# Another way of looking at Special Relativity

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25 minutes ago, RAGORDON2010 said:

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 thing is, he derived the Lorentz transform from the starting assumption that the laws of physics (ie. speed of light) are the same in all frames of reference. The fact that it then correctly relates measurements in those frames of reference is not surprising (in fact it is rather obvious).

I am not aware of Einstein ever using "a stripped down Michelson interferometer." (And I can't find any previous reference to it in this thread.)

25 minutes ago, RAGORDON2010 said:

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.

So the fact that the theory works, and correctly predicts the results of experiments, raises doubts in your mind? Do you understand how science works? (Based on your posts n this thread I am not sure you even know how SR works.)

25 minutes ago, RAGORDON2010 said:

I hope to develop these thoughts in future posts.

I suggest you do that in a new thread in the Speculations section so as to avoid breaking the forum rules.

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2 hours ago, RAGORDON2010 said:

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.

Your earlier comment sounded like you didn’t follow the derivation. Not being able to understand is not evidence of a flaw.

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On 2/22/2020 at 5:16 PM, RAGORDON2010 said:

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?

It was nice of you to thank me for my reply.

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On 2/23/2020 at 12:15 PM, Strange said:

I am not aware of Einstein ever using "a stripped down Michelson interferometer." (And I can't find any previous reference to it in this thread.)

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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28 minutes ago, RAGORDON2010 said:

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.

Click on your user name in some post. In the black stripe, near the top, on the right, there is a box called "see my activity"  Click it.

On the new page, on the left, under "all activity" you can choose between posts and topics

28 minutes ago, RAGORDON2010 said:

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.

It's almost as if you'd expect someone applying these concepts to kinematics would have based it upon the behavior of electrodynamics. Einstein, maybe.

Spoiler

On the Electrodynamics of Moving Bodies

Quote

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.

In SR, time dilation is purely a consequence of motion. What's different about motion of an unstable particle? Where is it specified that it wouldn't apply to an unstable particle?

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45 minutes ago, RAGORDON2010 said:

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.

That is how it was derived.

46 minutes ago, RAGORDON2010 said:

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.

OK. Thanks for confirming that you don't understand the basic concepts of special relativity.

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6 hours ago, RAGORDON2010 said:

.

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.

I am 100 percent positive that you have no knowledge of how the Breit Wigner function for the mass density distribution and survival probability works in the relativistic  form works in terms of particle decay for you to draw any valid conclusions of how it applies.

This distribution function also applies to Muon decay. Key note however this decay formula isn't the only one used. Though it is often applicable for unstable resonance.

You need the Feymann path integrals and how they relate to the CKM and PMNS mass mixing matrix to properly understand how scattering relate to decays. (S Matrix).

Edited by Mordred

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10 hours ago, RAGORDON2010 said:

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.

This isn’t just true for electromagnetism, but for all laws of physics, once written using the correct formalism. For example, the Standard Model of Particle Physics - including all parts that are not EM related - is fully Lorentz invariant. As is relativistic fluid dynamics. And relativistic mechanics. And so on.

The point is that all inertial observers experience the exact same laws of physics, not just the same propagation of light.

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On 2/24/2020 at 8:33 PM, swansont said:

In SR, time dilation is purely a consequence of motion. What's different about motion of an unstable particle? Where is it specified that it wouldn't apply to an unstable particle?

One reason for RAGORDON2010 might be that e.g. in the well-known example of muon-decay, there is no known mechanism that controls when the muon will decay. So there is nothing that can be effected by the muon's velocity. So one has to accept that it is magic. Or propose a new kind of interaction, which would be the scientific way.

What he obviously does not see is that the Lorentz transformations are rotations in spacetime. If I compare with a 'normal' rotation in space, we are used to the fact that objects themselves do not change when we look at them, even if they look shorter than their 'rest-length', when viewed under an angle. And just as we do not need to propose an interaction to explain this '3D length contraction', because for the observed object nothing changes, so we do not need one for the muon: in the muon's rest frame nothing changes.

@RAGORDON2010, as a question to think about: does the muon really have a longer half-life when it approaches earth with near lightspeed?

On 2/22/2020 at 6:16 PM, RAGORDON2010 said:

I had taken the idea from Einstein's light flash in his original paper.

Really? But then it should have been clear from the beginning that SR is based on how EM-phenomena are described by different inertial observers (as Swansont and Strange already pointed out). Why do you present us this as such an important insight:

On 2/24/2020 at 8:09 PM, RAGORDON2010 said:

In effect, it tells us that SR belongs firmly in the house of Electromagnetic Theory,

On 2/22/2020 at 6:16 PM, RAGORDON2010 said:

My story was part of a presentation I had developed some time back as a way of introducing high school students to Special Relativity.

Again: you should not teach SR to students, because it is obvious to us all you do not understand it yourself.

Edited by Eise
Typo

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(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!

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Posted (edited)

As this is your final post nothing you have stated changes any of my comments. Lastly physics didn't stop with Einstein.

It's too bad you teach outside the described curriculum in an institution. Quite frankly the job of a teacher is to follow the school curriculum despite personal opinion.

It's too bad you never examined the Lorentz transforms for relativistic vector addition. You would have realized you don't require the speed of light to apply them. One can show the speed limit of information exchange without using light from those transforms.

One thing about this thread is that you haven't applied the interval ct which is what gives  time dimemsionality of length nor have you mentioned coordinate time or proper time. Nor have I ever seen you apply the dot vs cross products with regards to velocity addition.

quite frankly all these little situations you describe never included any of the calculations...just verbal statements. That is insufficient to change my mind.

Edited by Mordred

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12 hours ago, RAGORDON2010 said:

(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.)

It's almost like he was applying that feature of electrodynamics to a kinematics problem. I guess he should have called his paper "On the Electrodynamics of Moving Bodies" or something like that.

As far as Einstein's derivation being similar to a Michelson interferometer, you need to look at the bigger picture — the common theme here is using orthogonal axes in your coordinate system. IOW, the similarity should be utterly unsurprising.

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Posted (edited)

16 hours ago, RAGORDON2010 said:

(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.

etc

You seem to have posted all this before and been told several times that modern authors have streamlined the presentation of Relativity over the now more than a century since inception.

Mordred and Marcus in particular ( +1) have tried to point to presentations that contains earlier and simpler theory as limiting cases so the theory of Newton and Galileo is a limiting case of the Special Theory which in turn is a limiting case of the General theory. We would expect this type of progression to continue wiht future developments.

Here is a mid 1960s version that demonstrates this, due to Wangsness.

You should take away with you this development along with the clear exposition of how it relates to Lorenz and modern versions of the two postulates of SR
The math is not too difficult.
But he does provide proper reasoning for each step taken (not always shown in shallow modern treatments).
I will just post the basic bit here, but he goes on in similar vein to eplore all the aspects of SR, inlcuding the electromagnetic ones.

Edited by studiot

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On 3/3/2020 at 8:43 AM, studiot said:

Here is a mid 1960s version that demonstrates this, due to Wangsness.

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.

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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”.

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Posted (edited)

The above implies you have a different set of mathematics for SR. So when are you planning on posting them ?

Quite frankly it is the math that counts not verbal word play.

Also what is wrong with using the correct terminology ?

Ie see here for ds^2

Or for s^2 the spacetime interval.

This Susskind video does an excellent job of showing how the invariance comes about.

Edited by Mordred

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17 hours ago, RAGORDON2010 said:

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.

Are you talking about some aspects of SR that can be observed without referring to different clocks etc., or are you saying all of SR can be understood without them?

If you separate two clocks and bring them together again, one might have measured a longer time than the other. Do you explain that without saying one ran faster or slower than the other, or is that not one of the aspects of SR that you're talking about, or are you saying that wouldn't happen?

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[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.]

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Posted (edited)
45 minutes ago, RAGORDON2010 said:

(t, x, y, z)

Before I try to comment:

45 minutes ago, RAGORDON2010 said:

time interval dt

There are some issues with the format of the math symbols when I view the post. Are you discussing a time interval $dt$ or proper time interval $d\tau$ ?

Edited by Ghideon
format

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2 hours ago, RAGORDON2010 said:

Key Assumption 1 - Assume that the elements dS and cdt can be viewed as minute, spatial intervals in 3-space.

Pretty obviously not since dS is a function of four independent variables !

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34 minutes ago, studiot said:

Pretty obviously not since dS is a function of four independent variables !

Sorry to say, this kind of thing always irks me a bit. You are absolutely right to say that d$$S$$ is a function of four independent variables, since $$S$$ itself is defined as a function with Minkowski space as its domain. But then the OP goes on to say that d$$S$$ is defined separately in some special way. That does not make sense. Once $$S$$ is defined, then d$$S$$ is supposed to be defined as well as the differential of $$S,$$ you do not get to make a separate definition, because if you begin to do that, then everything gets left in absolute confusion. I admit that I did not bother to check whether the OP's auxiliary definition of d$$S$$ does agree with the usual definition. If it does, he should not state it like it is a new definition. End of rant.

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16 hours ago, RAGORDON2010 said:

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

(dx2+dy2+dz2)

by (vdt)2 , giving

$$dS ^2 = (cdt)^2 - (vdt)^2$$.

Introducing v begs the question

As measured by whom ?

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On 3/20/2020 at 5:37 AM, studiot said:

Introducing v begs the question

As measured by whom ?

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.

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27 minutes ago, RAGORDON2010 said:

The experiment is reproducible, and the velocity of the particle at a given point in the motion can be objectively be verified.

27 minutes ago, RAGORDON2010 said:

time interval dt.

In addition to my question regarding proper time above I'm also interested in the following: What frame of reference*)  does the times, spatial distances and velocities belong to? Is it stationary to the lab equipment generating fields? Is it he particle frame of reference?

*) My question may be duplicate of  @studiot's question. I often interpret "Who" as "Who's frame of reference" when I read about relativity.

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5 hours ago, RAGORDON2010 said:

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.

Why charged and why do the fields have to be present?  None of that shows up in any of your equations. They would seem to be irrelevant.

5 hours ago, RAGORDON2010 said:

To review from Part 1 -

I personally find this annoying, as with the repeated previews of what’s to come. Get on with it already.

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