contact with each other brings about the well-known result. Both of them successfully reach the light barrier and convert into a pair of photons, which gladly fly away with the speed of light."

DVdm 17:21, 26 October 2006 (UTC)

You did not answer my question: what kind of verifiable reference do you need? Of course I can ask Dr. Vitaly Ginzburg and explain him this situation in Wikipedia. Please tell me to whom in Wikipeda his letter should be addressed and where to put the letter. I am not playing games here. In his book, Lev describes a number of imaginary situations to explain the underlying physics better, such as the existence ot aether, movements with the speed of light or faster, and so. Of course, he does not mean that all such things really take place, as I said about the aether. You should also remember that he had to simplify everything, because the book was written for beginners, althouth it actually takes a lot of effort and critical thinking to read and understand. One should read the entire text to understand, as peer reviewers did. Biophys 18:34, 26 October 2006 (UTC)

Dear Friends! I understand that you are very busy, and I am sorry for bothering all of you here. Please let me explain why I was so persistent. I dedicated my life to science, and I know how important and difficult the scientific knowledge is. I have PhD in physics and I red this book. I know (like Dr. Vitaly Ginzburg and others) that this book explains a lot of very important things that can not be found in any other sources. Unfortunately, very few people in USA or Europe knows anything about this book. This happened because Lev Lomize could not promote his ideas, because he worked in a secret institute, being prevented from any contacts with foreigners or leaving the country. He came to US too late. So what I actually wanted was to provide an opportunity for American and other scientists to learn about the existence of this book. Then, anyone could read it and decide for himself is it right or wrong. Remember, we are not talking here about anyone who is trying to disprove the Theory of Relativity or other crazy things. Biophys 18:34, 26 October 2006 (UTC)

I agree. Let's finish this discussion. I am not going to bother Dr.Ginzburg or others "just for kiks" as Tailpig said. I also agree with SCZenz who said that "a new approach to physics with no peer reviewed scientific journal publications would not be a good addition to Wikipedia". The point is: the approach Lev is talking about is not something completely new. This is something Hendrik Lorentz, Albert Einstein, Ives, Janossy, Feinberg and others were thinking about (as described by Lev). And as such, this subject certainly deserves a separate Wikipedia article. One of possible titles for such article would be "Relativity based on physical reality" as Janossy said. I hope that the book by Janossy is a good source for Wikipedia, although it is much more difficult to read and understand then book by Lev. Actually, this is not a single article, because each relativistic effect should be presented and explained as a separate sub-article. So, I have to ask for help from other people who are better qualified then me to do this job for Wikipedia, some of them perhaps from Russia and Israel Biophys 13:38, 27 October 2006 (UTC)

Physics in spacetime

Perhaps the article could be split by moving most of the Physics in spacetime section to a new article. Comments please. MP (talk) 14:55, 27 October 2006 (UTC)

Why? I do not see the point of making such a separation. However, if it is done, the subsection "Electromagnetism in 4D" (of section "Relativity and unifying electromagnetism") which was originally part of section "Physics in spacetime" should go with it because that subsection depends on notation defined in "Physics in spacetime". JRSpriggs 06:40, 28 October 2006 (UTC)

Why? - because the average reader of this article will not gain a true understanding of the role of spacetime physics in SR. There is just too much maths in that section (and possibly other sections) that for nonspecialists it will be off-putting. If we are to keep the section on spacetime physics, then a less mathematically verbose version should be used. MP (talk) 17:43, 28 October 2006 (UTC)

I think the issue boils down to what do you want a reader to be able to know/learn without vector-matrix/vector-tensor notation/mathematics which isn't already explained in previous sections of the article? As far as I can tell from a quick glance, the article covers the bases of relativity pretty well as far as physical description goes. The space-time sections don't, I think, add any new physical behaviors, but just give you a new and innovative way of looking at the physics (that is that positions in time and space and movement though the same can actually be thought of as pretty much the same thing, and not two different things). Perhaps there needs to be a better bridge between the sections on the algebraic, non-space-time way of looking at relativity and the sections on the tensor-based, space-time way of looking at relativity pointing out that though the math looks different, there's really nothing very new in terms of phenomena. Time will still dilate and lengths will still contract, and so forth. Without tensor math, space-time relativity could probably be summed up in a paragraph or two. But the real advantage is in using that math, and though the notation is odd to people who haven't seen it, it's not *that* complex (there's no calculus, the equations are pretty short, and they all correspond to algebraic equations which are already explained and which can be referrenced if necessary).

I should also point out that there is an article which is supposed to take an even simpler approach (the Introduction... article). But in the end, aside from perhaps something of a disclaimer saying "Okay, now things get a little more complicated, but this is all just a different way of writing what's already been explained in a way that suggests a different way of looking at things, which is...", where a couple of sentences explaining the gist of the space-time view follows. Anyway, that's my opinion. DAG 22:04, 28 October 2006 (UTC)

The articles in Wikipedia are supposed to address readers at many different levels of sophistication. Generally, one begins with the simpliest and most intuitive version and works up to the more esoteric and mathematical stuff. These sections are at the end of this article for that reason. And that is where they should remain. Nothing requires you to read the whole article. Just read until it becomes too difficult for you and then stop. JRSpriggs 11:04, 29 October 2006 (UTC) P.S. You can ask questions on this talk page, if something is unclear. Your question may also lead to a change in the article making it easier for everyone.