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Thread: Equivalence of electric and magnetic fields

  1. #1 Equivalence of electric and magnetic fields 
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    My question concerns the situation where two positive ions initially move in the same direction and are side by side in a vacuum. An observer also co-moves with the ions at their speed of v. I will refer to the co-moving observer as C and an observer at rest as R.

    1. C considers his moving frame to be at rest, so his clock is unaffected and the ions (which initially appear to be at rest) just have a rest mass of m0. C observes each ion accelerates at aC. From the equation F=ma we can say aC = FE/m0 where FE is the electrostatic repulsion.

    2. For R at rest there is a magnetic force of attraction which varies linearly with the ionsí speed of v. This can be expressed as kv, so for R the repulsion is reduced to FE Ė kv. The mass of the moving ions is increased by the Lorentz factor of γ or (1 - v2/c2)-0.5 to give a further reduction in acceleration. Hence R should see a lesser acceleration of aR = (FE Ė kv)/γm0.

    C and R see each otherís clock is slow by a factor of 1/ γ. So each can say the otherís measurement of speed is increased by a factor of γ and acceleration by γ2. But I donít see how non-linear factors can reconcile acceleration differences which basically vary with v. A single acceleration event seems to produce fundamentally different accelerations which cannot be reconciled by Lorentz transformations. Electric and magnetic fields are said to be the same when viewed from different frames, but the same fields would have the same effects. Where am I going wrong?


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    Quote Originally Posted by Andrew? View Post
    Where am I going wrong?
    You are not going wrong, your observation about Lorentz transformations not being able to reconcile these factors is accurate. That is because Lorentz transformations are maps between purely inertial frames only. If you want to consider situations that involve accelerated frames ( as in your example ), then Special Relativity is up to that task too, but the way frames are related to one another is more complicated than simple Lorentz transformations:

    SR treatment of arbitrarily accelerated motion

    So far as electromagnetism is concerned, what the theory of relativity is telling us is that electric and magnetic fields are actually just different aspects of the same underlying entity; mathematically, there is only one object, and that is the electromagnetic field tensor. This being a tensor, it is by definition invariant under changes in coordinate basis, so it is the same in all frames - regardless of whether they are inertial or accelerated. While the overall tensor is invariant, what does change when you go from one frame to another are its individual components - physically that means that different observers will see different "mixes" of E and B fields, which themselves are just aspects of the same entity. So basically, going from one frame to another means you are looking at the same electromagnetic field from a different angle. In the case of inertial frames this is meant quite literally, since a Lorentz transformation is nothing other than a hyperbolic rotation about some angle in 4-dimensional spacetime.

    So to make a long story short, electric and magnetic fields are not exactly equivalent, but they are two aspects of the same underlying physical entity, the electromagnetic field. So if you go from one frame to another, you are still looking at the same field, just from a different vantage point, so the mix between E and B which you see will be different.


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    Quote Originally Posted by Markus Hanke View Post
    Quote Originally Posted by Andrew? View Post
    Where am I going wrong?
    If you want to consider situations that involve accelerated frames ( as in your example ), then Special Relativity is up to that task too, but the way frames are related to one another is more complicated than simple Lorentz transformations
    Many thanks for your reply to my query. I assumed it had been buried underneath word games.

    It seems I hadn't described the situation clearly enough. I meant that the co-moving observer moves inertially with respect to the observer at rest. The protons accelerate symmetrically away from the co-moving observer who remains in a straight line with the protons. Both observers seem entitled to make measurements of the protons' paths, i.e. how long it takes them to reach various distances, and so can deduce the acceleration in each of the observer's frames. I have seen examples of this sort of thing when there is a neutral conductor rather than just a vacuum, and the acceleration of a charge has been calculated to be identical for both observers. Please let me know if I have not made it clear that the observers are moving at a constant relative speed.
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    I don't think I understand your scenario - one of the observers is comoving with respect to what ? The term "comoving" usually denotes that two or more objects are in the same frame of reference, or at least that their relative velocity is zero. However, then you talk about the protons accelerating away from both observers - I am pretty confused about this.

    The other thing is that, for relativistic velocities, F=ma does not hold in this form; instead it becomes

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    OK, sorry it wasn't at all clear.

    Imagine the two protons are initially moving parallel to each other vertically downwards at a speed of v, and so is the co-moving observer. So all three are initially in the same inertial frame. For convenience they could all be in the same horizontal plane. The co-moving observer could also be placed symmetrically at the apex of an isosceles triangle with the protons forming the base of the triangle. When the protons start to accelerate away, the co-moving observer continues downwards at v. The base of the triangle then gets wider. The stationary observer could remain somewhere along the line of motion of the co-moving observer. Does that help?
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    Ok, that is a bit clearer now. So what quantity exactly are you comparing between the two observers - the acceleration rate of the protons ?
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    That's right thanks. The co-moving observer just sees an acceleration due to electrostatic repulsion. For the observer at rest, the protons' motion leads to a magnetic force that reduces the acceleration.
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    Quote Originally Posted by Andrew? View Post
    That's right thanks. The co-moving observer just sees an acceleration due to electrostatic repulsion. For the observer at rest, the protons' motion leads to a magnetic force that reduces the acceleration.
    Just to clarify, you mean an acceleration due to the repulsion between the two protons, i.e. an acceleration perpendicular to their direction of motion, right ?

    I think from what you described, what you are trying to measure here is coordinate acceleration ( a 3-vector, or the norm thereof ), which is an observer-dependent quantity, and hence will naturally vary from one frame to another. This is not a contradiction, but expected behaviour; the two observers will not agree on this quantity. However, the two frames can easily be reconciled if we consider proper acceleration instead, which is a 4-vector. Physically, coordinate acceleration is what observers calculate, based on observations made from their own frames; proper acceleration on the other hand is what an accelerometer co-moving with the protons will physically measure, so of course everyone agrees on that.

    In general terms, and within SR, if we are dealing with 4-vector quantities in spacetime, no contradictions can arise, since such quantities are invariant between frames. We can still choose to use old-style 3-vectors instead, but then different observers will not agree on these quantities, because they do not share the same notion of time. At the end of the day there still will never be any physical contradictions, but that may not immediately be obvious, and might sometimes require some more or less awkward maths to demonstrate it. This is one of the reasons why in relativistic electrodynamics we try and go away from the E-B formalism, and use the tensor formalism with just the F field instead. Doing so means we don't have to worry about observer dependency. If needed, the E and B fields can always be recovered later.
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    Quote Originally Posted by Markus Hanke View Post
    In general terms, and within SR, if we are dealing with 4-vector quantities in spacetime, no contradictions can arise, since such quantities are invariant between frames. We can still choose to use old-style 3-vectors instead, but then different observers will not agree on these quantities, because they do not share the same notion of time. At the end of the day there still will never be any physical contradictions, but that may not immediately be obvious, and might sometimes require some more or less awkward maths to demonstrate it. This is one of the reasons why in relativistic electrodynamics we try and go away from the E-B formalism, and use the tensor formalism with just the F field instead. Doing so means we don't have to worry about observer dependency. If needed, the E and B fields can always be recovered later.
    Below is a link to a site which uses simple algebra to demonstrate the equivalence of the fields and frames. I don't see why simple algebra cannot be used to explain the even simpler situation I'm describing which lacks a conducting wire. As far as I know, SR was formulated before Einstein adopted Minkowski space and he liked to keep things simple where possible. Einstein described SR as “an outstandingly simple combination and generalisation of the hypotheses … on which electrodynamics was based.”

    Relativity of Electric and Magnetic Fields
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    The equivalence of the fields and frames is most easily shown by simply writing down the electromagnetic 4-force between the protons in covariant form :



    This is invariant when you go from one frame to another, so it is valid for all observers. No algebra required, no contradictions possible. How much simpler can it possibly get
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    It is simple for you to write down the covariant form but it means little to me. I can't see any accelerations in it for the two different frames.

    Equal accelerations can be shown using simple algebra if a conducting wire is present, so why isn't this the case for a vacuum?
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    Quote Originally Posted by Andrew? View Post
    It is simple for you to write down the covariant form but it means little to me. I can't see any accelerations in it for the two different frames.

    Equal accelerations can be shown using simple algebra if a conducting wire is present, so why isn't this the case for a vacuum?
    The relationship between 4-force and 4-acceleration is simply



    with m0 being the invariant mass. This is again an invariant expression, so it is the same in all frames. These are general laws, and valid regardless of what the accelerated object is ( wire or free particle ).
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    Thanks, but I still can't see the two accelerations. Ignoring all the maths, it seems obvious that the accelerations can only be reconciled using the particles in the wire. It doesn't work in just a vacuum.

    I think we are just repeating ourselves now. I'll try and find time to post a related idea of mine that you can get your teeth into.
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    Quote Originally Posted by Andrew? View Post
    Thanks, but I still can't see the two accelerations. Ignoring all the maths, it seems obvious that the accelerations can only be reconciled using the particles in the wire. It doesn't work in just a vacuum.
    As explained and shown, there is one 4-acceleration that all frames agree to; regardless if that is between free particles, or wires.

    I think we are just repeating ourselves now. I'll try and find time to post a related idea of mine that you can get your teeth into.
    You see, my perspective is different from yours; we haven't made the same choices. I'm just an amateur who does physics as a hobby; I started about 25 years ago, because I have a very inquisitive mind, and a strong sense of curiosity about how the universe works. I never went to university, but I invested portions of my free time to sit down and teach myself maths and physics - because I wanted to understand. In the beginning relativity seemed strange and full of contradictions to me also, but as I learned more and more about it, the clearer it all became. Now I'm fortunate enough that I can drill down all the way to the root causes of why relativity is necessary to have a self-consistent universe; it turns out that everything boils down to a fundamental principle in topology, which is as simple and intuitive as it is powerful and wide-ranging. That same principle also underlies electromagnetism and other classical field theories. I can follow the grapevine from that topological root, all the way up to how it manifests in specific phenomena, with maths and all; not only does it make sense, but it is above all also a thing of supreme beauty. Without the pressures of professional academia, I can now watch how that tree blossoms further and further in or modern understanding of the world - CDT, CFT/AdS, EPR=ER, causal sets, M-Theory, LQG, tensor networks, QFTCS/T, and so on. Many of these ideas will not stand the test of time, but that's ok - mistakes is how we learn and progress. In the meantime, we can use the theory of relativity in the real world to build computers, send probes accurately all the way to Pluto and the Kuiper Belt, predict the next solar eclipse right down to a scale of seconds, understand why gold has the colour it does, predict the outcome of particle collisions in accelerators to extremely high accuracy etc etc. All of this encapsulated in one simple, overarching framework - testable, repeatable, peer-reviewed, and put into practice countless times every day.

    In your case, it was obvious from the beginning that you started with a conclusion ( relativity must be wrong ), and then tried to support this in whatever way you could, instead of following the scientific method. There never were any genuine questions, since you had your mind already made up about which answer was acceptable to you, and which one not. Case in hand would be your assertion that there is no evidence for SR - whereas in fact it is arguably the single most thoroughly and extensively tested model in the history of science. And I don't blame you, this is how the human mind naturally works. I've been on Internet forums for a long time now, and the one thing I have learned is that everything is disbelieved by someone; this is why we have everything from flat earth to ancient aliens here. To a degree this is healthy and good, because oftentimes one has to put lots of thought into addressing such personal ideas. But ultimately the true test is always how an idea like that performs in the real world. Can you build a fully functional probe and achieve a stable orbital insertion around - say - Neptune, without reference to anything from relativity ? Will ancient alien tech get you there ? I very much doubt it, because relativity is in everything from the design of the circuit board, to the comms module, to the propulsion, to the orbital dynamics, to navigation. It's integral and fundamental to the universe; it picks up where Newton failed.

    Anyway, you have been polite and non-aggressive, and as a forum admin I appreciate that a lot - so please do continue on. The one thing I'd say to you - from a perspective of someone who has studied relativity in detail - however is that your understanding of the theory is incomplete and deeply flawed. This is quite separate from whether you agree with the theory or not. If you wish to show relativity wrong, you have to formulate an argument from a position of knowledge, not ignorance; hence I would advise you to study its foundations and principles a lot more. If you can make a clear, logic, mathematical argument, and support that with experiment and observation, then people will listen; but "it doesn't make sense to me" and "there's no evidence" will get you nowhere, trust me on this.

    if you want, I can recommend texts for study. It's up to you.
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    Thank you Marcus for your reply. I do not doubt your knowledge of physics or maths, and if there were a prize for explaining physics, I would gladly nominate you. My criticism of SR is based on the disproof of its predictions about distance contraction and the inevitability of symmetrical time dilation.

    In the link I previously quoted, simple algebra was used to calculate equal accelerations when a conductor is present. The reasoning was based on distance contraction in the conductor. In the absence of a conductor, the accelerations cannot be equal. If they were, it would be extremely easy for you to demonstrate this, again using simple algebra.

    The only other approach is experimentally. Does anyone know how feasible/expensive it would be to measure and film the accelerations?
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    Quote Originally Posted by Markus Hanke
    it turns out that everything boils down to a fundamental principle in topology,
    I would like to give your post a thousand likes sir!
    I would also like to know a name for the referenced topological principle, please?
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    Quote Originally Posted by GiantEvil View Post
    Quote Originally Posted by Markus Hanke
    it turns out that everything boils down to a fundamental principle in topology,
    I would like to give your post a thousand likes sir!
    I would also like to know a name for the referenced topological principle, please?
    Seconded. Markus also tells us that English is not his first language
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    Quote Originally Posted by Andrew? View Post
    Thank you Marcus for your reply. I do not doubt your knowledge of physics or maths, and if there were a prize for explaining physics, I would gladly nominate you. My criticism of SR is based on the disproof of its predictions about distance contraction and the inevitability of symmetrical time dilation.

    In the link I previously quoted, simple algebra was used to calculate equal accelerations when a conductor is present. The reasoning was based on distance contraction in the conductor. In the absence of a conductor, the accelerations cannot be equal. If they were, it would be extremely easy for you to demonstrate this, again using simple algebra.

    The only other approach is experimentally. Does anyone know how feasible/expensive it would be to measure and film the accelerations?
    I do not believe that one can show the invariance of the 4-acceleration between frames in any simpler way than I have done. If this does not satisfy you, then I'm afraid I cannot help any further, since I don't know how else to explain or show it. Maybe someone else here has an idea. For me personally, the equivalence is so obvious as to be almost trivial, even without using any maths.

    As for SR in general, again, you will see in the explanations given precisely what you want to see - that's not so much a criticism, as it is a statement of fact about human nature and psychology. I myself would be just as guilty of that in disciplines and areas other than physics. So if you think SR has been disproved for you, then neither I nor anyone else here will be able to convince you otherwise. You are ultimately the owner of your own knowledge and understanding.

    But regardless, you have been respectful and polite, so you are always welcome here to ask question if you like to. This is not contingent upon any agreements between us - I'm happy to offer up what I have learned, with the express understanding that what you do with that information is down to yourself entirely, even if that understanding is contrary to my own.
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    Quote Originally Posted by GiantEvil View Post
    Quote Originally Posted by Markus Hanke
    it turns out that everything boils down to a fundamental principle in topology,
    I would like to give your post a thousand likes sir!
    I would also like to know a name for the referenced topological principle, please?
    The principle I was referring to is that for a specific class of objects, "the boundary of their boundary is zero". For example, consider a ball in three dimensions - its boundary is its surface, i.e. a sphere. What is the boundary of a sphere ? It doesn't have one - it's smooth and continuous everywhere, without any edges, discontinuities, borders etc etc. So when you start with a ball, then the boundary of its boundary is zero - there is none. It turns out that this is a pretty general principle in topology.

    Now, it turns out that this seemingly simple fact is of fundamental importance to both electromagnetism and gravity. The problem for me now is just how to explain the exact connection in layman's terms, because that's not so easy ! For gravity, the mathematical restatement of that connection is in my signature below, but of course that will not be of much use to most people here.

    Allow me some time to think about how to explain it without incomprehensible maths. I'll come back to this.
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    Quote Originally Posted by Markus Hanke View Post
    Quote Originally Posted by GiantEvil View Post
    Quote Originally Posted by Markus Hanke
    it turns out that everything boils down to a fundamental principle in topology,
    I would like to give your post a thousand likes sir!
    I would also like to know a name for the referenced topological principle, please?
    The principle I was referring to is that for a specific class of objects, "the boundary of their boundary is zero". For example, consider a ball in three dimensions - its boundary is its surface, i.e. a sphere. What is the boundary of a sphere ? It doesn't have one - it's smooth and continuous everywhere, without any edges, discontinuities, borders etc etc. So when you start with a ball, then the boundary of its boundary is zero - there is none. It turns out that this is a pretty general principle in topology.

    Now, it turns out that this seemingly simple fact is of fundamental importance to both electromagnetism and gravity. The problem for me now is just how to explain the exact connection in layman's terms, because that's not so easy ! For gravity, the mathematical restatement of that connection is in my signature below, but of course that will not be of much use to most people here.

    Allow me some time to think about how to explain it without incomprehensible maths. I'll come back to this.
    Well, you have explained the meaning of "the boundary of a boundary is zero" in a specific case that you say is generalisable.

    That part clears up my lack of understanding of this apparently extremely important principle that I struggled with in one of your earlier threads.

    So thanks for putting that behind me.

    If you are able to address the relationship to EM and Gravity for Giant Evil I will try to pay attention although there may be very few pickings for me at that table with my level of appreciation.(but I look forward all the same)

    ps I think I may be right to recall that it was a Mr Wheeler who said that this principle was central to GR....
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    Perhaps I start with EM first, because that is simple and straightforward. You see, the electromagnetic field doesn't just appear out of thin air, but it arises from an electromagnetic potential, denoted by A. Without wanting to go into any overly complicated maths, I will just state that the concept of a "boundary" of something is encapsulated in a special type of operator, called the exterior derivative, written with a small "d". By definition, the electromagnetic field can be considered the "boundary" of the underlying potential, hence



    This is not just ad-hoc notation, but a formal, mathematical statement; this formalism is called exterior algebra, and you can find lots of textbooks on it, if you are interested. For here and now, we are interested only in what it means - that the EM field can be considered as the boundary of the underlying potential field.

    Now remember our topological principle - the boundary of a boundary is zero. Hence, if we apply the exterior derivative - which represents a boundary for us - twice, we should get zero : dd=0. And it turns out that this is indeed the case; this can of course be formally proven in terms of maths.

    So what happens if we look for a boundary in the electromagnetic field ? We can first try a straight application of our operator :



    by virtue of our "boundary of a boundary is zero" principle. Without proof ( see appropriate textbooks ) I state here that the object F physically represents the magnetic part of the electromagnetic field; it is called the Faraday form. Hence, the statement dF=0 means quite simply that the magnetic field has no boundary - magnetic field lines do not begin or end anywhere, they must either form closed loops, or extend into infinity, but there are never any magnetic charges ( magnetic monopoles ). If needed, one can go and explicitly write out the statement dF=0 in terms of E and B fields, and the result is precisely the first two ( "magnetic" ) Maxwell equations.

    So, starting from a potential, by simple application of exterior algebra and the aforementioned topological principle, we immediately find two of the four Maxwell equations ! Or to put it differently - magnetic field lines must be closed, because the boundary of a boundary is zero. The reason electromagnetic fields have the form they do, is because of a fundamental principle in topology.

    To find the other two Maxwell equations, we consider what is called the dual of the above object F. Geometrically, this can be thought of as the same object, only in a perpendicular direction to the original F. This dual form - called the Maxwell form - is obtained by applying an operator called the Hodge dual, denoted by a star symbol. If we do this, and then look for the boundary again, we get



    with J being current density. What this physically means is that electric field lines begin and end at electric charges, just as we would expect. If we were to write out the above in terms of E and B again, we'd get the other two ( "electric" ) Maxwell equations. Watch what happens now if we apply the exterior Derivate operator once more :



    which physically means that electric charge is conserved - and it is conserved because the boundary of a boundary is zero !

    So essentially, the structure that electromagnetism can take in our universe is determined by a topological principle; topology plays a fundamental role in the inner workings of nature. Of course. Maxwell did not know this - when he wrote down his equations for the first time, he did so based on experiment and observation only. The above relations actually follow from the theory of relativity, and the formalism employed above is equally valid in all frames of reference, even in the presence of gravity. These are fundamental laws and findings about EM which all observers agree on.
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    Thank you very much Markus!
    I'm only gonna have to read this about a 100 times, but I'm about half way there now.
    Looking forward to the "gravity" part.
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    None of the comments has addressed the question about the two accelerations that would be measured, so I will conclude by restating the problem:

    Two positive ions initially move vertically downwards side by side in a vacuum. An observer also co-moves downwards with the ions at their speed of v and is in the same horizontal plane. I will refer to the co-moving observer as C and an observer at rest as R.

    1. C considers his moving frame to be at rest, so his clock is unaffected and the ions (which initially appear to be at rest) just have a rest mass of m0. C observes each ion accelerates at aC. From the equation F=ma we can say aC = FE/m0 where FE is the electrostatic repulsion.

    2. For R at rest there is a magnetic force of attraction which varies linearly with the ions’ speed of v. This can be expressed as kv, so for R the repulsion is reduced to FE – kv. The mass of the moving ions is increased by the Lorentz factor of γ or (1 - v2/c2)-0.5 to give a further reduction in acceleration. Hence R should see a lesser acceleration of aR = (FE – kv)/γm0.

    C and R see each other’s clock is slow by a factor of 1/ γ. So each can say the other’s measurement of speed is increased by a factor of γ and acceleration by γ2. But I don’t see how non-linear factors can reconcile acceleration differences which basically vary linearly with v. A single acceleration event seems to produce fundamentally different accelerations which cannot be reconciled by Lorentz transformations. Conventionally there are said to be separate electric and magnetic fields that are the same when viewed from different frames. In which case these same fields should have the same effects for all values of v.


    The normal algebraic derivation of field equivalence is based on a fictitious contraction within a conductor. In the example I have given there is no conductor, so experiments in the frames of the two observers would show the accelerations are different. The equivalence of electric and magnetic fields according to SR is another false belief. When maths based on false beliefs confronts reality, reality wins.
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    The equivalence of electric and magnetic fields according to SR is another false belief
    SR does not postulate any "equivalence" between these fields; it states only that they are different aspects of the same underlying entity. This has been explained to you by different people here, and in considerable detail.

    None of the comments has addressed the question
    I think the issue is rather that you have chosen to disregard the answers that were given to you. But that is of course your prerogative.

    reality wins.
    Indeed it does ! Nature itself is always the final arbitror; we make models, extract numerical predictions from it, and then compare those predictions against experiment and observation. The scientific method. That is the reason why science went beyond Newtonian mechanics, because reality showed us that this - important, but ultimately flawed - approach works only as a first approximation under very limited circumstances.

    I am sure the irony of this discussion is not lost on the reader; after all, the computer you are typing on, the grid that powers it, as well as the Internet that connects it to the rest of the world, are all based on the very models the veracity of which you so vehemently deny - Maxwell's electrodynamics, and the theory of relativity. If they are all wrong, then ending up with something as intricate as a fully working computer seems like quite a strange coincidence, doesn't it
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    Quote Originally Posted by Markus Hanke View Post
    I think the issue is rather that you have chosen to disregard the answers that were given to you.
    I originally asked where my algebra had gone wrong for the situation involving a vacuum. The two accelerations seem unavoidably different. Elsewhere, the accelerations are depicted as being equal in the presence of a conductor. This depiction involves simple algebra. If the accelerations are the same without a conductor then this could be shown using simple algebra. None of your explanations about topology etc. have addressed my question, nor will any such future ones.

    Quote Originally Posted by Markus Hanke View Post
    I am sure the irony of this discussion is not lost on the reader; after all, the computer you are typing on, the grid that powers it, as well as the Internet that connects it to the rest of the world, are all based on the very models the veracity of which you so vehemently deny - Maxwell's electrodynamics, and the theory of relativity. If they are all wrong, then ending up with something as intricate as a fully working computer seems like quite a strange coincidence, doesn't it
    Modern technology makes use of asymmetric time dilation in bodies that acquire more KE. It does not make use of distance contraction or time dilation in equipment at rest because no such changes occur. As regards EM, I am merely saying the magnetic field needs to point in the correct direction - parallel with the electric field and the measurable force. This makes no difference to how computers work.
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    The two accelerations seem unavoidably different
    The acceleration you used is 3-acceleration, which is of course different, since it is frame-dependent. But if you use 4-acceleration, as I have done in my math snippet, then you will find that they are indeed the same. You will also notice that 3-acceleration is just the spatial part of 4-acceleration.

    It does not make use of distance contraction and or time dilation in equipment at rest
    Electrons are not at rest. And even if they were, they wouldn't have the properties they do without SR ( also true for all other particles ), so electronics wouldn't work the way they do. The behaviour of electrons and positrons is described by the Dirac equation, which is a relativistic wave equation, and essentially a restatement of the energy-momentum relation in SR.
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    None of your explanations about topology etc. have addressed my question
    That was intended only to show why Maxwell's equations ( and hence the topology of the EM field ) have the form they do. If you say that the magnetic field is parallel to the electric one, then you are claiming that topology is wrong. You are rapidly running out of disciplines to deny and reject In fact, I don't think there are any left, actually - you have presented a wholesale rejection of all of physics, maths, and chemistry, in one form or another. Essentially you have taken a step back into the late 1600s.
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    Quote Originally Posted by Markus Hanke View Post
    Essentially you have taken a step back into the late 1600s.
    Time dilation and E=mc^2 were unknown then.
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    Quote Originally Posted by Markus Hanke View Post
    The acceleration you used is 3-acceleration, which is of course different, since it is frame-dependent. But if you use 4-acceleration, as I have done in my math snippet, then you will find that they are indeed the same. You will also notice that 3-acceleration is just the spatial part of 4-acceleration.
    As usual I don't see how you are answering my question about the two accelerations. It seems to me that accelerations are, in general, 4 dimensional as they involve measurements over time.

    Are you saying that the accelerations would differ when measured in a laboratory but would be equivalent topologically or when viewed in Minkowski space?
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