Essay Abstract

This text demonstrates that how we think about both Mathematics and Physics can be influenced by the mathematical tools that are available to us. The author attempts to predict what Newton might have thought and done if he had known of the works of Euler and Hamilton and had been familiar with the matrix methods of Linear Algebra. The author shows that Newton would have come very close to Special Relativity.

Author Bio

I was educated as a Chemical Engineer with BS and MS degrees from Texas A&M University. My interests in Physics are the wave equation and Hamilton's quaternions.

Download Essay PDF File

Thank you for taking the time to read and consider my essay. I hope that you judge it to have been time well spent.

I was very casual in the presentation of the derivative of a quaternion with respect to a quaternion. For a more complete development of the subject, please refer to the following URL: http://vixra.org/author/gary_d_simpson. The title of that work is Quaternion Dynamics, Part 1 - Functions, Derivatives, and Integrals.

I want to elaborate some regarding Equations 11.1 - 11.4 and Equation 12. I did not think that I could do justice to the following ideas in the remaining amount of space for the essay. Also, I have not fully developed the ideas. Therefore, I stopped at a place that seemed to make sense.

Equation 12 is a quaternion representation of time. It consists of a scalar term and a vector term (i.e., a time vector). It is also presented as a quaternion version of the Lorentz Transform. A phrase such as "time vector" probably makes you a little skeptical. Would a phrase such as "the arrow of time" be more satisfactory?

When the dot product of two vectors is equal to zero, the physical meaning is that those two vectors are perpendicular. When the cross product of two vectors is equal to zero, the physical meaning is that the area of their associated parallelogram is zero. Typically, this means that the two vectors are collinear.

Equation 11.1 is the negative of the dot product of the velocity vector and the time vector. Setting it equal to zero implies that the velocity vector and the time vector are perpendicular. The definition of the time quaternion is such that the velocity vector and the time vector are perpendicular. Therefore, Equation 11.1 is automatically satisfied.

Equations 11.2 - 11.4 are a mixture of terms. I have grouped the cross product terms in square brackets to make it easier to understand. For the general case, the position vector is not aligned with the velocity vector. Therefore, Equation 1 requires the resulting time vector to be perpendicular to both the position vector and the velocity vector. For the special case where the position vector is aligned with the velocity vector, the time vector is zero and the time quaternion simplifies to a scalar value. Substituting the Lorentz Transform form of Equation 12 for the time vector for this special case causes the velocity-time cross product terms to become zero. Usually this would mean that the two vectors are collinear. But, the time vector cannot be aligned with the velocity vector because Equation 11.1 requires that they be perpendicular. My best guess is that the Lorentz Transform is applicable to the special case because the time vector is zero and hence it can "point" in any direction desired. I will explain this more clearly using one or more drawings in future work.

Best Regards and Good Luck to All.

Gary Simpson

Gary,

A very interesting essay, concluding with a quaternion representation for time that I will need to think about. I am a fan of Hestenes' writings on Geometric Algebra, and tried to keep his "Spacetime Physics with Geometric Algebra" in mind while I worked through your essay. [particularly his III 'Proper Physics and Space-Time Splits', page 8.] Somewhere along the way I got lost. Looking at your viXra papers, I now recall your earlier paper on 'spherical time'. Are the two related?

One of the very nice features of FQXi comments is that they give an author a place to expand on his essay. You derive the fascinating quaternion representation of time [your equation 12] and then you conclude the paper. I would be very interested in anything you might wish to say here about this view of time. [I've already read your comments above in which you do expand some.] You've obviously been thinking about this for a while, and perhaps have some useful remarks on seeing time in this perspective. It hasn't clicked for me yet. I'll try to read it again, and work through more of the calculations.

I was recently reminded that, while my time axis is perpendicular to the spatial dimensions in my rest frame, my time axis is not perpendicular to the spatial axes for an observer in an inertial frame moving with respect to my frame. This makes it little easier to see how "time dilation" occurs. I'm trying to visualize your quaternion representation of time in something like this perspective. Any comments? You define the velocity vector and the time vector as perpendicular. This seems to agree with also being perpendicular to the space axes of the above, but I'm not sure.

FYI, I just finished reading "Oliver Heaviside" by Paul Nahin. It's a very fascinating book and has a chapter on 'The Great Quaterniononic War' that you might enjoy.

Thanks for offering what seems like a unique view on things in an "alternate history" format [one of my favorite narrative vehicles.]

Best regards,

Edwin Eugene Klingman

    Edwin,

    Thank you for the comments. I had hoped that the alternate history vehicle would not be too tacky. I had just completed some work on quaternion functions and derivatives and I thought the subject matter was a good fit with the essay contest. I simply needed a way to convey the main result (that they differentiate exactly like real functions). Who better to convey that than Newton himself?

    The only connection between this essay and my previous FQXI essay is that it pointed me towards a deeper interest in quaternions and vectors as they pertained to differential equations.

    One thing that I wish to emphasize is that the i, j, and k vectors in the time quaternion are the same i, j, and k vectors that are in space. Also, the scalar term is not time or time like as with a four-vector. The view that I am presenting is that the time vector exists within 3-D space. This is not Minkowski space-time. The time vector has units of measure that are seconds or hours or whatever. Those units of measure are associated with the coefficients of the quaternion rather than with a fourth axis. The axes are viewed as giving direction only with no units of measure. So, in the purest sense, a vector divided by a vector should have no units of measure if the two vectors represent the same type of entity (ie, momentum, velocity, etc).

    The vector portion of the time quaternion will only be non-zero if the direction of motion is not aligned with the position vector. As an example, consider the moon moving about the earth. The position vector points from the center of mass of the earth to the center of mass of the moon. The velocity vector is almost perpendicular to the position vector since the orbit of the moon is nearly circular. The time vector is then perpendicular to the moon's plane of orbit. The scalar term of course has no direction.

    This brings out an interesting feature. Hamilton seems to allow curved motion without the need for acceleration. I will need to work and think about that for quite a bit.

    The time vector and the velocity vector are definitely perpendicular to each other. Of course, it is possible for many time vectors to exist in the same space. Regarding time dilation and other such effects from Special Relativity ... I am not certain, but I think that the Hamilton related transforms might be the inverse of the Einstein related transforms. I purposely stopped my essay short of that point because I have not done the work yet and because there is no reason to throw gasoline on the fire.

    I know a little about the great debates from 1890-1895. I cannot dispute that we have done a great deal without using the scalar term needed by quaternions. However, I think that a deeper understanding of the world will require that we revisit those debates. Geometric Algebra seems to me to be an essential and profound set of ideas.

    Best Regards,

    Gary Simpson

    Gary, this was a very interesting idea for an essay. I sometimes wonder to what extent pure mathematicians would discover ideas from physics if they had no input from physicists or observation. Complex numbers and quaternions are examples of mathematical ideas that were found to have applications in physics so I think if mathematcians were smart enough and left long enough they would become interested in many structures that were actually first found by physicists, such as Lorentz type groups. Your idea that this could have come about through a study of quaternions is very nice. I think a lot more could be said about what this implies for the relationship between maths and physics.

      Philip,

      Many thanks for the thoughts. I think the Mathematicians would be able to eventually figure it all out but it would take longer without input and feedback from Physics. It is curious though ... it seems to me that historically, the mathematicians have always been 50-100 years ahead of the application. When Einstein did GR, Reimann had already done the math. But today it seems like the math is lagging behind. Perhaps grad students in Physics should be encouraged to study math instead?

      Regards,

      Gary Simpson

      Duh ... you are so right ... and I know how to spell his name correctly too ... I will blame my fingers. Clicked "submit" without a proof-read:-(

      Regards,

      Gary Simpson

      I must read in detail, but I understand that the quaternion algebra can be used like operator of translation and rotation (I read the use in the spacecraft control); it is completely new to me the use in relativity.

      I don't understand - in this moment - if each Lorentz transformation can be a quaternion transformation, but I think can only a boost could be a quaternion transformation (the number of components of the Lorentz transformation are three only for a pure boost).

      It is all clear, and the extension of the use of directional derivative to derive the quaternion is interesting.

        Thanks for giving it a read. You have the main idea I think.

        Regards,

        Gary Simpson

        Gary,

        I found your formulation of a quaternion derivative intriguing. I recall that with complex numbers the derivative of the conjugate is zero: for a complex number z, dz*/dz = 0. This can be deduced from the Cauchy-Riemann equations for a complex number. Taking the derivative of the square of the norm zz* then gives dzz*/dz = z*.

        It turns out things are less complex (if you will excuse a pun) for a quaternion q. I was curious about the derivative of its conjugate, expecting more complexity, but it turns out (thanks Wikipedia) that the conjugate of a quaternion can be expressed entirely using multiplication and addition, q* = -(q iqi jqj kqk) / 2, so that it is easy to see that dq*/dq = 1. Taking the derivative of the square of the quaternion norm qq* gives dqq*/dq = q q* or twice the real part of q.

        Thanks for your essay which helped advance my understanding of quaternions.

        Best wishes,

        Colin

          Colin,

          Many thanks for having a read. Yes, quaternions have some very nice algebraic behaviors. You can sum a conjugate pair to eliminate the vector. You can take the difference between a conjugate pair to produce a vector. You can multiply a conjugate pair to produce a scalar.

          It principle, d/dx(uv) = (du/dx)v u(dv/dx) should be applicable but I have not gone through it in detail and the order of multiplication shoyld matter. My next effort will be developing identities.

          Best Regards,

          Gary Simpson

          Colin,

          Many, many, thanks.

          Thinking about your post some more has made me realize something interesting. The product of a conjugate pair is a constant scalar value (it is the sum of the four squares). Therefore, the derivative of Y = (Q*)Q with respect to Q must be zero (dY/dQ = 0). But if I think of this as Y = AQ and take the derivative with respect to Q, the result is dY/dQ = A. So if A = Q* then something looks to be amiss. I've been wondering what to do with Equation 3 in the work and if there are any Eigen-value type problems that need to be identified and resolved. You have given me a big clue.

          Thanks again.

          Regards,

          Gary Simpson

          11 days later

          Dear Mr. Simpson,

          Thank you for thoughtfully warning me that I needed to have a "knowledge of quaternian and Linear Algebra" in order to understand your essay. Unfortunately, Newton was wrong about abstract objects having the option of being stationary or in motion, and Einstein was wrong for assuming that it was abstract light that was capable of obtaining constant speed. It is the real surface of all real objects that is in the same constant motion at the same constant speed, and as light does not have a surface, light is the only stationary substance in the real Universe.

          Regards,

          Joe Fisher

            Joe,

            Many thanks for making the effort. I hope the experience was not too frustrating.

            Linear Algebra is not too bad but quaternions were unknown to me prior to three years ago or so. It took a lot of effort on my part to stop the voice in my head from telling me that it makes no sense to add a scalar and a vector. What finally convinced me that it was ok was simply using them for what Hamilton intended ... namely, the ratio between non-collinear vectors.

            It is interesting that you believe that light is stationary. In the 2012 FQXI contest, I presented a scalar solution to the wave equation that was precisely that. Also, if you examine Equations 11.2 - 11.4 of this essay and set v = c for any of the velocity components and then apply the Lorentz Transform, the result is that the change in position is zero. Note that I did not say velocity but rather change in position. I'm still pondering the meaning of both of these things.

            I'm catching up with my reading and should be able to comment on your work soon.

            Best Regards,

            Gary Simpson

            5 days later

            Dear Gary,

            Your essay is superb! I gave it a 10 with only a regret that 11 or 12 are not options. It is worth pointing out that your matrix equation for the inverse of y = Qx has Q^{-1} that is the same form as the electromagnetic tensor. This is why Maxwell formulated electromagnetic theory with quaternions.

            If at all possible you might find my essay

            http://fqxi.org/community/forum/topic/2320

            of interest. I did not make explicit references to quaternions, but they are lurking in the background in the discussion on Bott periodicity.

            LC

              Lawrence,

              Many thanks for the kind words. I am very appreciative of your enthusiasm. If I can supply someone with a new tool to use when attempting to solve some of these difficult problems then I will count myself as fortunate.

              Having said that, I should mention that it would be better not to indicate how you might have rated an essay. The administrators at FQXI might construe that to be vote trading and that could be cause for disqualification.

              One of my objectives is to get back to Maxwell in quaternion form. I need to spend some time working on identities and simple kinematics prior to that. I also think that it should be possible to formulate and solve a quaternion style wave equation. That should closely resemble Dirac's solution without the need for factorization using Clifford Algebras. Something that puzzles me regarding that work by Dirac is the equation ab = -ba. I understand non-commutation etc ... What seems odd to me is that the first thing that I think of is simply the cross product of two vectors. If a and b are both vectors then (a cross b) = - (b cross a) for any arbitrary vectors. Obviously, I need to study the subject some more.

              I have read your essay once. I will need to read it one or two more times before being able to make any meaningful comments. I see from your e-mail address that you are no stranger to quaternions.

              Best Regards and Good Luck,

              Gary Simpson

              The Dirac equation can be looked at as the multiplication of quaternions. The Dirac operator is a quaternion valued set of differential operators and the spinor field is also quaternion valued.

              What you might be pondering is the role of forming differential forms from quaternions that are antisymmetric and quantum commutators. One can think of the 1, i, j, k as one-forms that give wedge products that are a generalization of the cross product. I think this is some question with the relationship between the Heisenberg group and quaternions. I think this relationship involves the AdS_5 spacetime.

              My email address golden field quaternions refers to the 120 quaternions in the icosian group, that is half of the E8 lattice. That gets into octonions.

              Cheers LC

              Dear Gary D. Simpson

              Your essay is an example of how learning enables identifying or finding new problems and persistence leads to proposed solutions. Moreover, the presentation of your ideas was logical and accessible, which is considerate of the reader, likely reflecting a desire that reader can share in the benefit of your hard work.

              Does the following question in relation to your essay make sense? Can a space of points described by quaternions each of which isotropically scales by the same increasing (scalar) scale factor model the isotropic cosmological expansion of space?

              Regards.

              Bob Shour

                Dear Bob,

                Thank you much for your generous comments. You are most gracious. I try to write as simply as possible with short sentences in a linear, logical sequence with clear endpoints. Sometimes I have to leave an idea dangling so that I can merge it with something else.

                Regarding your question, I think that I understand your question and I think that the answer is yes. I'll expand on this a little. If you look at Equation 3 in my essay and you make the vectors collinear, the cross product term becomes zero. Now, you only have to worry about the scalar term. I have done the calculation using 13.8 billion light-years as the radius of the universe. This allows me to predict an expansion rate equal to 70.75 km/sec per Mpc. The observed expansion rate is 67.80 km/sec per Mpc.

                So it looks like I can get to within roughly 5% of the accepted value by using a fairly simple analysis. But I think there may be a problem. The way the cross product term is defined, it looks to me like linear velocity and angular velocity are linked (ie, they are not independent of each other). The implication is that if something is far away and moving linearly very fast then the space associated with it must be rotating fast.

                Best Regards,

                Gary Simpson