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The (E) theory: A new attempt to unify EM and gravity

  1. Nov 12, 2005 #1
    This paper (broken into three peaces; see attachments below) introduces a re-discussion of the Lagrange density. The tool to get this result is a new mathematical operation whose existence can be guessed from some considerations concerning the calculation of the variations along the time of the Poynting’s vector in any part of space-time without charge or current source. We show how it can be rooted in the tensor calculus. We demonstrate that it allows a re-writing of the very usual Maxwell’s EM field tensor and after that of the momentum-energy tensor carried out by the EM field. A part of it is calculated (the Lagrange density) and can be addressed to a spin-spin self-interaction. We suggest a connection with the variations of the 3-D volumes and with the QM representation. Keywords: polarizations; tensor calculus; reformulation of the Maxwell’s EM field tensor, reformulation of the Lagrange density, spin-spin interaction.

    1. The opening post must contain an abstract stating the results obtained and how the new theory is at variance with currently accepted theories.

    The introduction of what I have called "the (E) question" during the calculation of the partial derivates of the Poynting's vector with the help of the Maxwell's Laws for EM fields in vacuum yields an unexpected dynamical law... This was for me the starting point for series of explorations. After a systematic investigation of the question in any 3D space and a verification, I could rewrite my first intuitions under a better formalism and prove that the (E) question is connected to the introduction of a new mathematical operation that is an extension of the tensor calculus (actually only in French language).
    Also starting from the systematic exploration in any 3D mathematical space, I could win the irrational sensation that the Maxwell's EM field tensor should be in any manner connected to this extended vector product. This was the beginning for the development of the so-called Russian Dolls Method which, in turn, is now leading to a quite more general approach and formalism for this tensor.

    I have abandoned the obligatory and limiting reference to the esthetic of the tensor to concentrate myself on the idea that this tensor must have an alternative formalism connected with the existence of a special family of extended vector products. Doing so, I am slowly but surely suggesting the necessity to substitute the traditional EM potential vector by the cube* locally defining the extended vector product under consideration.
    This point is (I think) the detail where the present theory (still under construction) is at variance with currently accepted theories.

    2. The opening post must contain a section that either cites experiments that have been done that decide between the new and old theories, or it must propose experiments that could be done to decide between the two. If the submission contains a theory that is empirically equivalent to an existing theory, then this section may be substituted with a section that demonstrates the empirical equivalence and that compares and contrasts the insights gained from the submitted and existing theories.

    This is done with the document etgb43.pdf (not enougth place to upload, sorry; but was submitted in my first essay) which is practically devoted to demonstrate the ability of the present theory to absorb (incorporate) the question of the anomalous Hall effect. This section does not pretend to give an exact correspondence with the experiments made by the team cited in the reference. It is only opening a road.

    3. All references to relevant prior work must be documented in the opening post.
    4. Quantitative predictions must be derived, wherever appropriate, and mathematical expressions and equations must be presented legibly, using LaTeX whenever necessary. This should be done in the opening post. 9. External links will be permitted only for lengthy derivations and for diagrams. Any other expository text pertaining to the submitted theory must be posted at Physics Forums. Please note that this is a temporary Guideline that will remain in place only while we work on enlarging the maximum allowable attachment size in the IR Forum. Once that happens, we will require that all material pertaining to the theory be either posted at Physics Forums or attached to the thread.

    As recommended by the administrator of this website, I have broken my documents.

    5. New theories must not be already strongly inconsistent with the results of prior experiments.
    6. If a new theory is strongly inconsistent with prior experiments, but the theorist is insisting that the experiments were either misconducted or misinterpreted by the scientific community, then the thread will be rejected. Instead the theorist should rebut the contradicting scientists in an appropriate journal.
    7. Theories containing obvious mathematical or logical errors will not be accepted.
    8. Threads which contain obvious misrepresentations or gross misunderstanding of basic accepted science, especially when used in attempt to compare one's personal theory to currently accepted science, will not be accepted.

    Since this it is an allowed re-submission, I suppose that my theory is not containing too much errors (at least I hope it). Otherwise, I am spending an enormous quantity of time to built a coherent theory respecting all current admitted miles stones of physics. In this sense it must not be so original and interesting. But I am arguing that a re-lecture of a well-known reality with my new binoculars can perhaps open doors that was until now closed.

    This theory is indeed yielding a new formalism for the Maxwell's EM tensor. This formalism takes the local metric tensor into account without introducing any necessary choice for it; this is in accordance with the way of doing developed by the general relativity (GR).

    This new formalism also allows a reasonable connection with (and demonstration of ) the Lorentz-Einstein Law, provided a certain number of hypothesis which are in particularly reducing the number of independent coefficients in the local cube*. It can be shown that such reduction can transform the local cube* into a set of (4-4) matrices defined up to the term in position (0-0); that is a set of matrices with at most 15 independent terms. Such kind of reduction is transforming the definition of the extended vector product in a 4D space in such manner that it can be understood as very similar with the definition made intuitively in our systematic exploration for 3D spaces.
    The substitution (EM potential vector -> cube) is mathematically compatible with the use of the total differential of any quantity with two labels.

    It also allows the definition of the EM fields in terms depending on the coefficients of the cube and, as said above, to take in consideration the anomalous Hall effect. Some fundamental laws of electromagnetism are yet under examination; a first step shows the compatibility with the Kähler's metrics.

    Despite of its actual incompleteness, the theory provides a great number of investigations and the first interesting results in mathematics and in physics. Next steps will focus on the discovery of a group structure for the polarized matrices and on the construction of a perturbation theory based on the exploration of the Holonomy of the group. An original connection between conoides and spinors is under study.

    Best regards / Black forest
    Last edited: Feb 2, 2006
  2. jcsd
  3. Dec 14, 2005 #2
    First of all, want to say thank you for this posssibility to get an official discussion about my work; hope it will not be extremly short because of a stupid error from me. Thank you in advance for your critics...
  4. Dec 17, 2005 #3
    If someone wants to know more, he must please visit my homepage. Waiting for your opinions, jugements, critics or suggestions I can adress the first auto-critic to myself: the (E) theory only is a "theory" and because of this, it can be seen as a difficult speculation without connection with the reality; a kind of mathematical alternative presentation of the EM theory. For my defense and to justify a such complicated formulation (in comparison with the usual one), I am arguing that it naturally owns a formal possibility to incorporate the description of the Hall effects. Even if my concrete knowledges about this subject are too limited (I try to learn more), actual litterature introspecting f.e. photonic crystals can be regarded as a natural field of application for my research. So, I think it is in this sense a strategic item for the modern research (quantic computers, ...).
  5. Dec 17, 2005 #4
    Concerning this Hall effect I have proposed a formal investigation in etgb43.pdf
    Last edited: Feb 2, 2006
  6. Dec 17, 2005 #5

    Tom Mattson

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    I am digging into the first paper, and I will post questions once I formulate them.
  7. Jan 21, 2006 #6
    First commentaries

    One month later, the time is come for the first commentaries. In between I did appreciate the developments of the GEM Theory on the other sub forum and, specially, the help of Lawrence B. Crowell whose interventions can be helpful for both: the GEM and the (E) approach. Doug’s work as he likes to say it himself lies more on the real ground and mine is sometimes flying in the kingdom of the speculations.

    But I did have a great pleasure to read last commentaries concerning the point “bi-photons”. And I reproduce here this point for the comfort of coming discussions:
    “At this stage I would say with the GEM proposal that one of two things need to be done. Either the graviton and photon sectors, the abuse with the term graviton with standing, need to be clearly indicated in some way. This might be done with some tetrad formalism or with the embedding of GR and EM into some larger group. Another approach is to somehow show that the spin-2 field of gravity, say in particular in the pp-wave solutions, can be built up from some coupling of photons. A gravity wave is a bi-vector, and in quantum optics there are phenomena of photon bunching or "bi-photons" which are similar to gravity waves. I say similar, for they still interact with electric charges and so forth. In such a theory two photons with aligned spins would interact to form a graviton. Currently such an interaction for counter spin aligned photons generate the particle of weak interactions. In this way at very high energy, probably approaching the Planck energy, two photons would generate a graviton. How this would fit into Doug's theory is a bit unclear. Maybe if Randall et al. are right with so called "soft black holes" that occur at the TeV range in energy the other fields that the photon interacts with have some mass matrix so that there are oscillations between gravitons and photons. By this two photons correlated in a Hanbury Brown-Twiss manner will have some probability of being a graviton. Yet this is pretty speculative. The dust bin of physics is littered with a lot of quantum field theory speculations.”

    At this stage I must explain some choices I have done in the construction of my theory. For example and it is a difference with the Mortimer’s approach (other sub forum here) or with the Klein Kalusa approach, I decided intentionally to work so far as it is possible in a 4D frame. The reasons for this are connected with irrational ideas: something like: if the world we are living in really owns N dimensions, then our mathematics to describe it must also have a N-D underground; since the most part of the population can accept the fact that N = 4, … I agree it is not a scientific demonstration and the representation of the world that our brain is able to built is century dependent; take as example the slow introduction of the notion of time in physics. That’s true: the human brain only built a coherent image of the real world and it acts a little bit like a lens : real world ® image of the world on which the brain can work.

    The other point on which I want to insist after the commentaries of Lawrence B. Crowell is the capacity of my approach to built the Lorentz Einstein Law starting from a matrix representation of the EM field tensor [see etgb31c3.pdf in webpage; visit the panorama page; letter “O”]. This seems to be an encouragement to purchase the research concerning the “bi-photons”. The (E) approach is suggesting a strong mathematical connection between graviton and bi-photon (or something like that) within the frame of validity of the demonstration. It is the first time for me that I discover that my speculations have a correspondence in the reality and that’s why it is a great day for me.

    Best regards
  8. Jan 23, 2006 #7
    Hello Blackforest:

    I've noticed a behavoir in physicist: when they come to an issue they don't understand, they stop. I've seen this in my own work, where I try to explain something, they don't get this one point, and they do not get around that issue.

    I have been very fortunate. This has happened to me at least a half dozen times. It would often take me two or three months, but then I would see why they stopped. When I saw that, I could see that in fact, I was wrong. The fortunate part was that I found ways around the problem that initially was hard for me to see.

    In your paper, I get stopped write at the definition of the (E) question. I don't get simple things like why one would use ()'s. I don't get hard questions like how one could work in 3D when the deep lesson of the elegant theory of special relativity is 4D. I have taken classes in SR 3 times! It is very much against my outlook to embrace a 3D explanation of something that works brilliantly in Minkowski spacetime. I did not get passed 1.1.2.

  9. Jan 24, 2006 #8
    Thank you for the hints. One point is certain: English is not the language of my mother and I understand that it induces for me and you a hard challenge for a good communication. It is a pertinent critic, I accept it and one could say: I should present my work in France in French... In my next life, perhaps.
    But I am a stupid dreamer and I believe in the universality of the human being. Otherwise, I had until now no occasion to do what I should have done.
    The definition of the (E) question that I introduce in (1.1.2.) is a special and very reduced formulation of a general one that acts not only in a 3D space. If you own a kind of product on a vector space E of dimension N on K, call it "o" (that means a function o: E x E --> E) and if you take two vectors u and w on E, can you allways find a square matrix (N-N) with components in K call it [M] and a rest vector z of E so that u o w = [M]. w + z? That is the (E) question. If you make the particular choice of N = 3, o = cross product and if you do the calculation, you will find a trivial answer given by ( I hope you can understand me.

    Why do I make this simple choice? For the pedagogy of what is coming after this definition: the demonstration of the existence of an equation motion in Maxwell's vacuum.

    Sure, mathematically speaking, it is a reduced and simple formulation. But I think that it contains important informations for physics. Considering a classical point of view: why should we find any neutral stream in Maxwell's vacuum?
  10. Jan 24, 2006 #9
    Hello Blackforest:

    I was not, and will not, complain about your English. That would be unfair. Communication may be inefficient, but I can adjust to that. I think even in German that using a pair of () around a single letter is odd, in an almost English way.

    Math does translate pretty darn well across cultures. But then you must learn the proper math language. Take this line:

    >If you own a kind of product on a vector space E of dimension N on K, call it "o" (that means a function o: E x E --> E)

    This looks like basic group theory in some ways, the function o acting on elements of a vector space E stays within E. If this really is group theory, then you are obligated to use the language of group theory.

    >you allways find a square matrix (N-N) with components in K call it [M] and a rest vector z of E so that u o w = [M]. w + z

    This makes no sense to me. I have no idea what a "rest vector z of E" means. The function "o" is so general that the equality looks meaningless. I don't know if that "." is the end of a sentence or not. At this point I stop. If I were to continue, which I initially didn't, you claim now that the function o is the cross product. So the cross product can be written as a matrix, not news. I still see no logical connection to w and z. I see no question at all. That is my unvarnished reaction.

  11. Jan 25, 2006 #10
    Hello sweeter
    Nobody can force you to have an eye and a ear for my work. If you understand what I mean in reading my exposé despite a bad esthetical presentation, then you understand the message, the information, the most important part of the work. Outfit is another side of the whole thing; and for me, it comes only at the end, only if the main ideas own some value. If you stop the analyze just because of this point, I am sure you are missing a lot of beautiful and meaningful sides of the life. (Black forest; personal and general philosophy)

    Concerning a matrix formulation for the cross product, indeed, no news. The small difference is perhaps that we get a lot of possible solutions ([M], z) to this mathematical problem. Definition of a rest vector is given in the work (if you take the time to read it).

    It is not because a mathematical problem (question) is a priori embedding a great number of special cases that it is uninteresting. I think exactly the opposite.

    Take this other eventuality: The “o” product is supposed to be defined on E, N = 4 , [M] is a Lorentz matrix and z is a translation; then you remark that this special formulation of (E) question in a 4D space is inducing another one: “Can we find a subset of E, call it U = {u so that u o w is the transformed of w in SR}?” It is another way to scrutinize a well known problem in physics (look at the Poincaré transformations).

    But you are obliged neither to read nor to see. Thank you for the effort to make me better...
    Best regards.
  12. Jan 25, 2006 #11
    Hello sweeter
    Just for fun and to demonstrate that I can work with the usual laws of physics in a 4D space
    Best regards.
    Last edited: Feb 2, 2006
  13. Jan 25, 2006 #12
    Hello Blackforest:

    I think you are missing the point. In my thread, Careful thought what I had written was math nonsense. It is right on the first post: [itex](\nabla_{\mu}A^{\nu}+\nabla_{\nu}A^{\mu})[/itex]. It took me a while, but I eventually agreed with Careful: it is math nonsense. It was fairly easy to correct, just make all the indices be on the same level.

    I have no doubts you can work in 4D, and that you have put a big investment of time into your efforts. We have small side comments by two other people that indicate they may have printed out your pdf's, but so far have not posted here. They may be having a similar reaction: the (E) question is not a well-formed mathematical expression. Oh, it has parts of a well-formed expression, but that is like having parts of a well-formed sentence that ends in something that no one else can understand. With the LaTeX tools available here, you should be able to clarify what is the (E) question. At this point, I completely do not get it.

  14. Jan 26, 2006 #13
    I am mising something but I get the biggest difficulties to understand what. General presentation of the mathematical problematic is written pages 5-7 in the first document. I think the difficulty comes from the fact that one have in fact two questions in one: 1) the extended vector product that is an inner operation on E; 2) the problem of the decomposition of any extended vector product in a mixed matrix-language, i.e. the ([M], z) pairs. The frontier between these two sides of the problematic are perhaps a little bit unclear in my exposé. I have tried to give the essence of my thougths and probably I did not spent enough time in the important details of a more extensive presentation. Perhaps is it what you mean? Or do I miss again?
  15. Jan 28, 2006 #14
    Division algebras

    I still have no idea what the (E) question is. This may or may not be relevant, but I will say what I know about group theory and fields.

    The real numbers are a field. That means there is a group operation, lets call it plus, on any two members of the reals such that there is an identity (zero, because 5 + 0 = 5) and an inverse (5 - 5 = 0). There is another group operation on the real numbers without the additive identity, and that is multiplication. There is an identity (one, because 5 * 1 = 5) and there is always an inverse (5 * .2 = 1). Many folks do not realize that the foundations of calculus depend on the properties of mathematical fields.

    The complex numbers are also a field. The additive identity is (0, 0) on the R^2 manifold, and there is an inverse for any given complex number. If one excludes zero, then the multiplication operator has both an identity (1, 0) and an inverse (z*/||z||).

    What comes next? I hope you know the answer already, because if you are concerned about the completeness of number theory, this is a fundamental thing to know. The next field is known as the quaternions. For folks who have never heard of them, you can visit my web site devoted to the subject, quaternions.com. The words scalar, vector, curl, cross product, divergence, gradient, and curl were all coined by Hamilton for his description of quaternion operations. Gauss was the first one to find them, and Rodriguez put them to use independently for 3D rotations. The Pauli matrices are a bad copy of quaternions, where an extra factor of i is tossed in, making the Pauli matrices extremely convenient. Frobenius was able to prove that if you wanted to work with a division algebra over the real numbers, there were only three choices: real numbers, complex numbers, and quaternions. If you drop the requirement that multiplication is associative, then one can add octonions or the Cayley algebra to the list.

    Remember, I don't understand your question at all, so I am going to guess at what it proposes. It looks like you wonder if the cross product is enough in R^3 along with addition to make a mathematical field. If the question is something like that, the answer is known: that much algebra is too little. One needs to work in four dimensions. The product is more than just the cross product, because one would be leaving out the vector dot product and two scalar-vector products.

  16. Jan 29, 2006 #15
    philosophical motivations

    Hallo sweeter
    Concerning the first part of your intervention, basics knowledges about group theory, ring theory, “body” (corps; e.g. real numbers or complex numbers) are present in my small and personal package. Quaternion as well and I know an article demonstrating the possibility to write some fundamental equation of the general relativity (GR) with. You can actually read my efforts to develop the “extended vector product” like a group operation on my webpage in etgb54.pdf (news). It is not trivial at all but very interesting.

    The (E) question is not something like that you say in your remark; at least not in my head. I shall try to explain more precisely. The reason why I did start this mathematical exploration was the demonstration of the equation of motion in Maxwell’s vacuum. In the past (thirty years ago), when I have developed it, following the work of some physicist, I only was a student and did not own the necessary package to write these things in a 4D language. Despite of this lacuna, the demonstration exists, makes use of the (E) problematic with the cross product and tells the question of the existence of some streams in Maxwell’s vacuum. Today, this point is perhaps not surprising because of the development of the modern physics.

    There stay actually two interesting items for me; can we consider vacuum like a dielectric body and make one of the two hypothesis that was yielding the final result of my demonstration? Can we make use of this (E) question in the context of a vacuum; and if yes, can we develop it more generally; in a 4D language for example?
    You ask more or less directly if it is relevant. Because we now know the existence of neutral energetic streams in vacuum and because the quantum theory furnishes quite better tools to explain and explore this side of the nature, this quasi classical approach can give the sensation to be an old timer tool. Or can give the hope to connect two different approaches of the same thing.

    Concerning the 3D formulation, I think that one can argue that what happens to us takes always place in a local time slice of our own chronology. Thus, a apparently limited formulation in 3D must contain enough interesting informations about the whole thing occurring in a 4D world. It is our duty to guess, reveal and formulate the truth that is implicitly contained in a restricted formulation. The hidden side of the Maxwell’s equations certainly is the geometric context where they act. Despite the fact that we never take care of it because we usually don’t have to do it (due to the Euclidian nature of our direct environment), a more general theory must include the unconscious perception that our geometry is only one of the possible one. The (E) question is this crazy attempt to connect the way how a cross (or more generally an extended vector) product of two vectors can split [the ([M],z) pairs] in accordance with the local geometric nature at the place where this product is calculated. Thus, my theory says: if our world wouldn’t be Euclidian, then we would do all our mathematical operations in a different way; but there exists a tool to connect these different ways: this is the (E) theory. Do you follow me?

    And that’s why I think the (E) question is complicated, relevant and a kind of philosophical extension to the theory of the relativity. I really hope that I can convince you and some other people. Blackforest
  17. Feb 2, 2006 #16
    This is extremely difficult and disappointing to speak in the desert; it looks like if nobody here would be interested in my essay.

    As supplementary advantage concerning the development of the (E) theory, one can also remark that division of vectors does not directly exist inside the mathematics. We only own the concept of modulo-spaces ...

    Following some remarks made on an other sub forum here, one can remark that:
    1) in the LE2 basis of the (E) theory, i.e. basis where the Lorentz Einstein Law can be understood as a vector differential operator of second order (a method that I have proposed to develop an approach "à la Froebienus"), the set of all Christoffel's symbols (what I call a cube in my poetic language) is a Z2-grading set...
    2) Otherwise, basis of which the deformations are satisfying a description via the moving frame method by approximation up to the third order are naturally yielding a set of Graßmann variables...
    3) The extended vector product acts like a projection for positions vectors only in a LE2 basis
  18. Feb 2, 2006 #17
    Hello Blackforest,
    Interest must be cultivated. The mathematical context of your discussions with Sweetser is beyond me but putting that aside I think the problem with your essay is that it does not appeal right from the beginning, due to its layout.

    The first few sentences of your essay:
    The most important text in any article is the abstract. Reading your first lines, it seems that the abstract is supposed to be in the Introduction. So the essence of your story is expected there. The essence of your introduction is the statement that "A great number of equations is not necessary the best medium to explain a way of thinking", so for the unwary reader this obviously is what you are trying to tell him in your article. Next there is a great number of mathematical definitions where at no point it is clear what all these definitions are going to lead to, apart from the fact that it immediately contradicts your statement in the introduction. I think most readers drop out halfway your definitions.
    You don't have an abstract that gives a concise overview of the main ideas and points you want to make. Next, your introduction should at least give an overview of the chapters and their purpose, as well as perhaps some background that any reader should have. [EDIT: I must correct this slightly: there is an abstract in your opening post but that is not reflected in the article itself]
    In short: your article would not be accepted by any journal, purely because of its layout.
    I can fully understand Sweetser's remarks about not knowing what the (E) question is. I don't know either.
    Please take this as positive criticism.
    Last edited: Feb 2, 2006
  19. Feb 2, 2006 #18
    No problem; that's exactly what I wanted to ear: the truth! OK, fine; and thanks.

    So, considering that I shall never be able to master your language and the language in general (even in my own one), I shall consequently leave the discussion here because the challenge is too high and will bring me no ligth.

    Mentors of this forum can erase this sub-discussion.

    TP alias BF
  20. Feb 3, 2006 #19
    Does someone knows something about this new model suggesting:
    ..."that shocking a crystal will produce synchronized light".
    Materials Update alert 03 February 2006 (Nature review)
  21. Feb 4, 2006 #20
    Hello Mortimer, some more commentaries without importance for the theory:
    Black and Mortimer are two well known heroes in the world of the cartoons; I recommand you to read it to relax from your work. Your childrens will also certainly appreciate. Once more time the critic here lies on the layout, not on the ground of my work.

    I cannot believe that well educated people like you on these forums don't understand the simple (E) question. That's why I am now convinced that nobody does really want to discuss it and that you only want to see how far an amateur like me can go. And byside, if I sometimes have a good idea, you can always borrow it; isn't it? That's the reason why I shall no more take part to the pseudo discussions unless there is some one here to explain me a good reason to do it.

    Please take this also as positive criticism.
    Tchao bye
  22. Feb 14, 2006 #21
    <See "etgb531.pdf"> Extended Vector Product; (E) question; A.D.M. approach [12 - 14 February 2006; 3 pages; .pdf]
    Some readers seem to be lost in the (E) Theory because they don't understand its purpose or what one can do with it. As we could show in the "elementary properties of any extended vector product (EVP)"<see etgb54.pdf">, a given EVP is entirely defined by real physical circumstances. The EVP itself is a general mathematical concept. The cross product in a 3D space, the commutator of two basis vectors in any N-D space, ... are particular examples of a quite more general tool.
    I purchase here the development of the hypothesis that an actually unknown kind of EVP is defined in the universe that must allow the description of any EM field in accordance with the Maxwell's Laws. It has the important consequence that the Faraday Maxwell tensor should split under the: [F] = p.[G].[P] - q.[P]<sup>t</sup>.[G]
    formalism where (p, q) is a pair of scalars, where [G] is the local matrix representation of the metric tensor and where [P] is the local matrix representation of the split of the unknown EVP we are looking for.
    I consider here the A.D.M. slice between space and time and apply it to the research of the [P] (4-4) matrix for the case (p,0). I show that relatively simple solutions exist.
  23. Feb 15, 2006 #22
    still don't get it

    Hello Blackforest:

    I will give a few more hints about my complete lack of understanding of any of your work. I do not think it is fair to an audience in an http based forum to reference pages in a pdf. I for one do not have the time to print out and reference it. The basic idea should be easy enough to explain with words and a few equations. So you tried a number of posts ago...

    OK, "how a cross (or more generally an extended vector) product of two vectors" sounds like what is called an exterior product or Grassmann algebra. I do NOT understand the topic myself. If you are doing a variation on Grassmann algebras, you must stick to the language that those folks use. I have no idea what M is or z or why they should come as a pair. This phrase sounds undefined: " the local geometric nature at the place". Geometric nature? Is that topology? Is it the structure of the manifold?

    At this point, I do not know what the pair of things are, nor how they are to be evaluated in terms of geometry. So no, I do not follow you.

    Then I am confused by this statement: "if our world wouldn’t be Euclidian [sic]". Minkowski spacetime is not Euclidean, nor are the dynamic metrics of general relativity or my own work Euclidean (and by that I mean multiplying to vectors generates the sum of the squares of all the component parts).

    I consider it a warning sign when independent researchers use new abbreviations, such as EVP. I go to great effort to avoid them myself.

    For me, the E question remains poorly formed and not comprehensible.

  24. Feb 15, 2006 #23
    Hello Doug, at the end I find our dialog a bit funny and your behavior not only full of sympathy and patience for me but courageous.

    OK. But how do you want to discuss seriously about physics and mathematics only with words? I can agree with your principle that the author should be able to explain his theory with the usual vocabulary; conversely, you will probably also agree that an obligatory trip into the darkness of sometimes long and complicated calculations (despite my stupid introduction as noticed by Mortimer) cannot be avoid. This means: the documents that you can visit on my webpage are the illustrations of the thoughts I try in vain to explain here. Furthermore, I hope that my mathematics can equilibrate the defaults of my language. Or said with other words: reading my mathematics will perhaps give you the sentences that I am unable to formulate with the pencil or the tongue. I have at least hoped it. I thought that mathematics was a kind of international language for which you don’t need a translation in the literature or that the reader can translate itself in his own language. Apparently, I am thinking wrong or you don’t totally agree with my viewpoint.
    If you speak of Graßmann algebra, this means either that you visited my webpage (Thank you) or that my problematic implicitly contains this topic (I am glad of this). I must give it: this topic is also new for me. I don’t know how you proceed to built your own GEM, but since I am not a genius and since I only follow my intuitive representations of the universe, I progress by step. At each step I need mathematical tools of which I did not necessary knew the existence before. A typical sickness for an amateur, I suppose.
    I am now sure that the difficult things here is not my bad English but that I am following several roads at the same time. The question with the Graßmann algebra arises not from the article that I have proposed here on this forum for independent research, but from a parallel investigation concerning a construction of the general relativity (You remark: I didn’t write GR) based on a small personal treatment of the “moving frame method”. In this special investigation, the (E) question does not immediately plays a role.
    Of course “Minkowskian space time is not Euclidean” [sic], but the spatial part of it is … and your remark makes me totally clear that you do not understand the idea, the mathematical idea. If I could: I would cry! How can I explain it? Try to give a little bit liberty to your mind. Not for definitively abandon the rationalistic way of thinking; just to understand the idea.
    For example, try to remember all your basic lessons in mathematics and make a list of all mathematical operations you know on N, Z, R, C, and so and… After that look if you can find somewhere in the list “the division of two vectors”. As state in “Essential Mathematical Methods for Physicists” (Weber and Arfken; international edition; Elsevier; 2004), you will not; of course, there is not! The very initial idea on which lies the (E) problematic is based on this remark.
    We can multiply two vectors: it is the cross product (in N = 3 space) or the wedge product if you prefer the generalized version. And now take a vector, any one; take another one and try to find what the result of a division of the first by the second could be; try to find it in thinking that the wedge product between the hypothetic result we are looking for and the second vector you have chosen should yield the first vector. You intuitively guess that this result must (or should) be a combination of a rotation [the hypothetic M matrix] and of any translation (the z-vector; other said: the rest vector). If you don’t understand this step, I am afraid, I shall be obliged to give up my explanations. My neighbor who has a PhD level does understand my preoccupations and the main idea even if he says, like you, that the things are not optimal formulated. But try to do it yourself in German or in French language (joke!).
    If you accept this first intuitive introduction for the (E) problematic, then, yes, you can understand that I have tried to extend it in considering an extension of the wedge product that I have called the extended vector product; secondly you will rapidly understand that the split into ([M], z) is clearly depending on the topology of the space where it occurs.
    When an independent researcher use new abbreviations, such as EVP, it is only because he is describing his own concept in a more convenient manner… just because it realizes an economy of ink! But despite your courageous try to understand my way of thinking, I feel through your words that you are tired of it. Nicht desto trotz: Thank you for your efforts.
  25. Mar 5, 2006 #24
    The extended vector product: Definition

    Last edited: Jul 2, 2007
  26. Mar 5, 2006 #25
    You certainly understand that with this definition, any connection can be used to define a precise extended vector product and that if the extended vector product is define somewhere by the components of the connection, then the evp of two given vectors, say u and w, is the difference between the covariant derivative of w with respect to u and the usual derivative of w with respect to the vector u.
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