RAGORDON2010

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

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.

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.

(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.

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.

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.

[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.]

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.

test [math] ( S^2 ) = ( (ct)^2 ) - ( (x^2 + y^2 + z^2) ) [/math] Need help here. Why isn't this being transcribed?

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”.

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”.

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.

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.

(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!

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.

Strange, I appreciate the Latin - The burden of the proof lies upon him who affirms not he who denies. (Google translation) Here's the problem as I see it. Einstein's attempt to demonstrate that the Lorentz transformations correctly associate his "at rest" observer observations with those of his "moving" observer observations, which I discussed in an earlier post regarding his use of a stripped down Michelson interferometer, is fundamentally flawed. The fact that he immediately jumps from there into successfully applying the Lorentz transformations to many physical problems raises real doubts in my mind about the foundations of SR. I hope to develop these thoughts in future posts.

Studiot, thank you for your reply to my question - "Without invoking the mathematics of the the Lorentz transformations, is there a way to match an observation by observers in one reference frame with an observation by observers in the other reference frame?" To review, I used this question as a lead-in to my story about a charged ball on a speeding train exposed to an external magnetic field, and how the "at rest" observer sees that the ball begins to vertically loop, while the "moving" observer, riding on the train, sees that the ball begins to hop toward the rear of the train. I then wrote about how one could imagine inserting a firecracker in the ball and striking it with a laser beam to set off the firecracker. Both observers would then have a fixed point in spacetime about which they could compare notes. I had taken the idea from Einstein's light flash in his original paper. My story was part of a presentation I had developed some time back as a way of introducing high school students to Special Relativity. I worked around the idea of bringing the students into the story by having one group imagine that they were the research team in the "rest" frame that applied the magnetic field while the other group imagined that they were on the train with the ball. It turned out to be useful imagery, and I found myself falling back on it from time to time. in fact, I just referenced it again in a post I submitted last week to a thread discussing the SR twin story.

md65536 - Interesting observation. I wasn't aware of this distinction. Thank you for clearing it up for me.

Just a thought in passing - Twin tales have been part of the Special Relativity public-discussion canon going back to the original Einstein paper. The message is always that the observer in the “moving” frame is aging slower than the observer in the “rest” frame. In my studies of situations where I apply the Lorentz transformations to move from one frame to the other, I’ve noted that on occasion the result is reversed. Below, I present an example of each case. Application 1 - Retarded Decays of Speeding Unstable Particles Let’s imagine that the unstable particle is at rest on your train, which is moving at speed 0.8c relative to me standing on the platform. Suppose you observe that the particle decays over a time interval t’, while I observe that the particle decays over a time period t. If we position the particle at the origin of coordinates in your frame, we may set its x’-displacement equal to 0. Then the Lorentz time transformation from your frame to my frame takes the same form as the classic time dilation formula and for your decay time t’ equal to, say, 24 units of time with v = 0.8c, my decay time t equals 40 units of time. So, you, standing in the “moving” frame are aging slower than I, standing in the “rest” frame. Award One Point to the Twin Tales. Application 2 - The Exploding Ball I would like to return now to the a situation I described in another post. I am again standing on the platform while you are on a speeding train moving at 0.8c. Next to you is a charged ball containing a small firecracker. I apply a strong magnetic field across the track causing the ball to undergo a series of vertical loops as I perceive the motion, while you perceive the motion as a series of “pogo stick” hops. Once the ball starts looping as I perceive it, and hopping as you perceive it, I zap the ball with a laser beam and set off the firecracker. Pow!!! The ball explodes at an object point (t,x,y,z) in my frame and at the corresponding image point (t’,x’,y’,z’) in your frame, where the object-image coordinates satisfy the Lorentz transformations. Now let's make it easy on ourselves. Suppose I zap the ball at the bottom or top of a loop, so I have no x-displacement to deal with, and suppose my clock reads 24 units of time at the moment the ball explodes, in whatever units of time are appropriate to our measurements. Here, with a zero x-displacement, the Lorentz time transformation takes the same form as the classic time dilation formula referenced above, and so your clock will read 40 time-units. So, I, standing in the “rest” frame, am aging slower than you, standing in the “moving” frame. Deduct One Point from the Twin Tales. So perhaps the best that can be said is that whether the observer in the “moving” frame is aging slower than the observer in the “rest” frame, or vice-versa, depends on the specifics of the experiment under consideration.

I want to apologize to the Forum for the abrupt way in which I exited from this thread several months ago. Time has passed, whatever was troubling me has been resolved, and I would like to continue to offer my thoughts on Special Relativity. In particular, I would like to focus the attention of the Forum on the following question: What is the underlying physical foundation upon which SR rests? Or simply - Why does Special Relativity work so well? This will require a new thread, which I hope to begin sometime in the future.

The other day, my wife and I visited a coffee house near the University of Alabama campus. As we were leaving, I noticed a young coed slowly and very apprehensively turning over the cover page of a textbook. I saw that the title of the book was “Statistical Mechanics”. “Oh”, I exclaimed. “they’re still teaching that!” The young coed immediately looked up at me with an expression bordering on total fear. I have no idea what she was thinking, but I quickly said “That was my favorite subject! Good for you!” Her face exploded in one big smile, ear to ear. So, physics isn’t so hard. It just takes a little encouragement to bite into it.

Some year’s back, I had occasion to be hospitalized for 10 weeks or so while a bunch of doctors tried to figure out what ailed me. Finally, I was admitted to the Yale-New Haven Hospital in New Haven, Connecticut. The morning after my admission, a group of medical students on their rounds stopped by my room. Having been trained as a physicist and having not much else to say as they stared at me, I asked them if they had heard about Schrodinger’s cat. The reaction was “What?”. So I began to tell them about the cat, the box, the acid bath, the cyanide pill, and whether the cat was alive or dead while the box remained closed. I then explained that the probabilities that the cat was alive or dead were given by a mathematical construct know as a “wave function”, and until the box was opened, the state of the cat remained uncertain. Finally, the box is opened, the state of the cat is determined, and the wave function collapses. Needless to say, my story seemed to make little impression on the medical students. I then proceeded to explain that I was Schrodinger’s cat. I was suffering from some unknown ailment which could easily be confused with any number of ailments exhibiting similar symptoms. So, while the hospital staff considers these possible diagnoses, internally my body was moving from one disease state to the next, with relative durations set by the probabilities contained in my wave function. And then I looked directly at the medical students and told them that this state of affairs will continue until, at last, they opened me up (figuratively speaking I hoped) and arrived at a definite and correct diagnosis. At which point, my wave function will collapse, and my body will possess that and only that disease condition. Well, by and large, the students rolled their eyes, shook their heads and started for the door. As I called after them that they should read more physics, one young man stopped in the doorway and thanked me for my tale. Two days later, after the hospital staff had indeed arrived at a definite and correct diagnosis and set me up on a regimen that would keep me alive for the foreseeable future, I came across this same young student on my way out. Again he thanked me for the story and told me how much he appreciated the fact that I had brought Schrodinger’s cat to Yale-New Haven Hospital.

When I joined this forum, I was hoping for the chance to engage in an intelligent conversation about issues in Special Relativity that have confused new students for decades. Instead, I have found myself talking to walls. So I will be exiting from this forum, but I want to leave three parting thoughts: Thought 1 - Despite how many people say it for other people to hear, despite how many people write it for other people to read, despite how many people key it for other people to link to - The Fact of Nature that the speed of light is the same in all inertial frames plays NO role in explaining the successful application of Special Relatively to solving physical problems! It is a Red Herring! There is another Fact of Nature at work here. What other Fact of Nature? Well, I’ve hidden it in plain sight in Thought 2. Thought 2 - "Slip slidin' away. Slip slidin' away. The nearer your destination, The more you're slip slidin' away." - Paul Simon I've been hoping to offer some thoughts on "relativistic mass" - the notion that mass increases with increasing speed. It's common to come across the statement that accelerating a particle becomes more difficult as particle speed approaches c because "particle mass approaches infinity". I prefer to state the issue differently. I would say that accelerating a particle becomes more difficult as particle speed approaches c because the external field responsible for the acceleration loses effectiveness as the particle speed approaches the speed at which the field mechanisms function. This, of course, offers an explanation for why light speed forms a limiting speed in nature. An old boot can travel no faster through the water than the maximum speed at which the fisherman can reel in the line. Whenever I think about this phenomenon, Paul Simon's song comes to mind. The speeding particle slips and slides away from the grasp of the external field. Thought 3 - I can’t leave this forum without saying something about time dilation. It has always puzzled me that while the physics community easily accepts that time dilation effects in General Relativity relate in some way to the interaction between the time-keeping system and the surrounding gravitational field, the analogous time dilation effects in Special Relativity are viewed as “just so”. Well, I have never cared much for a “just so” story. But I do hold the view that Nature does not care at all for a “just so” story. Something is going on out there! In the most dominant example - the retarded decays of unstable particles moving at speeds close to light speed - I again must fall back on my belief that these effects are in some way a consequence, in ways not at all understood, of the rapid motion of the particles through surrounding electric and magnetic fields. I hold (and this is where Special Relativity exhibits its most severe vulnerability as it is commonly described) that no physical effect can occur as a consequence of merely moving at a uniform speed in an inertial frame of reference. And now, from sunny Alabama (a state in the USA, refer Rand McNally maps, circa 1934), I happily say GOODBYE, Y’ALL!! ROLL TIDE!!

Studiot, I appreciate your attempt to educate me on the history of Special Relativity. We all need occasionally to fill in gaps in our knowledge. However, I have been delving among the ins and outs of SR for maybe 5 decades now, and I'm afraid me ideas are pretty well set. There is a comment of yours, though, that I do wish to expand upon, to wit: [If you only have one frame of reference than you have the difficulty that Fitzgerald (and Lorenz) faced with the results of practical measurements on the propagation of light. This was that the Lorenz-Fitzgerald contraction was introduced as a mathematical formula which accounted for but did not explain the results of these experiments (in particular the Michelson and Michelson -Morley ones) Length contraction of the apparatus arms was a pretty heretical explanation. Einstein on the other hand, deduced the selfsame formulae from purely geometrical considerations of observing the same sequence of events in two frames.] Studiot, I have been "looking over Einstein's shoulder", in a manner of speaking, for some time now, and I want to call your attention to the manner in which he introduces the Lorentz transformations in his1905 original paper. (I have found the English translations of his 1905 papers 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.) 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. He also needs a source of light, say, a match. (Please note that there is more than a passing similarity here to Michelson's interferometer.) Einstein imagines that the device is placed in a "moving" frame labeled Frame k with the horizontal stick pointing in the direction of motion. The device is accelerated up to a speed v on the order of light speed and allowed to pass an observer in a "rest" frame, Frame K. When the vertex of the device comes up even with the Frame K observer, the match is struck. The observers in Frame k and Frame K, let's label them Observer k and Observer K, must then measure the times for the light rays emanating from the match to reach the two mirrors and the times for the reflected beams to return to the vertex of the device. As Einstein describes events, it appears that Observer k has the easier task. It is as if she struck the match. In fact, it's as if she is not moving at all. As she perceives the rays, one travels straight up to the vertical mirror and the reflected ray returns straight down to the vertex, the other ray travels in a straight line to the horizontal mirror and the reflected ray exactly retraces the path and returns to the vertex. Observer K has the more difficult task. It is as if he struck the match. As he perceives the rays, they must catch up to the moving mirrors, and the reflected rays must then meet up with the moving vertex. Einstein now sets out to demonstrate that the data obtained by the two observers satisfies the Lorentz transformations. 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. I have often wondered why the editors of the journal where this work was published didn't insist that he take a clearer approach in his analysis. I do intend to suggest one possibility in a future post, but there is no avoiding his conclusion that the horizontal stick as perceived by Observer K is Lorentz-Fitzgerald contracted. This is Michelson-Morley all over again! Nothing new here! That Einstein was able to proceed from this point and create his magnificent Special Relativity Theory is a tribute to his genius.

I too have been a teacher at various points in my life and, if on any given day, I found that I had held the attention of my students, prodded their curiosity, and possibly stimulated their imagination, I went home that night feeling that I had a very good day. I would imagine you have had days like that also. Teaching is not easy, particularly as the ideas become more abstract and removed from the students' day-to-day experiences. That's one reason I feel there is value in giving students alternative ways to view a problem. and the conceptual problems posed by Special Relativity definitely could use an alternative viewpoint. And if this viewpoint is grounded in their sense of two separate experiments conducted in the same laboratory, I believe a that good number of students would benefit from this viewpoint and find that they would be more open to arguments drawn from the conventional viewpoint of one experiment and two laboratories moving uniformly relative to each other at near-light speeds.