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RAGORDON2010

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  1. The Forum members are invited to visit two blogs I have created that expand on my earlier postings under the Special Relativity and Quantum Mechanics categories: “Special Relativity from the Inside Out” LINK DELETED “Introduction to Schrodinger Ensemble Theory” LINK DELETED
  2. Introduction to Schrodinger Ensemble Theory Some years ago, I chose to pursue a different approach to the study of the time-independent Schrodinger equation, particularly as it is commonly applied to the following situations: a particle in an infinite potential well, a particle in a finite potential well, the harmonic oscillator, the hydrogen atom. The first group of examples I will discuss are all one-dimensional. The work will generalize when I deal with the hydrogen atom. My concept is simple. For a given potential V(x), suppose \(\psi(x)\) is the solution to the Schrodinger equation in the form: \( (E)(\psi) = ((h/2(\pi))^2/2m)(d^2(\psi)/dx^2) + (V)(\psi) \). Suppose further that an ensemble of identical, non-interacting particles is distributed in real space at time t=0 such that the fraction of particles in the region (x, x + dx) is given by \( \psi\psi*dx \). Suppose, in addition, that these particles exhibit an initial momentum distribution such that the fraction of particles with momentum in the range (p, p + dp) is given by \( \phi\phi*dp \), where \( \phi(p) \) and \( \psi(x) \) are Fourier transforms of each other according to the usual rules. I then require that the fractional density functions be consistent across the two spaces - real space and momentum space. That is, I insist that the fraction of ensemble particles initially positioned in the region (x, x + dx) equals the fraction of ensemble particles with initial values of momentum in the region (p, p + dp). That is, I require that my consistency relationship \( \psi(x)\psi(x)*dx = \phi(p)\phi(p)*dp \) is satisfied. Finally, I use this consistency relationship to seek a momentum function p(x). On the one hand, it may be possible to find p(x) by inspection or via trial and error. Otherwise, it might be possible to integrate each side of the relationship separately and isolate p(x) from the result. Even then, there will still be some freedom left to decide on the direction of the momentum vectors. Please note that for these Schrodinger ensembles, total particle energy is not a "sharp" variable. The expectation energy averaged across the entire ensemble remains the eigenvalue E, but the energy of any individual particle is always computed from \( p(x)^2/2m + V(x) \) in the usual manner. Also note that since psi(x) and phi(p) only represent initial conditions placed on the ensemble, the subsequent development of the ensemble over time is determined by applying Liouville's theorem to the ensemble. I have not found a way to develop Schrodinger Ensemble Theory for the time-dependent Schrodinger Equation.
  3. Must be my browser. Regarding the comments, I ask the Forum to be patient. I think most concerns being raised will self-resolve eventually. A little more background on Related Experiments, and then I will focus on the question of invariance. (Most of us are home-bound anyway because of this damn virus, so it's probably healthy to have some anonymous person on the outside to argue with.) Note to all who view this thread - The count of views to this thread surpasses 60,000. I take this view count very seriously. It is my intention that every one of my posts be an accurate and clear reflection of my thinking. To this end, if I draft a post on, say, Monday, the draft is read/edited and read/edited until, say, Thursday or Friday when I finally submit it to the Forum Unfortunately, this discipline does not hold for my responses to individual comments from Forum members. Those responses tend to be “off the cuff” and ill thought out. In particular, my disrespectful comment on a frame-driven physics in the context of the expression: \( (dS/2)^2 + (vdt/2)^2 = (cdt/2)^2 \). I interprete this expression as pointing to the formation of “Minkowski ellipsoids” that mark off the progress of a particle as it moves along its path of motion under the influence of applied fields. Instead of commenting the way I did, what I should have said, upon reflection, is that neither these ellipsoids nor their defining expressions are intended to be viewed as transformation invariants across a pair of related experiments, or in the conventional sense, across the associated “rest” and “moving” frames of reference. I hope all of this will become clearer to the Forum in my future posts. Special Relativity - A Fresh Look, Part 5 This post begins with the Related Experiments treatment of the “In-Line” Relativistic Doppler Effect and follows with the Related Experiments treatment of the “Transverse” Relativistic Doppler Effect. In his 1905 paper, Einstein* begins his analysis by imagining a monochromatic source placed at rest at a point some distance from the origin of his “rest” frame, Frame K. If we only wish to focus on the in-line Doppler effect, we may limit the positioning of the source to somewhere along the Frame K negative x-axis. *(ref. “Einstein’s Miraculous Year - Five Papers That Changed the Face of Physics", Edited by John Stachel and Published by Princeton University Press, 1998, pgs. 146-149.) Following Einstein’s approach, we write the wave function argument for a light wave emanating from the source and traversing in the positive x direction with frequency f, period T = 1/f, and wavelength w = c/f = cT as it would be recorded by a stationary detector positioned at the origin: \( 2(\pi)(f)(t - x/c) \). For our Related Experiments analysis, we assign the above set-up to our image experiment. We place a monochromatic source with frequency f’, period T’ = 1/f’, and wavelength w’ = c/f’ = cT’ at rest at a distant point somewhere along the negative x’-axis and we place a stationary detector at the origin. We expect that the detector will record a wave function argument equal to \( 2(\pi)(f’)(t’ - x’/c) \). Moving over to our object experiment, we use the a similar set-up, but here we place the source in motion with velocity v in the direction of the stationary detector. We now determine what the detector would record in the object experiment as follows: We substitute for t’ and x’ in the argument \( 2(\pi)(f’)(t’ - x’/c) \) using the Lorentz transformations in the form: \( t’ = (\gamma)(t - vx/c^2) \), and \( x’ = (\gamma)(x - vt) \), with \(\gamma\) defined in the usual way. After some simplification, we will find that the stationary detector records a wave with argument: \( 2(\pi)(\gamma)(f’)(1 + v/c)(t - x/c) \), giving a frequency of \( (\gamma)(f)’(1 + v/c) \). This represents the relativistic Doppler shift for a source moving toward a fixed observer (or equivalently, for an observer moving toward a fixed source.) To determine the frequency transformation for the case where the source moves away from a fixed observer (or equivalently, where the observer moves away from the fixed source), we need only replace v in the above with -v. For the case of the Transverse Relativistic Doppler Effect (TDE), we follow Einstein’s general analysis, but, for Frame K, we position the source at the origin and place the detector at rest at an arbitrary point, point P, on the positive z-axis some distance from the origin. For source frequency f, period T = 1/f, and wavelength w = c/f = cT, we would expect that this detector will record a plane wave emanating from the source with argument \( 2(\pi)(f)(t - z/c) \). For our Related Experiments analysis, we assign Einstein’s Frame K set-up to our image experiment. We place a monochromatic source with frequency f’, period T’ = 1/f’, and wavelength w’ = c/f’ = cT’ at rest at the origin, and we place the detector at rest on the positive z’-axis at a point P some distance from the origin. As in the Einstein model, we expect that this detector will record a plane wave emanating from the source with argument \( 2(\pi)(f’)(t’ - z’/c) \). Moving over to our object experiment, we use the a similar set-up, but here we locate the source somewhere along the negative x-axis and set it in motion with velocity v in the positive x direction. We now ask how the wave emitted by the moving source as it passes the origin would appear to the detector at point P. We substitute for t’ and z’ in the argument \( 2(\pi)(f’)(t’ - z’/c) \) using the Lorentz transformations in the form: \( t’ = (\gamma)(t - vx/c^2) \) and z’ = z, with \(\gamma\) defined in the usual way. After some simplification, we find that the detector records a plane wave with argument \( 2(\pi)(f’)(\gamma)(t - (vx/c + z/(\gamma))/c) \). This represents a plane wave with frequency \( f = f’(\gamma) \) and direction cosines, (l, m, n), with l = v/c, m = 0, and n = \(1/(\gamma)\). We see that the light detected by the receiver is blue-shifted by a factor of gamma. Also, we see that the light beam will appear to be emanating from a displaced source, an example of “aberration”. Let \( \theta \) = angle between the light beam and the x-axis. Let \( \phi \) = angle between the light beam and the z-axes. Then \( cos (\theta) \) = l = v/c, and \( cos \(phi) = n = 1/(\gamma) \). Since \( \theta \) and \( \phi \) are complementary angles, \( cos (\phi) = sin (\theta) \), and we would expect \( (l)^2 + (n)^2 = 1 \), which is true here.
  4. (1) With reference to my expression - \( (dS/2)^2 + (vdt/2)^2 = (cdt/2)^2 \), aren't \( ... \) proper [math] tags? I'm missing something here. (2) In regard to your comment "SR applies equally to uncharged particles. Charge has nothing to do with SR." I believe SR is a theory of electromagnetism - in particular, a theory of interactions between electromagnetic fields and particles carrying electric charge and magnetic moments, i.e., spin. That's where I am mentally at the moment. One more post beginning with a review and then no more. Promise. (1) I've tried to explain my position as clearly as I can at the end of tonight's post. I guess the bottom line is that I am simply not interested in a frame-driven physics. (2) My son living in Cambodia has been tracing our family roots. I'm unaware that he has found a branch in Aberdeen, but maybe one will turn up. I would be most pleased, though, to learn that years from now, a university would be named in my honor. Anyway, back to work. In these difficult times, it's good to keep the mind active. Special Relativity - A Fresh Look, Part 4 I want to begin development of the close relationships between EMF theory, the Lorentz transformations and object/image experiments. But, first, I want to review the differences between Related Experiments as I have defined them in former posts, vs. the conventional Special Relativity viewpoint of a single experiment viewed by observers in a “rest” frame and by observers in a “moving” frame, a frame moving uniformly with respect to the rest frame. In the related experiments point of view, we have an “object” experiment consisting of a particle with initial velocity \(v_0\) free to move under the influence of applied external fields E and H, where (t, x, y, z) represent the 4-space motion of the particle with respect to an origin at (0, 0, 0, 0). We also have an “image” experiment consisting of an identical particle initially at rest and free to move under the influence of applied external fields E’ and H’, where E’ and H’ are image fields of E and H transformed under the Special Relativity field transformations with velocity parameter \(v_0\). In this experiment, (t’, x’, y’, z’) represent the 4-space motion of the particle with respect to an origin at (0, 0, 0, 0). The 4-space observations (t, x, y, z) are related to the 4-space observations (t’, x’, y’, z’) by the Lorentz time and space transformations with velocity parameter \(v_0\). These object and image experiments can be carried out in laboratories widely separated from each other in distance and in time. In the conventional point of view, the frames move uniformly relative to each other at velocity \(v_0\). Once the experiment begins, the particle is acted upon by applied fields E and H as determined by observers in the rest frame, and applied fields E’ and H’ as determined by observers in the moving frame, where E, H, E’, H’ are related by the Special Relativity field transformations with velocity parameter \(v_0\). The progress of the particle is tracked with 4-space coordinates (t, x, y, z) by the rest frame observers, and 4-space coordinates (t’, x’, y’, z’) by the moving frame observers, where (t, x, y, z), (t’, x’, y’, z’) are related by the Lorentz time and space transformations with velocity parameter \(v_0\) (Note, there are some situations where the object/image experiments and rest-frame/moving-frame experiments must be structured differently, e.g., experiments illustrating relativistic Doppler shifts. I will present examples of these situations in my next two posts.) Needless to say, I am not a fan of the conventional point of view. In particular, I cannot “compute" the notion that Frame K observers and Frame k observers move relative to each other at speeds close to light speed. This idea is so bizarre that I often wonder how it found its way into legitimate scientific discussions. Secondly, I tend to look at physics through the eyes of an experimental physicist. What’s the set-up? What will be measured? What’s the underlying theory, and how will the data be analyzed? Conventional SR theory doesn’t offer me much help here. One Forum member has asked why I am spending so much time critiquing a subject 115 years old. My first response is that I do not accept a physics where Special Relativity stands apart from modern quantum views of electromagnetic field/particle interactions such as QED and QFT - Nature cannot be that fractured. My second response is that young students inclined to pursue physics deserve a presentation of the science which doesn’t leave them shaking their heads, rolling their eyes and heading for other fields of study. But, I suppose my best response is to cite the following from a NYT obituary, October 3, 2014, for the physicist Martin Perl: “In a blog post last year, he wrote: 'The time scale for physics progress is a century, not a decade.' “ Given all of the above, I will continue to discuss EMF theory solely in the context of the Related Experiments point of view.
  5. I continue to maintain the position that Special Relativity theory is an offshoot of Electromagnetic Theory. I cannot speak of particles and forces within the context of SR without assuming the particles carry an electric charge and/or a magnetic moment, and without assuming the forces stem from electromagnetic fields. I introduced particle/field interactions in Part 2 when I referenced the thinking of Michael Faraday. My reviews are intended to hold my presentation together so readers will not have to search through my earlier posts each time a post a new one. Tonight I am posting Part 3. For the present discussion, please assume all measurements are made in the laboratory frame. As for the passage of 115 years, I believe that it's time to go back to basics and re-examine the fundamentals. I intend to show that much of real importance has been overlooked. Special Relativity - A Fresh Look, Part 3 To review from Parts 1 & 2 In Part 1, I considered an experiment consisting of a particle with initial velocity \(v_0\) free to move under the influence of applied external fields E and H, where (t, x, y, z) represents the 4-space motion of the particle with respect to an origin at (0, 0, 0, 0). I then defined a “Minkowski differential interval” as \( dS^2 = (cdt)^2 - (vdt)^2 \), where v is particle velocity as the particle moves over a small time interval dt. Over the course of Parts 1 and 2, I introduced four key assumptions - Key Assumption 1 - I assumed that the elements dS and cdt can be viewed as minute, spatial intervals in 3-space. Key Assumption 2 - Borrowing an idea from Einstein’s 1905 paper on Brownian motion, ”On the Motion of Small Particles Suspended in Liquids at Rest by the Molecular-Kinetic Theory of Heat”, I assumed that the time interval dt is very small compared with observable time intervals but still large enough that the motions performed by a particle during two consecutive time intervals can be considered as mutually independent events. Key Assumption 3 - I assumed that the Minkowski differential interval, when written in the form \( (dS/2)^2 + (vdt/2)^2 = (cdt/2)^2 \) may be thought of as an ellipsoid of revolution (“Minkowski ellipsoids”) having elliptic cross-sections with length of major axis cdt, length of minor axis dS, and distance between foci vdt. A characteristic of these ellipsoids is that the total distance from one focus to a point on the boundary of the ellipsoid and back to the second focus equals the length of the major axis. Key Assumption 4a - Building on ideas put forth by Michael Faraday, I assumed that electromagnetic fields interact with charged particles and with particles carrying magnetic moments via a stimulus/response interaction. That is, the presence of the particle in the field initiates a stimulus disturbance that travels outward from the particle at a fixed speed and initiates response disturbances from those elements of the field affected by the stimulus. These response disturbances, in turn, travel back to the particle at the same fixed speed and, arriving at the particle, exert a force or moment on the particle. Key Assumption 4b - I assumed that stimulus and response disturbances travel through electromagnetic fields at light speed c. In this post, I will build on these assumptions and sketch out the rudiments of a new scientific theory within the framework of classical physics - The Einstein, Minkowski, Faraday Theory of Electromagnetic Field, Charged Particle Interaction”, or EMF Theory for short. To begin, EMF Theory gives form and substance to Michael Faraday's vision. We can now speak of a distinct field-particle event beginning with the particle positioned at one focus of a Minkowski ellipsoid. Stimulus disturbances travel outward from the particle at light speed c, triggering response disturbances from the affected field elements. These response disturbances return to the particle also at light speed and contribute to a force or moment acting on the particle This singular event transpires over a time interval dt, during which time interval the particle arrives at the second focus of the ellipsoid after traveling a distance vdt. All of the field elements, and only those field elements, that could possibly participate in the event are enclosed inside the ellipsoid. I will refer to these discrete events as “Faraday events”. Close analogies mirroring the inner workings of a Faraday event are submarine sonar and bat echo-location. For sonar, the speed of sound in water plays the role of light speed. For bats, it's the speed of sound in air. The sequence of Minkowski ellipsoids and associated Faraday events mark off the progress of the particle as it moves along its path of motion under the influence of the applied fields. Each ellipsoid is positioned with major axis along the tangent line to the curve the particle is tracing out at the specific moment. What we have arrived at here is a theory that exactly straddles the boundary between Classical Physics and Modern Physics. In many ways, it is a “missing link”. Personally, I find it to be quite astonishing that the Theory of Special Relativity, a theory so firmly rooted in Classical Physics, can penetrate so deeply into the secrets of Nature In my next post, I will begin to develop the close relationships between EMF theory, the Lorentz transformations and object/image experiments Why are my LaTex math expressions enclosed in \( ... \)as in \( (dS/2)^2 + (vdt/2)^2 = (cdt/2)^2 \) not being transcribed properly? Please advise.
  6. I'm presuming that the experiment that I am defining, which consists of a charged particle moving under the influence of applied fields, is carried out in a controlled laboratory setting. The experiment is reproducible, and the velocity of the particle at a given point in the motion can be objectively be verified. I'm aware that I'm wandering into forbidden territory here when I speak of the "definite motion of a particle" but I hope to be able to resolve these issues to your satisfaction at the tail end of my analysis. I appreciate your continued interest in my postings. Special Relativity - A Fresh Look, Part 2 To review from Part 1 - In Part 1, I imagined an experiment consisting of a particle with initial velocity v free to move under the influence of applied external fields E and H, where (t, x, y, z) represents the 4-space motion of the particle with respect to an origin at (0, 0, 0, 0). I then defined a “Minkowski differential interval” as \( dS^2 = (cdt)^2 - (vdt)^2 \), where v is particle velocity as the particle moves over a small time interval dt. Next, I introduced two key assumptions - Key Assumption 1 - Assume that the elements dS and cdt can be viewed as minute, spatial intervals in 3-space. Key Assumption 2 - Borrowing an idea from Einstein’s 1905 paper on Brownian motion, ”On the Motion of Small Particles Suspended in Liquids at Rest by the Molecular-Kinetic Theory of Heat”, assume the time interval dt is very small compared with observable time intervals but still large enough that the motions performed by a particle during two consecutive time intervals can be considered as mutually independent events. End of review of Part 1 There is a simple 3-space geometry that relates the three minute, spatial intervals dS, cdt, and vdt. Key Assumption 3 - The Minkowski differential interval, when written in the form \( (dS/2)^2 + (vdt/2)^2 = (cdt/2)^2 \), may be thought of as an ellipsoid of revolution having elliptic cross-sections with length of major axis cdt, length of minor axis dS, and distance between foci vdt. A characteristic of these ellipsoids is that the total distance from one focus to a point on the boundary of the ellipsoid and back to the second focus equals the length of the major axis. I will label these ellipsoids “Minkowski ellipsoids”. It now remains to give physical significance to the Minkowski ellipsoid. To this end, with great admiration and respect, I reference the ideas of electromagnetism put forth by Michael Faraday (1791-1867). Near the end of his career, Faraday proposed that electromagnetic forces extended into the empty space around a conductor (Michael Faraday, Wikipedia). [NOTE: I am under the impression that at some point in the past, while I was perusing Faraday’s writings and biographies, I came across the concept I describe below. If a Forum member can point me to a citation, it would be greatly appreciated.] Key Assumption 4a - Building on ideas put forth by Michael Faraday, I assume that electromagnetic fields interact with charged particles and with particles carrying magnetic moments via a stimulus/response interaction. That is, the presence of the particle in the field initiates a stimulus disturbance that travels outward from the particle at a fixed speed and initiates response disturbances from those elements of the field affected by the stimulus. These response disturbances, in turn, travel back to the particle at the same fixed speed and, arriving at the particle, exert a force or moment on the particle. Key Assumption 4b - Stimulus and response disturbances travel through electromagnetic fields at light speed c. In my next post, I will build on Parts 1 and 2 and sketch out the rudiments of a new scientific theory within the framework of classical physics - The Einstein, Minkowski, Faraday Theory of Electromagnetic Field, Charged Particle Interaction”, or EMF Theory for short.
  7. [md65536, I hope to revisit questions about measurements of temporal and spatial intervals once I complete my "SR - a Fresh Look" presentation. Please bear with me. (Incidently, my grandaughter, who seems to have a knack for finding interesting books for me, recently gave me a copy of "What is Real" by Adam Becker (Basic Books, NY 2018), which addresses in some detail the history of the "measurement problem" in quantum physics. Interesting reading.)] Special Relativity - A Fresh Look, Part 1 Consider an experiment consisting of a particle with initial velocity \( v_0 \) free to move under the influence of applied external fields E and H. Let (t, x, y, z) represent the 4-space motion of the particle with respect to an origin at (0, 0, 0, 0). Now define S, the “Minkowski interval”, by the expression \( S ^2 = (ct) ^2 - (x^2 + y^2 + z^2) \). Next define dS, the “Minkowski differential interval”, by the expression \( dS ^2 = (cdt)^2 - (dx^2 + dy^2 + dz^2) \), where the particle may be thought of as moving at velocity v from point (x, y, z) to a neighboring point (x + dx, y + dy, x + dz) over the time interval dt. For sufficiently small dx, dy and dz, we may replace \( (dx^2 + dy^2 + dz^2) \) by \( (vdt)^2 \), giving \( dS ^2 = (cdt)^2 - (vdt)^2 \). Little attention has been paid over the years to the physical significance of the quantities dS and cdt. Which brings me to my Key Assumption 1: Key Assumption 1 - Assume that the elements dS and cdt can be viewed as minute, spatial intervals in 3-space. Regarding the time interval dt, we find in the translation of Einstein’s 1905 paper on Brownian motion, ”On the Motion of Small Particles Suspended in Liquids at Rest by the Molecular-Kinetic Theory of Heat” (ref “Einstein’s Miraculous Year - Five Papers That Changed the Face of Physics", Edited by John Stachel and Published by Princeton University Press, 1998, pgs. 85 - 98), that Einstein describes a time interval \(\tau\) "which is very small compared with observable time intervals but still large enough that the motions performed by a particle during two consecutive time intervals can be considered as mutually independent events.” (pg.94) I now submit Key Assumption 2: Key Assumption 2 - That the time interval dt is very small compared with observable time intervals but still large enough that the motions performed by a particle during two consecutive time intervals can be considered as mutually independent events. [I am working on Special Relativity - A Fresh Look, Part 2 and will post it when finished.]
  8. It looks like my expression was transcribed as I intended, but I have too many (...). This is test 2 \( S ^2 = (ct) ^2 - (x^2 + y^2 + z^2) \) Got it! Thanks.
  9. test [math] ( S^2 ) = ( (ct)^2 ) - ( (x^2 + y^2 + z^2) ) [/math] Need help here. Why isn't this being transcribed?
  10. I wish to continue presenting my insights into a different view of Special Relativity. I took a cue from an invitation from Swansont to open a new thread in Speculations where I posted the following. I found today that Strange has stopped that thread and it seems that I am being directed back to this one. So, for the sake of consistency, I am repeating this post here and will follow up shortly with another one. Special Relativity - a Fresh Look: Overview A fresh look at the underpinnings of Special Relativity is merited for the following reasons - 1. In earlier posts, I’ve shown how to view SR applications as Related Experiments - a pair of matched experiments in which charged particles are subjected to external electromagnetic fields. In the object experiment, the particle is given an initial velocity v and subjected to fields E and H. In the image experiment, fields E’ and H’ are applied to the particle at rest, where E’ and H’ are the transformed images of E and H under the SR field transformations. The 4-space motion of the particle in the image experiment (t’,x’,y’,z’) will then match up with the transformed 4-space motion (t, x, y, z) of the particle in the object experiment under a Lorentz time and space transformation with parameter v. In approaching SR this way, we avoid any discussions or dependencies on clocks that run slow or fast, and meter sticks that shrink or grow, as we move from one experiment to the other. 2. The Relativistic form of Newton’s Second Law of Motion is a Classical Physics formulation. We are given a set of initial conditions, a set of prescribed forces and a differential equation from which we can compute the position, velocity and energy of the particle for any time in the future to any degree of accuracy, and, if we insert negative values of time, we can compute the position, velocity and energy of the particle for any time in the past to any degree of accuracy. This is classical Classical Physics - given knowledge of the initial conditions and applied forces, the entire past and future of the particle is completely determinable. Contrast this with the stochastic behavior of Modern Physics, where SR plays a major role in nuclear physics, the physics of high energy particle collisions, and quantum field theory (QFT). 3. The mention of QFT brings me to my final point - QFT speaks of relationships between particles and fields characterized by a series of minute, discrete interactions in which the particles are accelerated slightly or decelerated slightly and/or deflected slightly and/or rotated, twisted or spun slightly. In contrast, conventional SR theory is marked by functions that are everywhere smooth and continuous. I intend to develop a model of SR which addresses all of the above, stays well within conventional bounds of discussion on the subject, and, here and there, introduces key, defensible ideas. Finally, I ask that the Forum members allow me to retain control over my terminology. For example, I shall refer to Minkowski’s S function as a “Minkowski interval”, and I shall refer to his dS function as a “Minkowski differential interval”.
  11. A fresh look at the underpinnings of Special Relativity is merited for the following reasons - 1. In earlier posts, I’ve shown how to view SR applications as Related Experiments - a pair of matched experiments in which charged particles are subjected to external electromagnetic fields. In the object experiment, the particle is given an initial velocity v and subjected to fields E and H. In the image experiment, fields E’ and H’ are applied to the particle at rest, where E’ and H’ are the transformed images of E and H under the SR field transformations. The 4-space motion of the particle in the image experiment (t’,x’,y’,z’) will then match up with the transformed 4-space motion (t, x, y, z) of the particle in the object experiment under a Lorentz time and space transformation with parameter v. In approaching SR this way, we avoid any discussions or dependencies on clocks that run slow or fast, and meter sticks that shrink or grow, as we move from one experiment to the other. 2. The Relativistic form of Newton’s Second Law of Motion is a Classical Physics formulation. We are given a set of initial conditions, a set of prescribed forces and a differential equation from which we can compute the position, velocity and energy of the particle for any time in the future to any degree of accuracy, and, if we insert negative values of time, we can compute the position, velocity and energy of the particle for any time in the past to any degree of accuracy. This is classical Classical Physics - given knowledge of the initial conditions and applied forces, the entire past and future of the particle is completely determinable. Contrast this with the stochastic behavior of Modern Physics, where SR plays a major role in nuclear physics, the physics of high energy particle collisions, and quantum field theory (QFT). 3. The mention of QFT brings me to my final point - QFT speaks of relationships between particles and fields characterized by a series of minute, discrete interactions in which the particles are accelerated slightly or decelerated slightly and/or deflected slightly and/or rotated, twisted or spun slightly. In contrast, conventional SR theory is marked by functions that are everywhere smooth and continuous. I intend to develop a model of SR which addresses all of the above, stays well within conventional bounds of discussion on the subject, and, here and there, introduces key, defensible ideas. Finally, I ask that the Forum members allow me to retain control over my terminology. For example, I shall refer to Minkowski’s S function as a “Minkowski interval”, and I shall refer to his dS function as a “Minkowski differential interval”.
  12. The article that has had the greatest effect on my thinking about physics over the years is “Extended Mach Principle” by Professor Joe Rosen, then at Tel-Aviv University, Israel (AJP, Volume 49, March 1981, pp. 258-264). Of all the fundamental principles Professor Rosen addresses, these three stand out for me - The origin of all laws of physics lies with the universe as a whole. Every single physical property and behavior aspect of isolated systems is determined by the whole universe. If the rest of the universe is taken away leaving only an isolated system, all laws of physics will cease to hold for it, and even space and time will lose their meaning for it. What I would like to do here is pursue these principles with regard to the following two questions: What role does the surrounding universe play in the decay of a single unstable particle at rest in an inertial frame of reference? What role does the surrounding universe play in the retarded rate of decay of these unstable particles as they move rapidly within this inertial frame of reference? I’ve always thought that the decay of an unstable particle is the strongest illustration of Eddington’s “Arrow of Time” - There is a BEFORE, there is an instant of NOW, and there is an AFTER. With respect to my Question 1, it is not hard to point to numerous examples of interactions where particle decay is in some way connected to the surrounding environment - an atomic pile comes to mind, so do particles struck by random photons, neutrinos, or miscellaneous other particles, real or virtual, that “exist” in the wilds of the universe. I intend instead to focus on Question 2. Retarded rate of decay as a function of pure motion is defined by Einstein’s time dilation formula appearing in his Theory of Special Relativity and also by an identical formula appearing in his Theory of General Relativity. Interestingly enough, the time dilation formula applies to any type of unstable particle regardless of mass, charge, spin or any of the other parameters generally applied to unstable particles by particle physicists and depends only on a relative velocity v and light speed c. If we exclude the class of retarded decay rates associated with General Relativity on the basis that the Universe is interacting with these unstable particle through gravity, we are left with the class of retarded decays associated with Special Relativity. I’ve made the point in earlier posts that I believe that Special Relativity Theory belongs firmly in the house of Electromagnetic Theory, including phenomena related to light such as the constancy of light speed in any inertial frame and the Relativistic Doppler Effect. Given this, I would think that retarded decay of speeding unstable particles, a hallmark of SR time dilation, would be in some way connected to the charge and/or magnetic moment, i.e., spin, of the unstable particle. The theoretical physics community has a large storehouse of weaponry with which to attack this phenomenon - QED, QFT, the Standard Model with its quark/gluon interactions, interactions with the universal background radiation, interactions with fields of passing neutrinos, interactions with the Higgs Boson, the influence of Dark Matter and/or Dark Energy. Just for starters, there are the unstable particles detected down here on the Earth’s surface that are created in collisions between atoms in the upper atmosphere and high-energy particles and gamma rays coming in from outer space. They should decay long before reaching the Earth’s surface. If we approach this problem from the point of view of interactions with external electromagnetic fields, then we might look at interactions with the Earth's magnetic field as well as with miscellaneous electric and magnetic fields in the upper atmosphere. Another aspect of the problem is that particle decay is a stochastic process. Any single unstable particle can exist over a range of time intervals - all that can be determined in the laboratory is the mean time to decay from observations of many instances. Accordingly, the application of the SR time dilation factor has to be applied to the mean time observation, which becomes even more tenuous when we account for the fact that there will be some statistical distribution in the velocities of the observed particles relative to the laboratory frame. Be that as it may, as I’ve pointed out in earlier posts, I am still hoping for an explanation for retarded particle decay times that goes beyond simply stating that Special Relativity requires it.
  13. Studiot , thank you for bringing the Wangsness material to my attention. Oddly, I think his spherical light shell approach to deriving the Lorentz transformations was the vehicle I was first introduced to as a freshman undergrad. I never was happy with it, and I think this dissatisfaction was a prime motivator for me to seek out Einstein’s original 1905 paper to read what the master actually wrote. Currently, I am working on a post targeted for the General Philosophy category. Perhaps we’ll meet up again over there.
  14. (This will be my final post to this thread.) Back in November, I submitted a post to the Forum where I discussed Einstein’s approach to deriving the Lorentz transformations in his classic 1905 foundation paper on Special Relativity. A number of Forum members raised questions at the time about my submission. I would like to try to address those questions by going deeper into the analysis. (I have found the English translation of Einstein’s 1905 paper in the book "Einstein's Miraculous Year - Five Papers That Changed the Face of Physics", Edited by John Stachel and Published by Princeton University Press, 1998, to be particularly helpful, and I will be referencing this translation where necessary.) I wrote in my November post - “One almost has to read between the lines, but it soon becomes clear that Einstein is working with a central device consisting of two sticks joined at the ends to form a right angle, one vertical and the other horizontal, with a mirror at the free end of each stick.” Einstein sets this device in motion in the horizontal direction at speed v with respect to an observer in a “rest” frame, identified as Frame K. The frame in which the device is embedded, the “moving” frame, is identified as Frame k. (I noted that there is more than a passing similarity between this device and Michelson's interferometer.) The key sections in the Einstein paper supporting my descriptions may be found in Stachel, pg.132, where Einstein introduces a “light ray” directed from the origin of his “moving” frame, Frame k, along the X-axis to a point x’ and reflected from there back toward the origin. Later, on the same page, he speaks of light ray propagation and reflection along the Y- and Z- axes. (I did not refer to Z-axis propagation in my earlier discussion because Einstein’s Z-axis analysis parallels his Y-axis analysis.) Einstein then proceeds to focus on how the paths of these rays, in terms of distances traveled and durations, would appear to an observer at rest with respect to the device in Frame k, and also to an observer in Frame K with respect to whom the device is moving. (In my earlier post, I said that the these light rays emanated from a match struck at the moment that the Frame k and Frame K origins coincide. I apologize if this imagery confused anyone. It was my intent to make clear that Einstein was speaking of one and the same light rays as viewed by the observers in the respective frames.) From the top of pg. 132 over to the middle of pg. 137, Einstein uses the observations of these observers as a means to derive the Lorentz transformations. Comment 1 - In my November post, I wrote the following: “In other writings, I have described Einstein's analysis as ‘unnecessarily and uncharacteristically opaque’. After plunging into the depths of an argument that I have never been able to parse, he finally breaks through the surface of the water proudly holding in his hand a Lorentz-Fitzgerald contracted horizontal stick as perceived only by Observer K… This is Michelson-Morley all over again! Nothing new here!” I stand by these remarks. The key mathematical relationships appearing in Einstein’s analysis are by and large the same as those that appear in many first-year textbook discussions of the Michelson-Morley experiment. There, the speed parameter v plays the role of the ether wind. It’s common in those discussions to compare the ether wind to a river current. The times required for a swimmer, who swims at speed c in quiet water, to swim against the current across the river and back, and up-river an equal distance and back, are compared to the times required for the swimmer to swim those distances in quiet waters. It is then argued that the distance swum up-river must be Lorentz-Fitzgerald contracted in the case of the non-zero current in order to insure that the times to swim across and back, and up-river and back, agree. As in the case of the ether wind situation, Einstein requires that his horizontal stick be Lorentz-Fitzgerald contracted as perceived by the observer in Frame K so that this observer will observe that the two light beams return to the vertex of the device together. (This is critical to Einstein’s analysis because, as I pointed out above, the observers in Frame k and Frame k are observing the very same light rays. It’s easy to show that the observer in Frame k will perceive the two light beams returning to the vertex of the device together, i.e. the Frame k observer will observe what we may label a “coincident event”. I say more on this in the following.) Comment 2 - In another post, I referred to Einstein’s analysis as flawed. I stand by this position for two reasons. The first is strictly mathematical - Einstein begins his analysis by treating x’, the horizontal distance traveled by the Frame k light beam, as an infinitesimal. This is clearly unnecessary and incorrect, and only serves to mislead the reader. The second reason is that there is a fundamental paradox in the insistence that the observers in the two frames agree that the light beams return to the origin of the device together. In the traditional ether application, the swimmers are actually stand-ins for the light beams that originate by placing a light source beneath a thin sheet of glass with a thin silver coating applied on one side. Part of the light is reflected off to the right and part is transmitted through, thereby producing the pair of light beams that move parallel to the arms of the interferometer. It is essential that the swimmer swimming across the river, and the swimmer swimming up-river, return to the “dock” together so that (1) the light beams they represent return to the viewing eye-piece in phase and (2) their interference pattern remains unaltered as the interferometer moves through the ether and though space. In Einstein’s application of this model, the reflected light rays as perceived by the observer in Frame K similarly must arrive back at the vertex together. This can only happen if the horizontal stick is Lorentz-Fitzgerald contracted as perceived by this observer. (Note also that the total time traveled by the light rays as perceived by the observer in Frame K is time-dilated with respect to the total time traveled by the light rays as perceived by the observer in Frame k.) Suppose we take Einstein’s picture a little further and insist that the return of the light rays to the vertex together as perceived by the Frame k observer triggers a Fourth of July rocket. KaBoom!!! Red, white and blue flaming pieces of metal across the sky!!! Now suppose that the time and distance intervals for the observer in Frame K are not, respectively, time-dilated and Lorentz-Fitzgerald contracted so that the observer in Frame K does not see the coincident arrival of the reflected light flashes at the vertex of the device. Are we to say that the Frame k observer will see the rocket explode, but the Frame K will see nothing? Ridiculous! If we don't concede that time and distance intervals differ across uniform motion boundaries, then we must concede that stepping across such a boundary takes us to a Parallel Universe! This is an untenable choice. Comment 3 - Enough criticism. What’s to be done? Now, one hundred and fifteen years later, - nothing. But I do like to play with the idea that I could take a trip back in time and advise the young Einstein on the structure of his paper. I would advise him so: “Albert, enough with attempting to derive the Lorentz transformations by building on Michelson-Morley. Begin this way - First, review the definition of an inertial reference frame and introduce the postulate that the Lorentz transformations serve as a bridge between measurements of time and space intervals made by an observer in one inertial frame and those of an observer in a second inertial frame moving uniformly with respect to the first. Next, review the fact that Maxwell’s equations hold their form under a Lorentz transformation and point out that the transformed fields are those that would apply in the frame targeted by the Lorentz transformation. Finally, use these facts to assert that electromagnetic theory in all of its aspects, including the fact that the speed of light is c, holds in any inertial reference frame. (Note - Einstein’s additional postulate (Stachel, pg. 124) that the speed of light in empty space is independent of the speed of the source always struck me as unnecessary. I’m under the impression that this is already a feature of Maxwell’s equations.) And now you are ready to move on and describe the ways in which your insights into the application of the Lorentz space, time, and field transformations brilliantly allow the solution to so many important physical problems. Congratulations!
  15. Strange, one of the problems I find with the structure of this Forum is , unlike Twitter, I can't go back to a single location and find all of my posts listed together, one after the other. I'm sure I mentioned the "stripped down Michelson interferometer" in an earlier post. But be that as it may, I hope to return to Einstein's analysis in a future post and be more explicit about how I interpreted the work. Also, I believe your question addressed to me - "Do you understand how science works? (Based on your posts n this thread I am not sure you even know how SR works.)" - deserves a forthright reply. And my reply is the following - I believe that the fact that under a Lorentz transformation, Maxwell’s equations retain their form tells us something very fundamental about the place of Special Relativity Theory in the construct of modern physics. In effect, it tells us that SR belongs firmly in the house of Electromagnetic Theory, including phenomena related to light itself such as the Relativistic Doppler effects - both in-line and transverse. The only substantial outlier that I can point to is the retarded decay of a speeding unstable particle. I hold that this phenomenon has not received adequate attention from the theoretical physics community. Simply to say that time dilated decay is a “kinematic” effect - that is, purely a consequence of the motion - is to invoke a paranormal influence having no bearing on legitimate science.
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