Relativity fails with Magnetic Force

You've completely missed the point of the paper. Levy-Leblond derives all the possible "reasonable" relativity theories that don't contradict everyday experience. A characteristic invariant speed appears in one of the two possibilities he derived. At the time (the 1970's) it was quite well established among the paper's intended audience (research physicists) that Galilean invariance had been ruled out experimentally as a possible relativity theory. The paper doesn't prove that physics is Lorentz invariant. It just shows that the only other possibility is "no invariance". If anyone ever demonstrates a Lorentz violation experimentally, we'll have to throw out the idea of relativity theories altogether, since Levy-Leblond's paper proves we've already exhausted all the possibilities.

So? That's exactly how and why I introduced the paper: a derivation of the Lorentz transformation that doesn't depend on the invariance of c as a postulate.

No, we use standard reference systems (eg. clocks and rulers) to define inertial coordinate systems. We've been making measurements this way for thousands of years now without any problem of self-reference.

Er, that's exactly why Einstein shouldn't (and didn't) attribute any physical significance to the Galilean transformation. Why would he want to formulate a theory that contradicts itself?

No, I think it's a false analogy because you've merely invoked it without justifying it.

Well light doesn't seem to have any trouble reflecting off moving mirrors in my experience, so I don't think your point has much to do with reality.

There's no cheating. The velocity is defined as the time derivative of position where t and x are the coordinates of an inertial frame. Inertial frames are defined as coordinate systems in which the laws of physics take their canonical form (in practice as measured by standard reference systems used as measuring instruments). Historically it was assumed that inertial coordinate systems were related by Galilean transformations. The theory of relativity is a revisitation of that assumption.

Your problem seems to be that you insist on building the Galilean transformation (and by consequence, Galilean velocity addition) into the definition of what constitutes an inertial frame. This is your own non-standard definition and it has nothing to do with the invariance of c. Experiments that measured invariance of c did not use your definitions. No steps were ever taken to ensure the measurements were made in coordinate systems related by Galilean transformations.

So what does this have to do with accelerator experiments, in which particles are accelerated in an electric field, collide, and produce new particles which provoke cascades in calorimeters (or heat up the beam dump if they don't collide)?

You haven't actually suggested any new physics here. All you've done is re-express things in terms of the old Newtonian definitions. You can still talk about the "Newtonian energy" and the "Newtonian momentum" of a particle, or the "Newtonian force" (defined as the rest mass multiplied by the "Newtonian" momentum) if you really must. That doesn't mean it's a good idea.

I repeat a quote from one of your own sources:

That doesn't sound like recital of theory to me.

 

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And why would I want to do this in general? The fact that you can fiddle a bit with your invalid operations on partial derivatives just to force them to come out with the right answer doesn't make the fiddling valid in general.

More importantly, you still haven't explained how your equation 11 can be correctly derived from the synchronicity condition when their solutions aren't the same. You've admitted that any multiple of the Lorentz transformation satisfies the synchronicity constraint. Yet you're claiming you can also show, via "deriving" equation 11, that nothing can satisfy the synchronicity constraint. There's a really bizarre duality going on in your thinking here.

 

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Sure it does. If you want to hold the submerged object in place and prevent it from accelerating, you'll have to counteract the same buoyant force you would if the object were submerged in a tank.

Have you ever actually seen a derivation of the buoyant force on a submerged object? Its origin is in the pressure gradient present in a static fluid in an external force field, and has nothing to do with treating the object as "balanced against" a "displaced fluid element" as on a set of scales.

I was addressing your general statement that the solutions to the time-independent equation had nothing to do with time evolution. It's incorrect: you can obtain all the time evolving states as linear combinations of the stationary states.

There's a possibility you didn't think about: it's perfectly possible to have a stationary current - i.e. a zero-divergence current such that the particle probability distribution does not evolve in time. This is the case for the stationary solutions for quantum tunneling. Quantum theory is quite specific about this. The probability current associated with a given wavefunction \(\psi\) is given by:

\( \bar{j} \,=\, -\frac{i\hbar}{2m} \Bigl[ \psi^{\ast} \bar{\nabla} \psi \,-\, \psi \bar{\nabla} \psi^{\ast} \Bigr] \)​

Calculate this for the textbook solutions to the quantum tunneling problem and you'll find that the current is non-zero. The stationary states each describe a delocalized particle with a net current leaking through the potential barrier.

So? The point is that the Lagrange formalism stands just fine on its own as a general framework for defining theories. Again, classical mechanics is just the special case where you set L = T - V. Specifying a different Lagrangian defines a different theory with different equations of motion, so Lagrange methods are applicable to a much larger family of theories than just classical mechanics.

No it's not. You're correct when you say that Noether's theorem only necessarily applies to theories with a Lagrange formulation. It's your implication - that physicists are really stupid for carelessly assuming Noether's theorem always applies - that isn't. The requirement of a Lagrange formulation in practice turns out not to be a significant restriction.

For instance, at the moment the two fundamental established theories in physics are the standard model and general relativity. Between them they describe most of what's known about the four fundamental interactions seen in nature. General relativity has a Lagrange formulation (the Einstein-Hilbert action) from which you can derive the Einstein field equation. And the only existing formulation of the standard model is its Lagrange (or Hamiltonian) formulation.

Try doing quantum theory without energy and momentum.

 

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For me there is no question that a preferred reference frame has to be defined physically. You were claiming that the lack of this definition (and thus the resulting frame-dependence) is deliberate. I objected that you don't have any evidence for this. For what reasons or motives the definition is lacking does not touch on my view that the Lorentz force is ill-defined unless additional constraints are introduced.

You shouldn't assume that everybody who has a different opinion than you about the interpretation of a theory has failed to understand the latter. Maybe it is just the other way around. As I indicated already, I did not have any difficulties in 'learning' physics (the physics being taught that is). But this didn't prevent me later on to reflect on certain issues more deeply, and it is only then that problems became apparent to me which are usually not addressed in courses or books.

Violating causality is only a special case of violating locality (introduced by the time arrow). Essentially the argument is the same.

Well, you provided the irony yourself by starting a paragraph supposed to make the point that you are a thinking person as well with the phrase "I don't think.." . But anyway, this was only a side-note to counter your attempt to draw the scientific discussion into a pointless debate about personal competence.

Well, I reckon everybody has a story to tell how he got to learn a certain theory. The question is whether this 'learning' was just within the rigid framework of pre-existing arguments and concepts given to you (usually by the educational system or popular resources) or whether this involved a conscious attempt to transcend the subject matter by trying to analyze it in a critical way. Only the latter can result in a true 'understanding'.

This is exactly why I am saying that a strictly local physical mechanism (e.g. as suggested by the induction equations) is not possible. As mentioned already, the magnetic or electric field a given point can only depend on the physical state of the matter at all other points.

As I mentioned already, the fact that an infinitely long wire has no well defined total charge is exactly the excuse for violating charge invariance in these derivations. But there is really no need to discuss this any further. As far as I am concerned, all the relevant arguments have been mentioned already, and the neutral reader can make up his/her own mind about who is right here.

Thomas

 

If the decision between the Lorentz- and Galilei transformation has to be made by experiment, then this can hardly be a called a 'derivation' of the corresponding transformation. So again, it is misleading and incorrect if the paper claims to be a derivation of the Lorentz transformation.

As I said, the paper doesn't derive the Lorentz transformation. That could only be done by indeed having the invariance of c as a constraint. But as mentioned already, if done correctly, that would result in the identity transformation and not the Lorentz transformation .

But the clocks and rulers are in effect coordinate grids that are given a priori. There is nothing (or at least ideally there should be nothing) amongst the physical matter they map that would affect them in any way.

Why did he formulate a theory at all if his only aim was to not contradict himself? The point is, in a physical theory there should not be any equation that does not have any physical meaning.

The point is you can not tell how it is reflecting off the moving mirror unless you use detectors in that mirror's reference frame (i.e. detecors that are at rest with regard to the mirror). Trying to use detectors at rest in a different reference frame will inevitably give wrong results because of the invariance of c.

No, the problem is that you seem to be insisting that the coordinates of a material object should transform in the same way as those of a light signal. This is a paradox assumption given the fact that the speed of the latter is invariant with regard to moving reference frames but the former not. As mentioned already, only two different transformations can describe the two cases correctly (the Galilei and the identity transformation).

First of all, it is not exactly an original piece of information to merely confirm the theoretically expected value. What I am saying is that generally speaking I would not value information issued by the military as highly as from other sources, because as I said, complete public transparency contradicts in principle the nature of the military. What's more, as I pointed out already, the standard GPS system is run in a way (namely by comparing timings from different satellites rather than comparing a satellite with a ground clock) so that the 38 microsec/day offset is irrelevant for position determinations (as it would only lead to an error of about 1 cm)

Thomas

 

You gave a specific example which you claimed would in general invalidate my operations. I pointed out that your example was incorrect in as far you did the variable substitution only with regard to one variable. My example shows that if you do the substitution with regard to both variables and average the two solutions, you get the correct total differential.

There is no duality in my thinking. I derived the conclusion that the constraint can not be satisfied (unless v=0) by differentiating it with regard to x' (applying the chain rule correctly). If Einstein applies the chain rule incorrectly and gets a different result (namely that the Lorentz transformation satisfies the constraint) then it his result that is incorrect, not mine (you can see this actually from the inconsistency that any value for gamma would satisfy the constraint but not any value for the synchronicity offset (whilst both depend on v according to Einstein's solution).

Thomas

 

The buoyant force is not the issue of interest here. It is the acceleration due to this force, and that depends obviously on the displaced mass. Whether the buoyant object falls or rises (and with what initial acceleration) depends exactly on the displaced mass. In this sense, the situation is in no way different from a mechanical scale (where one side of the scale represent the buoyant object, the other the displaced mass).

As indicated already, any stationary solution misses the whole point about the tunneling, namely that initially the electron is on one side of the barrier, and after some time on the other. Essentially, the stationary solution assumes, contrary to the initial conditions, that the electron is distributed with the same probability throughout space. This uniform distribution is then merely modified by the barrier. It does not mean that any electron would have gone from one side of the barrier to the other. Note that actually the wave functions in the different regions are 'stitched' together by the assumption of continuity (see e.g. http://en.wikipedia.org/wiki/Rectangular_potential_barrier . It is only by this procedure that a pseudo-causal connection between the regions is established. In reality however there is no connection between the regions if the barrier energy is higher than the particle energy (in a similar way as it is not possible for an atomic electron to spontaneously go from a lower to a higher level).

The whole point of having Lagrangians (or Hamiltonians) is to have a formulation analogously to Classical Mechanics. If you look at the Lagrangians in various areas of physics given at http://en.wikipedia.org/wiki/Lagrangian then you can see that all these are also the difference of two terms analogously to T-V in Classical Mechanics. The point is that there is no reason to assume that such an analogy must exist, i.e. the 'mechanization' of physics in this sense is completely unjustified.

First of all, quantum theory was developed about 10 years after the formulation of Noether's theorem, so it could not have had any impact on the development of the latter. As I said, if the concepts of energy and momentum are relevant in quantum theory, then this must be considered as a mere coincidence. It certainly does not strictly follow from Noether's theorem.
Moreover, note that energy and momentum in quantum theory are equivalent to frequency and wave-number respectively (E=h*nu, p=h/2pi*k). If for instance an atomic transition results in a light emission with frequency nu, then by means of E=h*nu one formally associates with this a quantum mechanical energy difference E for the two atomic levels involved, but in principle there is no necessity to do this (and in fact this energy is in general different from the classical energy difference between the two states).

Thomas

 

No, they aren't equivalent.

In QM, energy and momentum are expressed in terms of the probability of a particle being in one place rather than another. If a photon has momentum it carries "energy" from one place to another as information, therefore the information content of a photon's wavefunction is the probability it will be detected anywhere/somewhere in space and time. Momentum and position are separate parts of a wavefunction, what we "see" is an event that locates the momentum, up to probability.

 

Well maybe you should start questioning this, especially given your inability to incorporate your interpretations into a consistent theory.

I've told you: electrodynamics is Lorentz invariant. This is a provable and proven symmetry of the Lorentz force and Maxwell's equations. If the Lorentz force and Maxwell's equations are valid in a source's rest frame, they are automatically valid in any coordinate system related to the rest frame by a Lorentz transformation. You can apply electrodynamics in a source's rest frame if you want. The point is, you don't have to. Seriously, pick up a book on electrodynamics and read it sometime.

I'm not assuming anything. You are demonstrating that you do not understand relativity, electrodynamics, and partial differentiation. It's not just in your opinions. It's in the way you insist on setting up your arguments. If you'd actually studied the relationship between relativity and electrodynamics you simply wouldn't be bothering with finite wires at all - it's about the least productive approach imaginable. The level of this discussion is really quite mundane, and it's being set by you. Every time I try to discuss the problem at a more advanced level, I'm silently ignored. Discussing relativity with you is at least as painful as discussing classical physics with someone who refuses to learn calculus and vector notation.

It is plainly obvious that you have not read relevant textbooks which would answer your questions. You just don't seem interested in learning anything that doesn't conform to classical physics. Even quantum mechanics you try to dismiss as a "tool for predicting the energy levels of atoms" or something of the sort.

You haven't answered my question: are you still maintaining that the induction equations violate causality?

Either way this argument doesn't help you. The induction equations are still incompatible with invariant electric and magnetic fields. Calling the fields "macroscopic" doesn't change anything.

Er, in case you missed it the irony is that I stated "I don't think you're a particularly good judge of what constitutes logical thinking" and you went ahead and replied with a (particularly dishonest) logical fallacy. And you just did it again. If we count that Einstein quotation of yours, that's at least three out of context quotations. Please, stop making logical fallacies. Then my ad hominem won't be an accurate assessment of you any more.

It's not pointless. I have identified your lack of competence in relativity, electrodynamics, and partial differentiation as sources of much of your problems. You should really either take steps to correct this or leave the evaluation of relativity to people who are competent. Most of the rest of your problems really just consist of you throwing the words "inconsistent", "unallowed", or "illegal" at any theory that contradicts an assumption you're unwilling to part with.

Please, take it for granted from now on that I've analysed relativity critically. And in much more depth than you seem willing to.

No it doesn't. On the one hand, you have your own definition of "strictly local" which no sensible theory satisfies. At the opposite extreme we have non-local theories that specify relations across finite distances in an irreducible manner. The kind of "locality" we're interested in in physics is something in between, as I've already described. If you like thinking in terms of "infinitesimals" like dx and dt then, in the context of the hyperreals, infinitesimal quantities are located in between zero and the finite reals in terms of magnitude.

No, it means there's nothing to violate.

We shouldn't be discussing this at all. The webpages you linked to prove that relativity and electrodynamics are compatible only for the special case of an infinitely long wire - nothing more. As I've said many times now, this is not intended as a general proof. I've already told you that the best way to prove the Lorentz invariance of electrodynamics in full generality is to express the theory in covariant form. It's done in any reasonably complete course or textbook on electrodynamics. It's second-year undergraduate physics material. I keep telling you this and it keeps falling on deaf ears.

 

I'm not getting dragged into a pointless debate over what the word "derive" should or shouldn't mean. The paper stands on its own merits whatever you call it, and the conclusion stands: Lorentz invariance or no invariance.

If you mean that their lengths and evolution rates should be unaffected by relative motion, then you are making an a priori assumption - for which you have no experimental evidence - that relativity explicitly abandons. This point just doesn't seem to be getting through to you: you are merely restating an intuitive assumption that almost everyone who has ever studied relativity will have considered at some point and subsequently dismissed.

Where did you get the "only" from?

Er, why?

In any case the Galilean transformation isn't crucial to Einstein's derivation. He just substitutes it because it simplifies the intermediate stages of his derivation a little. Just compare the synchronicity constraint on Einstein's paper (in terms of x') with yours (in terms of x). They're completely equivalent and you'll get the same result from either. Einstein's approach just saves ink.

But there's nothing preventing us from observing and attributing coordinates to the interaction between light and moving objects. Saying we're "not allowed" to do this sounds more like a cop-out than a physical explanation. The only way I can make sense of your idea is if you've got some rule predicting that light emitted from (say) the origin will only interact with an object moving at velocity v at space-time coordinates which satisfy :

\( x \,=\, (v + c) t \)​

wherever this intersects with object's trajectory (this "law" is Galilean-invariant and reduces to x = ct in the object's rest frame). But making up a rule like this and actually having a theory of light that predicts this behaviour are two different things. The point with relativity is that it doesn't require made-up rules like this: it requires no modification to electrodynamics.

Wrong. The Lorentz velocity addition formula is consistent with both the velocity transformation of light and that of everyday objects. This is a counterexample to your naive claim, which you should really stop repeating.

And here we go again. So what are you accusing the U.S. military of? All you've done so far is vaguely insinuate that because of the military involvement, maybe you get an excuse not to answer an otherwise difficult question I asked you.

Your webpage proves that a fictional setup involving perfectly stationary satellites would be immune to timing errors. GPS satellites aren't stationary. They orbit Earth twice a day (ie. they're not in geostationary orbit) and in different orbital planes. So both their positions in the sky and their positions relative to one another are constantly changing. Then there's also the fact that the orbits are slightly eccentric so their speeds (and therefore time dilation rates) vary. Where do you address all this? Have you even made any effort to look up how the GPS system *actually* operates and how relativistic corrections are implemented?

 

It did. I showed that it lead to a contradiction. And you, in general, have not justified your operation. You fabricated it. And it just doesn't seem to be bothering you that your invalid operation is leading you to an incorrect result.

Which is exactly what you do on your website. In your equation 11, the variable x has completely disappeared.

And since the correct result is:

\( \tau \,=\, \alpha \bigl( t \,-\, \frac{v}{c^{2}} x \bigr) \)​

for arbitrary \(\alpha\), your conclusion is wrong. This is verifiable by direct substitution into the synchronicity constraint. Please don't tell me you actually need me to prove this algebraically for you.

No. Einstein's result is correct. Alternative methods produce the same result. For instance as I said Einstein most likely applied a Taylor expansion. You could also just set:

\( \tau \,=\, A x \,+\, B t \)​

plug this into the synchronicity constraint, and see what the constraint between A and B is. Finally, as I said, you can simply plug the Lorentz transformation into the constraint and check that it is at least one solution.

 

It is precisely this unjustified and unsupported gross misconception of fluid mechanics that I'm criticising. It's always the same with you: you present a misconception, I correct it, and then you just repost your misconception as unquestionable fact. Where is this critical thinking you keep going on about? You say that a submerged object is analogous to a mass balanced against the "displaced fluid" on a set of scales. Why should anyone believe you?

No, I've just given you the correct interpretation of the stationary solutions. Please re-read my last post on the subject.

You're still demonstrating you don't understand quantum mechanics. There's no issue of assumption. The stationary states are *possible* solutions that describe a net current leaking through the potential wall. I'd expect they're also the relevant solutions to most experiments and practical applications. If your initial condition is a localized particle then in principle you can obtain the evolution for that by decomposing your initial condition on the energy eigenstates, multiplying each mode by \(e^{-i\omega t}\), and then adding them up again.

Note that a localized particle doesn't actually have a well-defined energy. Only the stationary states do. All you could really say about a partially localized particle is that it has a higher probability of crossing a potential barrier than a classical particle with an equivalent probability distribution in energy, due to the momentum carried through the barrier by the lower energy eigenstates.

So, basically, quantum tunneling is wrong because it contradicts your preconceived classical expectations?

No. Not similar. The stationary states *are* the energy levels. Quantum tunneling does not predict any "jumping" from one stationary state to another.

No, the point of having Lagrange formulations is explained right here on the Wikipedia site you linked to.

It's not an assumption. Those are the Lagrangians needed in order for the theories to make the predictions they do. Experiments dictate which Lagrangians we need. The justification for the application of Lagrange methods in all these cases is the same as the justification for any idea in science: we tried out the idea and it worked.

I really don't see what you're hoping to gain by continuing to argue this point. We know Lagrange methods apply to relativistic mechanics, classical electrodynamics, the entire standard model (including quantum electrodynamics), and general relativity because we have an explicit Lagrange formulation for each. So "coincidence" or whatever name-calling you like to resort to, Noether's theorem gives us an explicit energy conservation law for each of these theories. That's unarguably more general demonstrated applicability than just to classical mechanics.

No, quantum mechanics is explicitly formulated that way. The Hamiltonian operator appears explicitly in the Schrodinger equation and generates time-evolution. If the Hamiltonian depends quadratically on the conjugate momenta (the case for theories of practical interest), then the Lagrangian appears explicitly in the Feynman path integral formulation of QM.

No, it's the other way around. Noether's theorem (or rather a quantum variant of it - these things do receive explicit attention, you know) applies because the formulation of quantum mechanics is explicitly Hamiltonian based.

The energy of an individual particle is (in which case, try doing quantum theory without frequency and wave-number). The Hamiltonian operator isn't.

Since when? Why do we need two different concepts of the energy difference between states when one does the job just fine?

 

The problem I mentioned (the insufficient definition of the velocity in the Lorentz force) has as such nothing to do with the Lorentz transformation. It exists already for the formulation of pre-relativistic electrodynamics, in fact even if you consider the Lorentz force law as purely empirical.

I always try to make the subject matter look mundane, simply because maths and physics is at the root mundane. Fancy mathematics, that you seem to prefer, only tends to distract from the real issues. It may impress the easily impressed, but lastly it misses the point.

Most of the things that we have discussed in this thread are not even remotely addressed in any textbook I am aware of. So I don't know why you keep on suggesting them as resources here. As I said, I am trying to go beyond the textbook knowledge and get to the root of certain issues. If it is too inconvenient for you to put your own 'thinking cap' on in this respect, then I can't help it, but don't blame me for not reciting textbook resources in this context.

Yes, I would say they violate causality, because the differential df(t)/dt can not be consistently defined both as the limit of (f(t+dt)-f(t))/dt and (f(t)-f(t-dt))/dt . Since both definitions must be essentially equivalent, but the first option requires knowledge of the function at the later time t+dt to calculate the derivative df(t)/dt, this violates then causality.

I wasn't specifically referring to the logic of your statement but the negated form. Just imagine Descartes would have said "I don't think I am not" instead of "I think, therefore I am". I doubt that the alternative version would have had much impact in philosophy. There is just a subtle difference between thinking something and not thinking the opposite.

When was this? I haven't seen many signs of a critical attitude towards Relativity in this thread from your side.

Thomas

 

That's simply incorrect (and you should know it by now). On its own merit, the paper merely arrives at the conclusion that both the Galilei- and Lorentz transformations are acceptable solutions to the initial assumptions they make.

As I said already, it would be a circular definition if the length of the measuring stick depends on the same effect it is supposed to measure. One must a priori assume some length to be invariant (in mathematical terms your equation must have an independent variable in order not to represent a circular definition).

You asked "Why would he want to formulate a theory that contradicts itself?" that is why somebody would publish an inconsistent theory with intent (maybe as a practical joke?). I was just picking up on this and remarked that somebody without such an intent might just as well publish nothing at all (rather than a consistent theory).

That equation would represent the path of the light signal in the objects reference frame (where its velocity is c) projected (i.e. Galilei transformed )into the rest frame of the origin. This however is physically irrelevant and indeed incorrect because of the invariance of c. Any row of detectors in the origin's rest frame must also register a velocity c. This is better to see from the viewpoint of the detector: because of the invariance of the speed of light, it can not possibly matter whether the light source is moving or not when the signal is emitted, it will always reach the detector after a time t=x/c where x is the distance between the light source and the detector at the moment the signal is emitted.
So in this sense, this is not a made-up rule but a direct consequence of the constraint that the speed of light should be invariant in different (moving} reference frames.

First of all, the velocity addition formula does not contain the full Lorentz transformation. And as pointed out earlier by me, the latter does not preserve symmetry in all reference frames when reflecting the x-coordinate about the origin (which contradicts the postulate of the invariance of c).

Secondly, merging the transformation formulae for two completely separate things into one formula doesn't mean the latter is consistent with reality. As an example, assume you have a law that on a motorway normal cars are allowed to go with any speed they like, but trucks are forced to go at just one fixed speed. Now why would you want to have a transformation formula that combines the two laws into one? In this case it would lead to the absurd (and indeed incorrect) conclusion that the speed of the cars is affected by the circumstance that trucks are only allowed to go at one particular speed.


Thomas

 

Yes, but as I mentioned already, for this general equation (which does not specify the derivative) it does not make a difference whether you substitute just for one variable or for both and average the results. It only does if you apply the derivatives to specific functions like in the example you gave.

The original constraint 1/2*tau2=tau1 (Einstein's 'master equation' and Eq.(7) on my page http://www.physicsmyths.org.uk/lorentz3.htm ) is satisfied if you leave alpha undefined. In order to define alpha, you have to differentiate the constraint (which is the crucial point about Einstein's derivation). This explicitly brings in the space-time coordinates of the reflection and detection points of the light signal (Eqs.(9) on my page) through the inner derivatives. And it is only at this stage that any solution with v#0 has to be ruled out.

Thomas

 

Because it is an unquestionable fact. I don't why you are still opposing this. Fluid mechanics is still mechanics, and the essential features are identical. If an object in a fluid is lifted up against gravity, then only because an equal volume of fluid is falling down under gravity (or vice versa). This is essentially how a set of scales works.

No, as I said, it is wrong because the interpretation of the stationary solutions as a tunneling effect is just a consequence of having matched the unnormalized solutions in the different regions such that a continuous function results. So the assumption of tunneling has simply been put into this by default.

As I said already, Lagrangian, Hamiltonian and Newtonian mechanics are exactly identical. They are merely different forms to describe the same thing (there are textbooks that deal with this point in some depth, and I also gave already the web reference http://www.mathpages.com/HOME/kmath523/kmath523.htm ) .
The question is then on what basis do you justify the use of Lagrangians in areas of physics that are quite evidently not describable in terms of Newton's laws?

Thomas

 

The problem is that you refuse to look at the Lorentz force in context. You try to interpret the v in the Lorentz force as invariant in order to be able to interpret the electric and magnetic fields as invariant. I've already shown that this leads to a contradiction with the induction equations, so the idea of interpreting Maxwell's equations with invariant quantities dies right there. If you want a theory with an invariant v and invariant fields, you'll have to propose it yourself. If you can't, you have no business calling your interpretation "obvious".

This is the same silly response that mathematically inept individuals always seem to resort to - "I don't understand it, therefore it's 'abstract' and meaningless". I know plenty of people who'd find the level of math on your website (algebra and calculus) "fancy". Would you like to be dismissed on those grounds?

You've never heard of Jackson's Classical Electrodynamics or The Feynman Lectures on Physics?

This is why both James R and Alphanumeric first reacted to your "Physics Myths" website by questioning your qualifications. You just don't seem to know anything about many of the topics you specifically decided to put a webpage up about. You grossly overestimate your levels of knowledge and understanding of these subjects, and you show no interest in learning more about them before judging them.

Do you also think this equation:

\(\bar{F} \,=\, \frac{\text{d}\bar{p}}{\text{d}t}\)​

violates causality? My impression here is that reality doesn't seem to care much for the concept of causality you're describing.

What are you going on about? I expressed a perfectly clear opinion and you selectively quoted "I don't think", deliberately distorting what I said by omitting the context. Just don't do that.

About three or four years ago, when I learned, and (I emphasize) understood the Minkowski four-vector formulation of relativity and it's application to Maxwell's equations. It's a way of expressing equations in a way that's Lorentz-invariant by construction. If you choose not to make the effort to learn this for yourself (it's not complicated, but it's a bit longer than what I'd like to have to explain on an internet forum) then you are simply not equipped to understand my perspectives on relativity and electrodynamics.

I can assure you, you will have the same problem with anyone with a reasonably strong background in physics who happens to stumble upon your website.

 

And since Galilean invariance has been experimentally ruled out, what does that leave you with?

This makes no sense. If the length of a measuring stick "depends on the effect it is supposed to measure", then that's precisely what allows you to measure the effect: by comparing a measuring stick in motion with one at rest.

If you really want to take that attitude then it works against you. The metre is currently defined as the distance light travels in 1/299,792,458 of a second. Personally, and unlike you, I'd much rather discuss why that definition makes sense than argue that I'm right because I hereby assume I am. But if you want an a priori assumption, you've got it. We can just as well take the Lorentz transformation as a standard as any other coordinate transformation, and you are wrong by the standard accepted by the scientific community.

You stated that Einstein should attribute physical significance to the Galilean transformation. There's no reason he should do this. The whole point of relativity is that he gives the Lorentz transformation that significance instead.

Then I can't make any sense of your idea and, as I said it sounds more like a cheap cop-out ("c is invariant because I'll complain if anyone describes or measures anything that contradicts that") more than anything else.

According to the equation I gave you, it does. If the detectors are at rest then v = 0 and you can use them to measure that the speed of light is c. If not, then v = 0 in the detectors' rest frame so they measure the speed of light to be c in their rest frame. In the origin's rest frame, the detectors are moving. The light interacts with them along the line x = (v + c) t, but we're unlikely to try to measure the speed of light with moving detectors (and you've made statements to the effect that this is "not allowed") so we never measure anything different than c.

You're still assuming x (and therefore also t), individually, have to be invariant. As I've said, we expect them to be Doppler-shifted.

No it doesn't. The Lorentz transformation is inherently asymmetrical - it's a boost in one direction and not in the other.

No, but it's still a counter-example to your claim that we *have* to apply the identity transformation to light and the Galilean transformation to everything else. You really don't like it when people think of other options, do you?

 

No, it's satisfied for any specific value of \(\alpha\). Try \(\alpha = 1\) or \(\alpha = \gamma(v)\) for example. All of these solutions satisfy the synchronicity condition Einstein described at the beginning of his paper.

You've tried that argument already, remember? Differentiating a constraint will never produce a new constraint, any more than adding 2 to both sides of the equation does. Einstein determines \(\alpha\) later in his paper by imposing reciprocity on the Lorentz transformation. You can take any particular solution to the synchronicity constraint (ie. some particular value of \(\alpha\)) and see what happens when you differentiate it. If you evaluate the left and right hand sides of the synchronicity constraint, you have the equality:

\(\frac{\alpha x^{\prime}}{c} \,=\, \frac{\alpha x^{\prime}}{c}\)​

The only correct result you can obtain by differentiating both sides with respect to x' is:

\(\frac{\alpha}{c} \,=\, \frac{\alpha}{c}\)​

This is still an equality, and I don't need to set v = 0 to obtain it, so the general contradiction you claim you've found simply doesn't manifest itself when we work with this concrete, specific solution. Also as you can see and as I just said, the differentiation still doesn't determine \(\alpha\), which is why Einstein can't (and doesn't) try to determine \(\alpha\) that way.

 

And that's the basis it's derived from. The Navier-Stokes equation is Newton's second law for fluids.

No, there's a massive difference. A set of scales has one degree of freedom. A fluid has an infinity of degrees of freedom: it's described by a velocity field \(\bar{u}(\bar{x};\, t)\) and it can take on any velocity at any point. If you treat the fluid as incompressible then you can impose \(\bar{\nabla} \cdot \bar{u} = 0\) (volume conservation), and that's about it.

No, the submerged object starts to rise because the water exerts a net upward pressure on it. As it rises it will have to push fluid out of the way (and suffer the reaction force, slowing it down) while fluid flows in behind it as it rises. The point is there is no reason to assume the volume of fluid set in motion is the same as the volume of the submerged object, or that it's all moving with the same velocity everywhere.

What, solving a differential equation correctly is unjustified? The solutions in the different regions, alone, aren't global solutions to the Schrödinger equation.

There's really no reason to be mystified by quantum tunneling, since it has a classical analogue in, for example, the evanescent waves in optical fibres. Loosely speaking, if you've got a theory in which particles are allowed to exhibit "wave-like behaviour", you shouldn't be too surprised when the theory predicts wave-like behaviour.

I know what you said. And since we have Lagrange formulations of electrodynamics, relativistic mechanics, and general relativity, what you said is misinformed.

No, the derivations in textbooks and on your web reference (and in my class notes - I already know this stuff, you know) show that the Lagrange formulation of classical mechanics, with L = T - V, is equivalent to Newton's laws. We can also show that the Lagrange formulation of electrodynamics is equivalent to Maxwell's equations, that the Lagrange formulation of general relativity is equivalent to the Einstein field equation, and so on.

I told you: you can describe a lot more than just Newton's laws with a Lagrangian. I've cited specific, well-known examples that prove this.