Relativity fails with Magnetic Force

I am not aware that Maxwell, Lorentz etc. would have have explicitly defined the velocity v in the Lorentz force law as frame dependent. This interpretation merely seems to have been adopted retrospectively in view of the conceptually sloppy formulation of the law. It was simply observed in lab experiments that a charge q moving with velocity v in a magnetic field B was experiencing a force F=q*vxB, but v is here strictly speaking the velocity in the lab frame (i.e. relative to the magnet), but not frame independent.

Obviously, you can kinematically describe the velocity of a particle in any reference frame you want, but that doesn't mean the velocity in the Lorentz force has to be interpreted in the same way.

On the contrary, there is no stringent case for the current explanations if there is no positive evidence for the existence of a spin of a free electron. Postulating that something exists but that we can't see it for this or that reason is hardly a conclusive argument. It might just as well be that the 'something' does not exist in the first place.

1) why would you want to it to be locally defined (especially when (as you yourself seem to admit) the magnetic field arises from non-local causes?
2) even a frame-dependent velocity would be a non-local reference, as it refers to the origin of the corresponding coordinate system.

What you proved is that the Lorentz transformation, correctly applied, does preserve charge invariance, and that can only mean that there is no wire charging (the wire being finite or infinite). When these derivations discussed claim to show a charging effect, then only because the Lorentz transformation is incorrectly applied to the problem.

So where is the charging here? In both reference frames the total charge is zero. As mentioned before, the configuration with the end-points oppositely charged would merely create a quadrupole field, not a monopole field as required.

As I said earlier, I am not referring to the force associated with 'turning the corners' in the current loop (which doesn't involve any work as it merely changes the direction of the velocity vector), but with that due relativistic slow-down/speed-up when entering from the vertical into the horizontal sections and vice versa in the boosted frame. This clearly violates energy and momentum conservation.

No, I am saying that your suggestion that the upper/lower horizontal sections of the loop have higher/lower charge densities due to a lower/higher electron velocity in the boosted frame is an unphysical setup.

So you are saying yourself that the differential length contraction for the ions and electrons is nonsensical?

Thomas

 

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In the Levy-Leblond derivation (Ref.), the existence of a general limiting velocity would have to be postulated to distinguish between the two acceptable solutions corresponding to the Galilei and Lorentz transformation (only the latter contains a free parameter c with the dimension of a velocity).

No, according to Einstein's own approach, the constraint is x=ct <=> x'=ct' and x=-ct <=> x'=-ct' . You can't have different signs in the primed and unprimed system. Otherwise you would have the non-sensical situation that the light signal would travel into opposite directions in the primed and unprimed frame (and with the quadratic form of the constraint, they could even travel into any arbitrary directions).

You probably misunderstood the proposed setup. There is only one line of detectors here. Assume, as you suggest, that these have their x-coordinate painted on them (both in the positive and negative direction). If you then send a light signal from the origin propagating in the positive and negative direction, you will find that the the detector with the label +x will show the same detection sign as that with the label -x, so And because of the invariance of the speed of light this must hold independently of whether or not the light source is in motion (relatively to the detectors) when the signal is emitted. If you designate the case of the moving light source with primed coordinates, it is thus evident that x=ct <=> x'=ct and x=-ct <=> x'=-ct (I have set t=t' here because the detector clocks can't possible be affected by the circumstance that the light source has been moving when the signal was emitted). And this obviously is the identity transformation.

The error in your above conclusion for the 'two-frames' example "This implies that each detector in the "rest" frame must pass the moving detector with the same coordinate when both detect the light pulse" is to assume that the locations in the two frames must be connected by a velocity dependent transformation, because this would exactly contradict the invariance of c. It only holds if the velocity is not invariant i.e. for material bodies

It is a matter of logical and mathematical consistency which should be self-evident: if the function tau is supposed to transform the coordinates from the unprimed to the primed system, then obviously the arguments of the function must be coordinates in the unprimed system. Inserting a coordinate of the primed system would be non-sensical and yield incorrect results.

Where is the difference? This is effectively identical to my Eqs.(7)-(9) on my page regarding Einstein's 1905 Paper (if you set the starting values t=0 and tau(vt,t)=0). And as shown on my page, the Lorentz boost satisfies that constraint only for v=0.

The point I was making is that (if you restrict t to positive values), the function f = ct + x is only defined on the light cone (f=0) for negative x, and g = ct - x =0 only for positive x. This constraint means that in general f and g are defined on different (mutually exclusive) sets of coordinates. Hence you can't add or subtract them as Einstein did.

I don't see a fundamental difference here. Once in place, the Ptolemaic system also could make predictions (with regard to the positions of the sun, moon and planets), and these predictions were quite accurate (more accurate even than the predictions of the early versions of the heliocentric system).
In either case you have mathematical relationships that predict certain things, but that does not really touch on the ultimate physical interpretation of these mathematical relationships. And as I have indicated above, the physical interpretation of phenomena e.g. like clocks going slower and charged particles being more difficult to accelerate by electromagnetic field at high velocities may (and in my opinion should) have a a different physical explanation than that implied by Relativity.

And I am not impressed by alleged experimental proofs of a theory which is mathematically and conceptually inconsistent. Whilst any theory can easily be checked and evaluated by everybody, it is difficult (if not impossible) to verify in retrospect that experimental data have been obtained and evaluated correctly. Even if the data are genuine and correct, the question then only can be why they seem to confirm an inconsistent theory. I have given some speculative suggestions for that, but that goes really beyond the fundamental theoretical points I have been making, and can probably only be answered by suitable additional experiments (or a re-analysis of old experiments).

Well, this could be experimentally tested. You just would have to measure the lifetime of a muon with a fixed velocity in dependence of the magnetic field strength. If there is dependence, then Relativity would be wrong.

How would they know the actual power of their beams? Normally, you measure the energy and momentum of a charged particle by the deflection in an electromagnetic field, so if the interaction force decreases with increasing velocity, you will correspondingly overestimate the power as the particles are deflected less.

The GPS argument is actually a myth. As shown on my page Global Positioning System (GPS) and Relativity, there shouldn't a build-up of inaccuracies in the standard operational mode (which consists of evaluating time differences for different satellite clocks rather than time differences to a ground clock). The error due to Relativity for everyday GPS should be only about 1 cm.

If I had the time and motivation to do this. But I would gladly leave this to other people.

Thomas

 

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Generally "velocity" simply means a particle's velocity and velocities are always frame-dependent. Anything more is explicitly stated. Take your favourite example for instance. The first sentence on Wikipedia's page on friction states:

The word "relative" appears a total of fifteen times on that page. It is only you who is assuming electrodynamics is poorly formulated. This is completely unjustified: you cannot interpret the velocity in the Lorentz force any differently without interpreting the whole of electrodynamics as a fundamentally different theory than the one that Maxwell etc. proposed. I also don't see how you can seriously accuse electrodynamics of "poor formulation" and then describe an alternative that truly is poorly formulated in terms of unspecified parameters and vague statements about the velocity of the "system" producing the field.

This isn't 1900 anymore. Do you really want to get into a discussion of the validity of the \(- e \bar{\Psi} \cancel{A} \Psi\) interaction term in one of the most precisely verified theories in the history of physics? If there was a problem with the electron-photon interaction term, accelerator experiments would have revealed it a long time ago.

So how do you explain the Zeeman effect or the fact that only two electrons can ever occupy a single orbital? Nobody's ever seen an electron or hydrogen atom, and nobody ever will: atoms are hundreds of times smaller than the wavelengths of visible light. You could suggest that we don't really know that atoms exist - only that current atomic models correctly predict the periodic table, Brownian motion, and the blurs we see on images obtained by electron microscopy. The point is, stating that current models *only* predict the right numbers and might be wrong is easy and is always a possibility in physics anyway. You're not saying anything particularly remarkable in this respect. Actually providing an alternative model that's simpler or works better than the mainstream one isn't so easy.

Where did I say the magnetic field arises from non-local causes? Magnetic fields are produced locally by moving charges and, as part of the electromagnetic field, obey a propagation (wave) equation.

No it doesn't. If you move the origin then you change a particle's position by a constant. The derivative of a constant is zero, so translations of the origin do not affect velocities.

I did show wire charging. In the rest frame the body of the wire has a charge density of zero. In the moving frame it has a charge density of \(\lambda \,=\, -\gamma \frac{jv}{c^{2}}\). This will obviously affect the electric field near the body of the wire, which is the point of the derivations you linked to.

Why is a monopole field "required"? These multipole expansions of the electromagnetic field are only defined for static fields produced by static sources. The only way you can make a sensible comparison of the asymtotic behaviour of the field between two frames is if the multipole expansion is defined in both frames. This is only the case if a source is static in two frames, which is in general only true for the very situations you're trying to avoid - namely infinite homogenous systems such as infinitely long wires, infinite charged plates, etc.

Also, even if you ignored the fact that the wires I described aren't static charge distributions, the asymptotic behaviour of the field would be dipolar in both frames since the endpoints are oppositely charged.

The force accociated with "turning the corners" is the only force that acts on the electrons. In relativistic mechanics the velocity difference between the electrons and the wire has no required physical significance. If the force guiding the electrons around the corners originates from Lorentz invariant physics, then it will by definition predict that the electrons have the velocities given by the Lorentz velocity addition formula in the moving frame compared with the predicted velocities in the rest frame. It's clear you're only finding a problem because you're implicitly assuming that the physics producing the force on the electrons is Galilean invariant rather than Lorentz invariant.

Note by the way that the "work done" on the electrons turning the corners is frame-dependent, even in Newtonian mechanics.

No, I'm saying that applying more than one coordinate transformation between two frames is nonsensical. In doing so you are effectively claiming that two events could coincide in one frame (ie. have the same space-time coordinates in one frame) but not necessarily in another frame. I would consider this a much stranger conception of space and time than anything suggested by special or general relativity, if not an outright mathematical error.

 

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What the Levy-Leblond derivation shows is that no conceivable universe could exist that possesses an invariance other than Galilean or Lorentz invariance that is consistent with the postulates he's made. The fact that this is as far as one can go based on "reasonable" postulates doesn't mean that this is as far as physics can go. As far as our universe is concerned, the possibility of Galilean invariance has been ruled out experimentally and the invariant speed c has been measured.

It seems I did, but then you're describing a setup that completely ignores the problem. If we only had to explain the independence of the speed of light signals on the speed of the source then there wouldn't be a problem - it was always the general assumption of the physics community anyway. Relativity is an explanation for the observed independence of the speed of light on the motion of the detectors as well as the motion of the source. This isn't so trivial to explain. You seem to be completely avoiding this issue.

It's not an assumption. The use of detectors as I described them defines a coordinate transformation between frames. Comparing the x coordinates painted on each detector and the time shown on each one's clock as they pass one another defines the transformation. And we know experimentally that this transformation must leave the speed of light invariant. That's the problem and that's what all the fuss was about in the late 19[sup]th[/sup] century.

No, assuming coordinate systems are related by Galilean transformations contradicts the invariance of c.

As far as mathematical consistency is concerned, it is completely irrelevent what x' is. The only consistency requirements are that the relation between (x, t) and (x', t) should be invertible and differentiable. I'd have thought this would be self-evident. All Einstein is doing is highschool-level variable substitution.

It seems I read your "x1=x'+v.t1" simply as "x1=v.t1". But as I said your conclusion is falsified by the very existence of this function:

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

which anyone with a pen, paper, and a little spare time can check satisfies (using the notations on your site) this constraint:

\( \frac{1}{2} \tau(x_{2},\, t_{2}) \,=\, \tau(x_{1},\, t_{1}) \)​

For the Lorentz transformation, both sides of the constraint equation simply evaluate to \(\frac{\gamma x^{\prime}}{c}\).

Since the original synchronicity equation admits solutions that your equation (11) doesn't, you've derived a differential equation that is not equivalent to the original constraint, and the only question is where you've made your error. Looking more closely it seems that the problem stems from some misconceptions regarding partial derivatives. As I explained earlier Einstein probably did Taylor expansions, but if you insist on using the chain rule then regarding this:

In an expression like \(\frac{\partial \tau}{\partial t}\), the t is just a dummy variable indicating that you're differentiating the function \(\tau\) with respect to its fourth argument. You're right that there's a subtlety in comparing the derivatives on the left and right hand side, but it's in where the derivatives are evaluated. The chain rule, applied to the constraint equation and with the evaluation points explicitly written, yields:

\( \frac{1}{2} \biggl( \frac{\text{d}x_{2}}{\text{d}x^{\prime}} (x^{\prime}) \, \frac{\partial \tau}{\partial x} (x_{2},\, t_{2}) \,+\, \frac{\text{d}t_{2}}{\text{d}x^{\prime}} (x^{\prime}) \, \frac{\partial \tau}{\partial t} (x_{2},\, t_{2}) \biggr) \,=\, \frac{\text{d}x_{1}}{\text{d}x^{\prime}} (x^{\prime}) \, \frac{\partial \tau}{\partial x} (x_{1},\, t_{1}) \,+\, \frac{\text{d}t_{1}}{\text{d}x^{\prime}} (x^{\prime}) \, \frac{\partial \tau}{\partial t} (x_{1},\, t_{1}) \)​

where the parameters x[sub]2[/sub], t[sub]2[/sub], x[sub]1[/sub], and t[sub]1[/sub] are treated as single-valued functions of x'. The partial derivatives on the left hand side are evaluated at (x[sub]2[/sub], t[sub]2[/sub]) while they're evaluated at (x[sub]1[/sub], t[sub]1[/sub]) on the right hand side. There are two indepedent justifications for equating the partial derivatives on the LHS and RHS. The first is that the expression holds for all values of x' including in the limit Einstein considers:

The second is that Einstein states at various points that he's looking for a linear function, meaning that the partial derivatives of \(\tau\) are actually just constants independent of where they're evaluated. You could even call the partial derivatives A and B, set \(\tau \,=\, Ax + Bt\), and see directly what the constraints between A and B are without ever differentiating anything.

Since x and t as they appear in the differential version of the constraint are independent parameters of the function \(\tau\), the last step of your derivation doesn't make any sense. Also while it isn't specifically the problem (the last step of your derivation is invalid anyway), I should also point out that the notation you employ in the above quote and here:

is dangerous and suggests a misconception. Unlike total derivatives, partial derivatives don't behave like normal fractions. Writing \(\partial x_{1} \,=\, v \partial t_{1}\) is mathematically meaningless, and it is an error to try to "substitute" such an expression into a partial derivative. This is most obviously illustrated by the fact that partial derivatives don't reciprocate like fractions do - ie. if y[sub]i[/sub] are functions of parameters x[sub]j[/sub] then in most cases,

\(\biggl( \frac{\partial y_{i}}{\partial x_{j}} \biggr)^{-1} \,\neq\, \frac{\partial x_{j}{\partial y_{i}}\)​

In general inverse partial derivatives are obtained by inverting the entire Jacobian matrix.

Again, as I said, this issue is settled by the fact that the Lorentz transformation actually satisfies the constraint you say nothing can satisfy.

There's no such restriction. As I just stated, light cone coordinates are defined for all t and all x. Einstein's derivation is stating that if the general transformation of coordinates between frames takes the form

\( \begin{align} f^{\prime}(f,\, g) \,&=\, A f \\ g^{\prime}(f,\, g) \,&=\, B g \end{align} \)​

for constants A and B, then (for example) the particular trajectory f[sub]0[/sub] = 0 will map onto f'[sub]0[/sub] = 0.

The difference is the theory's extrapolative power. Generally you know you're on the right track when your theory successfully predicts something beyond the data set it was originally developed to explain. The Ptolemaic and early heliocentric models were carefully fitted to model the quasiperiodic movement of the sun, moon, Mercury, Venus, Mars, Jupiter, and Saturn, and could only make predictions about the sun, moon, Mercury, Venus, Mars, Jupiter, and Saturn. Even in their limited domain of applicability they failed whenever the precision of astronomical measurements improved, and epicycles had to be added.

Contrast that with early "problems" with the predictions of Newtonian gravitation, which led to the discovery of Neptune exactly where the theory said it should be. Electromagnetism provided a natural description for the propagation of light despite only being developed to explain the results of experiments involving currents and magnets. Relativistic quantum mechanics predicted the very existence of antimatter. Electroweak theory correctly predicted the existence of the W and Z bosons, the interactions they mediate, and the order of magnitude of their masses fifteen years before they were identified experimentally. These are the kinds of predictions successful theories make. Here, only the first example - Newtonian gravitation - is not Lorentz invariant.

You haven't shown any inconsistencies in relativity. All I see on your website is you misinterpreting relativity in every possible way and concluding that the problems you find are relativity's problems. For some reason you are choosing to interpret x = ct <=> x' = ct' as an identity constraint. Why? It is so obviously inconsistent with the rest of Einstein's derivations as well as the actual problem he's trying to solve (which you've also apparently misunderstood). And since applying multiple different Lorentz transformations between different frames is not relativity, what is this still doing on your site?

Most accelerator experiments include calorimeters which make more-or-less direct independent measurements of the energies of particles produced during collisions by absorption of their energy. Engineers designing detectors will also be familiar with the rate at which damage builds up over time and how this depends on the energies involved.

For high energy experiments, beam disposal is also a significant technical challenge. At the LHC for example, the proton beams will have to be dumped and replaced after each ten hour run. "Dumping" a beam means aiming it into a block of graphite composites, and the operators have to defocus the beam and move the aim around to prevent it from boring through the material too quickly (the LHC currently isn't operational, but beam tests were performed in August and September last year). There are also known examples of the kind of damage that the beams can cause, for example from the 2003 accident at the Tevatron. Some background info here. These beams have lethal destructive power.

I don't think you can plausibly claim accelerator beams are orders of magnitude less powerful than the engineers think they are.

Er, not that I'm saying it's accurate, but in the setup you describe on your site you should find that the positional error is zero. If both clocks run at the same slowed rate, then as you say both transmitters will emit their signals simultaneously at some (irrelevant) delayed time. For some reason, you applied a correction factor to the signal propagation times - which have nothing to do with the clock rates.

Why are you focusing exclusively on positional errors though? The GPS clocks are atomic clocks which are accurate to about a nanosecond a day. A 38 microsecond discrepancy/day is enormous and quite easily observable compared to this. One of your references (the other - "Ref. 1" - seems to have expired by the way) states:

 

Errata:

Apparently a syntax error slipped in when I edited the indices on this expression. It should be:

\( \biggl( \frac{\partial y_{i}}{\partial x_{j}} \biggr)^{-1} \,\neq\, \frac{\partial x_{j}}{\partial y_{i}} \)​

 

You mean Enistein's theory of relativity? No! No! It can't be!!! NOOOOO!! Please somebody tell me this is incorrect. Please!

 

Well, I mean that.

The link has changed: http://www.anewlightinphysics.com/sections/Section1-1_Considerations_against_Relativity.htm

 

The two that stand out the most in my memory from general browsing are:

on this page about energy/momentum conservation. (the "assumptions" in question being that Lagrangians and Hamiltonians are only defined in terms of classical kinetic energies and conservative interaction potentials).

and:

as a criticism of quantum mechanics on this talk page.

 

Sorry, but I don't understand the point you are trying to make: I have in fact earlier mentioned the friction force as an example where the relative velocity is defined physically (e.g. the friction force on an object moving within a medium). In this case the velocity is obviously frame indepedent. What I am saying is that the velocity in the Lorentz force has to be interpreted similarly in a frame-independent way in order to have any physical significance.

Just assume for the sake of the argument it is the year 1900, and try to give a justification for the interpretation of the velocity v in the Lorentz force law as an frame-dependent (i.e. not physically defined) quantity.

As mentioned already, these are effects associated with electrons in atoms (or more generally speaking in electric fields), not with individual free electrons.

In post #79 you said "I'm not arguing against the fact that events in one place can affect events somewhere else (at a later time)."

As I said already, nothing is produced 'locally', as that would strictly speaking imply a zero volume, but a zero volume can't produce a magnetic field.

Making the origin undefined doesn't mean that the reference to it becomes local.

This is incorrect. The diverse derivations of the charging effect assume an infinite wire, that is the charge density is changed everywhere along the line of real numbers R. There can therefore be no endpoints that are oppositely charged, i.e. charge invariance is violated.

We discussed that already, and rather than warming up my response to that again, I would like to ask you to take the treatment on my page Magnetic Fields and Lorentz Force and show in a similar mathematically complete and coherent way how the consideration of non-static effects would invalidate the conclusion given there.

As I said, there is no work done by turning corners. The kinetic energy before and after is the same. The energy would be different however if the electrons move slower/faster in the top/bottom horizontal section than in the vertical section (as you claimed it is the case in the boosted frame).

But the latter is effectively what you would be claiming if a section of the electron line charge would get length contracted differently to the ion line charge (an electron ion/pair coinciding in one frame would not coincide anymore in the other frame).

Thomas

 

It is only the invariance of the speed of light that has been measured, not the existence of a general limiting velocity for material objects. And only the latter would justify the solution corresponding to the Lorentz transformation in the Levy-Leblond derivation.

So then where is the problem? You are always free to choose your coordinate system such that the observer is at rest.

The point is that by 'comparing' the coordinates of events you are implicitly making the assumption that it is allowed to do this. There is however nothing that could a priori guarantee that the sequence of events in one reference frame can be obtained by taking the sequence of events in another reference frame and geometrically projecting it in the first one according to the relative positions of the two frames. Experience tells us that it is OK to do so for material objects, but then their velocities are obviously frame dependent. For light signals the latter would however contradict the invariance of c, so in this case it is not possible to project their coordinates from one system into the other, but you have to physically measure the coordinates in each reference frame explicitly (e.g. by using two separate rows of detectors.


But as I suggested, the problem is much easier to understand if you have just one row of detectors (i.e. one reference frame) and let instead the light source moving with different velocities (which is fully equivalent).

No, any velocity dependent transformation would contradict the invariance of c. So as indicated earlier, this leaves only the identity transformation.

According to Einstein's own words "To any system of values x, y, z, t, which completely defines the place and time of an event in the stationary system, there belongs a system of values xi, eta, zeta, tau, determining that event relatively to the system k, and our task is now to find the system of equations connecting these quantities".
So it is obvious that the arguments of xi, eta, zeta, tau must be (x,y,z,t) but not (x',y,z,t) (x' doesn't make any sense at all as an argument here, as it is the fixed length of the moving rod (or to be more precise the distance between the light source and the reflecting mirror)).

Yes,

(2) tau(x,t)=gamma*(t-vx/c^2)

satisfies the equation

(1) 1/2*tau(x2,t2)=tau(x1,t1)

but it does so for any value of the constant gamma. In order to specify gamma you need a further equation (after all, there are two unknowns: tau and xi). This the reason why Einstein differentiates Eq.(1) with regard to x'. The point is that he differentiates it incorrectly. As shown on my web page http://www.physicsmyths.org.uk/lorentz3.htm , a correct differentiation would lead to gamma=1 (i.e. v=0).

On the contrary, Einstein's differential equation is not consistent anymore with the original constraint for the invariance of c (x=ct<=>x'=ct' and x=-ct<=>x'=-ct'). I have given earlier already a mathematical example to illustrate this point, and I repeat it here:



consider the equation

(I) y(x)=ax + b

with the constraints

(C1) y(1) = 1
(C2) y(-1) = -1

The task is to determine the coefficients 'a' and 'b' by applying the constraints to (I). Now since (C1) results in 1=a+b and (C2) in 1=a-b, it is obvious that this requires b=0. But since b=0 is not what Einstein likes, he decides to modify the constraints such that (I) is valid for all b, i.e. he changes (C1) and (C2) to

(C1') y(1) = a +b
(C2') y(-1) = -a +b .

"Fine" Einstein says, "now I have a system of equations that is consistent for all 'b' (and 'a' at that)", but unfortunately it has nothing to do with the problem anymore. The task was not to find a set of constraints that are consistent with (1) irrespective of the value of the coefficients, but to apply the constraints (C1) and (C2) to (I) and thus to find the coefficients.

The function tau is supposed to be linear in all variables, so the derivative should be the same everywhere.
But the point is can't be, because we have

(1) t1= x'/(c-v) = x'*a1

(2) t2= x'/(c-v) + x'/(c+v) = x'*a2

so

(3) dt1/a1 = dt2/a2

but

dtau2 = 2*dtau1

So there can not possibly exist a single linear function tau(x,t) .

We are not interested in general functions. The functions here are (see my page http://www.physicsmyths.org.uk/lorentz3.htm )


x1=x'+v*t1 , t1=x'*a1 i.e. x1=t1*(1/a1+v)

and

x2=v*t2

and thus

dx1 = dt1*(1/a1+v)
dx2 = dt2*v

So we don't even need to bother about partial or total derivatives here, because we have just got constant factors to the variables.

As I said, that's not possible. Taking your two functions f and g, the light cone is defined by f = ct + x =0 and g = ct - x =0 . If both equations would hold simultaneously, this would mean that at the same time x=ct and x=-ct, which is an algebraic impossibility unless t=0.

No theory can really have any extrapolative 'power'. A theory is based on a certain observational data set, and as such can only be a representation of these data. If other data are consistent with this theory, then this is strictly speaking a coincidence. They could as well not be consistent with the theory for any number of reasons. As you may well know, some people suggest for instance that Newton's law of gravity has to be modified for small gravitational accelerations (MOND theory) which is supposed to explain a number of observational data like galactic rotation curves. Not that I personally support this theory, but it just goes to show that in principle there is no extrapolative power of a theory. Any set of new data could require a modification of the theory or even a completely new theory.

That's your opinion based on your own point of view. My point of view (obtained after having thought about this issue for a long time and having already discussed it with numerous people) is different, and I have tried to justify this in detail here. If you consistently fail to see my point, then there is lastly little I can do about it.

As I said already in the course of this thread, this is merely supposed to invalidate the 'wire charging due to differential length contraction' argument used in various resources. It points out an inconsistency in that argument, and is not supposed to represent what a strict application of the Lorentz transfornation would yield (which in fact would not lead to any wire charging, as mentioned by me earlier).

The energy absorption per unit volume depends on the product of the energy of the incident particle and the absorption cross section. If you scale both quantities inversely, there shouldn't be any difference for the overall energy absorption.

No, have a look again at my GPS page. From Eq.(4) it is evident that if you change the clock rate of both satellite clocks by the same amount, there is a corresponding error in the absolute position (an inaccuracy for the value of the speed of light would have the same effect).

I do not question the 38 microseconds/day discrepancy, but a) it is insignificant for the everyday GPS positional error (as shown above), and b) it may have nothing to do with Relativity at all, but could be for instance simply due to the Lorentz force on the satellite clocks as they are moving in the earth's magnetic field.

Thomas

 

There shouldn't be too many errors. The whole purpse of the site is to point out (and if possible correct) what I consider errors in various physical theories I came across in the course of my own work or in other contexts. There is no other motivation for the site. I am not aiming at fame or the Nobel prize like most scientists, but I merely want to initiate a critical debate about issues that are normally taken for granted.

No, strictly speaking I don't actually have a Masters and PhD. I have Physics Diploma and a Doctorate in Astronomy (obtained in Germany), but these qualifications are accepted as being equivalent to a Masters and PhD respectively.

But anyway, I mentioned my qualifications on my site only so that people do not assume I don't know what I am talking about. I have all the necessary background to allow myself a professional opinion, even though it may be a dissident one.

I hope this answers your question.

It is not worth starting another thread for this because there is nothing wrong with my claims. There is in fact nothing controversial about this. It is is a straight application of Newtonian physics that is just being neglected in standard textbook treatments of buoyancy.

You are welcome to point out anything you think to be in error.

Thomas

 

Precisely. Part of the point I was making is that when a relative velocity (ie. with reference to some system) is meant, it is stated explicitly. The rest of the point is simply that an invariant velocity in the Lorentz force makes no sense: one of the frequently touted properties of electrodynamics is it's absence of so-called "action at a distance", and directly coupling a charge's velocity to the velocity of a system a long way away would directly contradict this. The point is that there is no justification to your claim that electrodynamics was poorly or ambiguously formulated (a seperate issue to whether it is correct or not).

Why? Quantum mechanics and quantum field theory are part of my general knowledge about physics. Why should I entertain conclusions you are drawing based on an incomplete and often superficial understanding of modern physics?

I've already given several theoretical justifications. Mostly they boil down to the fact that they needlessly complicate the theory (you lose locality, your undefined "interactions" between charged particles if we also address your attempt at an invariant current, ...). So far the only hint of a justification that you've given for modifying electrodynamics is that you personally don't like its implications as is. Since I don't share your prejudices, I don't share your motivation.

What is this supposed to mean? The vast majority of velocities and quantities in physics are frame-dependent and perfectly measurable. Why are you picking on the Lorentz force?

So? Do you have an alternative to quantum mechanics or not? It is an easy and weak argument to claim there could be an alternative to physics you don't know anything about. But this isn't physics I know nothing about: I've learned quantum mechanics. In principle, I know how to apply it to calculate things like the Zeeman effect or spin precession, and I don't think it's "obvious" that there must be an alternative explanation to these effects.

So? If there were no such effects there would be no physics. Locality means you can express a theory in terms of local couplings and differential equations (which at most refer to the state of a system an "infinitesimal" distance away). The best definition I've seen was when learning quantum field theory. Locality was the requirement that a theory's Lagrangian could be expressed as an integral of a Lagrangian density:

\( L(t) \,=\, \int\text{d}^{3}x \mathcal{L}(\bar{x}, \, t) \)​

where the Lagrangian density \(\mathcal{L}(\bar{x}\, t)\) only refers to the various fields at the point \(\bar{x}\) at time \(t\). When you quantify such a theory, the interactions between particles become local, eg. an electron won't interact with a photon a kilometre away. Classical and quantum electrodynamics are local theories in this sense. Your direct coupling between a charge and the velocity of a system a finite distance away violates locality in this sense.

The fact that velocities are independent of where you place the origin means that where you choose to put the origin is physically irrelevant.

You seem to be resorting to outright denial here. I specifically gave you a derivation that does not assume an infinitely long wire, and shows that the wire will gain a charge density along the body between the two endpoints.

I told you: even if you ignore the fact that the wires are moving in most frames, so comparing the asymptotic behaviour of the fields between different frames isn't necessarily meaningful to begin with (this alone completely invalidates your approach), you transformed the wires incorrectly and your statements about their multipole moments are simply incorrect. Your webpage therefore proves nothing.

Rather than trying to justify a completely flawed investigation of relativistic electrodynamics, why don't you learn the Minkowski formulation of electrodynamics and try to explain what's wrong with that? It is standard undergraduate physics material and should be discussed in any good textbook on classical electrodynamics. You aren't really qualified to discuss relativistic electromagnetism in the first place if you're unfamiliar with this material.

There is no work done in turning the corners in the rest frame. Work, given by:

\(\text{d}W \,=\, \bar{F} \cdot \bar{v} \text{d}t\)​

is not invariant. Even in Newtonian mechanics, unless the force is zero, dW is nonzero in all but a class of reference frames in which the force happens to act perpendicular to the velocity. In Newtonian mechanics, if the electrons move through the wire with speed u relative to the wire, and the wire itself moves with velocity v, then while an electron is in the vertical segment its total speed (the norm of its total velocity) is \(\sqrt{u^{2} \,+\, v^{2}}\) and it's kinetic energy is:

\(T_{\text{vert}} \,=\, \frac{1}{2} m \bigl( u^{2} \,+\, v^{2} \bigr)\)​

After it turns the corner and is in the horizontal segment its total speed is \(u + v\) and its kinetic energy is:

\(T_{\text{horiz}} \,=\, \frac{1}{2} m \bigl( u \,+\, v \bigr)^{2}\)​

They differ by \(T_{\text{horiz}} - T_{\text{vert}} \,=\, m u v\). In the boosted frame, the electron's energy changes as it turns the corners. In relativistic mechanics the numbers are different but the qualitative result is the same: work done is frame-dependent.

Only if you pretend relativity of simultaneity isn't a part of relativity.

 

What do you mean by "limiting velocity"? The Levy-Leblond derivation only distinguishes between the presence or absence of an invariant speed. We've most certainly ruled out the latter possibility, and we've observed examples of the former.

Why shouldn't we be "allowed" to do this? A moving observer could mark out their coordinate system grid with a bunch of bouys each equipped with a clock. All I have to do is measure the trajectory of each bouy in my own coordinate system and build up a conversion table between the coordinates I measure in my frame and the position/time indicated on each buoy. That's the coordinate transformation between the frames.

You're forgetting that we don't just have a priori knowledge about the coordinates of light pulses. Operationally, if a light signal is located at some coordinates, it means that a detector (or someone's retina) placed there would detect the light pulse. Any measured coordinates of a light pulse are actually the coordinates of a detector (a material object) when it signalled a detection. Unless you want to claim that a light pulse can coincide with a signalling detector in one frame but not in another (precisely what I meant earlier by notions of space-time stranger than anything suggested by special relativity), you're forced to conclude that the coordinates of light signals will transform the same way as the coordinates of detectors.

No, this avoids the problem of finding two coordinate systems moving with respect to one another which share an invariant velocity c. In general you cannot pick a reference frame in which both are at rest unless you're addressing a completely trivial problem to begin with.

Here's a velocity-dependent transformation that satisfies the invariance of c:

\( \begin{align} t^{\prime} \,&=\, \gamma \bigl( t \,-\, \frac{v}{c^{2}}x \bigr) \\ x^{\prime} \,&=\, \gamma \bigl( x \,-\, vt \bigr) \end{align} \)​

Could you please stop pretending this transformation doesn't exist? It may not satisfy all the constraints you keep making up, but like it or not it leaves the speed of light invariant.

You invented this rule. Only the final result needs to be expressed in terms of x, y, z, and t. In the intermediate stages of the derivation, whatever physical interpretation you give to x' can have no bearing on the mathematics of variable substitution.

If you have an equality f(x) = g(x) and derive both sides you get another equality:

\(\frac{\text{d}f}{\text{d}x}(x) \,=\, \frac{\text{d}g}{\text{d}x}(x)\)​

so any solution to a constraint equation will automatically be a solution to the differential version of the constraint. Einstein is not attempting to produce a new constraint here - his motivation is simply that the differential version of the constraint is easier to work with.

You can see what happens when you substitute \(\tau\) directly into the constraint equation:

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

Differentiating both sides with respect to \(x^{\prime}\) just gets you:

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

It obviously works in this particular case (I'd have thought this should be trivially obvious), despite your claim that this can't work in general.

As I've previously stated, Einstein most likely never differentiated anything at all. He states "Hence, if x' be chosen infinitesimally small", which suggests he most likely performed a Taylor expansion. This is simpler than attempting to apply and justify the chain rule.

Einstein is not attempting to impose the invariance of c at this stage of the derivation. You've obviously misunderstood the paper here. As I've already repeatedly explained, what you call the "master equation" is just a mathematical statement of the synchronicity constraint described in §1 of his paper. It is not intended to fully constrain the problem. Einstein only introduces the invariance of c condition a little later in the paper.

If (I) is supposed to be an analogue to the identity constraint or a simultaneous x = +ct and x = -ct constraint then I've already told you where you've gone wrong.

The "light cone" (in two dimensions and centered on the origin) is defined by f = 0 or g = 0. It's the union of the lines f = 0 and g = 0, not the intersection.

There is no requirement for f = 0 and g = 0 simultaneously. You made this constraint up yourself. The so-called "light cone coordinates" aren't even restricted to the light cone: they're functions defined over all x and all t. Simply denying this won't make it false.

Yet time and again good theories have shown that they do have considerable extrapolative power. Even if they do eventually fail, if they predict many results beyond their original dataset they're at least regarded as a step in the right direction. That's why we scrap the Ptolemaic model of the solar system and the caloric fluid theory of heat, but we don't throw out Newtonian gravitation or classical mechanics. Even though we know they fail in general, Newton's theories have still proved their usefulness, including for purposes Newton may never have envisaged (such as getting astronauts safely to the moon and back).

Good theories are expected to be able to predict a few things beyond the exerimental results they were originally intended to explain. This is one of the main ways of distinguishing "good science" from ad-hoc modification and curve-fitting in physics.

Simply making up constraints for relativity to contradict isn't making a point. It is knocking down a strawman. It's a simple fact that, whether you like it or not, a) the Lorentz transformation leaves the speed of light invariant (I can't believe you're still trying to deny this) and b) we have successful Lorentz invariant theories. That's really all there should be to it.

At the level of individual ions and electrons, "differential length contraction" essentially is what's behind wire charging. What's incorrect is your naive application of the length contraction formula to an entire line of electrons in a wire segment. It's only valid for electrons that always stay in the same line - ie. if you consider a line of electrons entering a wire and leaving the other end.

So just jigger with the numbers and the problem goes away? The interactions of high energy particles with matter are extremely well studied. You are never going to be able to simply wave away a three order of magnitude discrepancy this way. The absoption cross section in particular isn't something you can play around with: the calorimeters are designed to absorb basically everything (except some muons and neutrinos). And since the calorimeters resolve the energies of individual particles (they'd be next to useless if they couldn't), there's no cross-sectional calculation involved. You could only risk underestimating the total collision energy this way if more particles escape undetected than anticipated. Here you're just dismissing a branch of experimental physics you're obviously completely unfamiliar with.

More importantly, do see what you've just done here? You originally only knew about measuring a charged particle's momentum by deflection in a magnetic field and suggested making ad-hoc modifications to the Lorentz force. I tell you about calorimeters and now suddenly they too are untrustworthy. Do you see the pattern here? Whenever a branch of physics produces results you don't like, you rush to conclude that yet another field of physics is wrong. Is that a good way to do physics in your opinion?

Then you agree that the GPS clocks run fast by the amount predicted by general relativity - which is what I'd consider the relevant point here.

And it's just a coincidence that a theory formulated in 1916 made correct predictions about GPS clocks put in orbit after 1970? Can your alternative theory match this quantitative prediction?

 

Note/Addendum:

Previously I stated that the exact place you made an error on your page was in your last step:

I suppose I haven't really explained why this is an error yet, and it's probably best done with the help of a simple example that isolates it. Consider a two-parameter function f = f(x, y) which will play the analogue of \(\tau\) here. In principle this function is arbitrary and almost any nontrivial function will do for the purpose of this example, but for simplicity I'll take:

\(f(x,\, y) \,=\, x \,+\, y\)​

so its partial derivatives are:

\( \frac{\partial f}{\partial x} \,=\, \frac{\partial f}{\partial y} \,=\, 1 \)​

Now I'll define the path:

\( \begin{align} \tilde{x}(t) \,&=\, t \\ \tilde{y}(t) \,&=\, 2t \end{align} \)​

f evaluated along this path is just:

\(f\bigl(\tilde{x}(t),\, \tilde{y}(t)\bigr) \,=\, 3t\)​

and the derivative with respect to t is simply 3. In general the chain rule yields:

\( \frac{\text{d}}{\text{d}t}f\bigl(\tilde{x}(t),\, \tilde{y}(t)\bigr) \,=\, \frac{\partial f}{\partial x} \,+\, 2 \frac{\partial f}{\partial y} \quad (\ast) \)​

which unsurprisingly evaluates to 3 for the partial derivatives of f = x + y.

For the path defined above we have \(\tilde{y}(t) \,=\, 2 \tilde{x}(t)\). Now, you called the partial derivatives the equivalent of \(\frac{\partial f}{\partial \tilde{x}}\) and \(\frac{\partial f}{\partial \tilde{y}}\). According to you, I'm allowed to say \(\partial \tilde{y} \,=\, 2 \partial \tilde{x}\) and "substitute" this into a partial derivative in (*). This could either yield:

\( \frac{\text{d}}{\text{d}t}f\bigl(\tilde{x}(t),\, \tilde{y}(t)\bigr) \,"="\,\frac{\partial f}{\partial \tilde{x}} \,+\, \frac{\partial f}{\partial \tilde{x}} \,=\, 2 \frac{\partial f}{\partial \tilde{x}} \)​

which gives 2, or (keeping y instead):

\( \frac{\text{d}}{\text{d}t}f\bigl(\tilde{x}(t),\, \tilde{y}(t)\bigr) \,"="\, 2 \frac{\partial f}{\partial \tilde{y}} \,+\, 2 \frac{\partial f}{\partial \tilde{y}} \,=\, 4 \frac{\partial f}{\partial \tilde{y}} \)​

which gives 4.

Neither 2 nor 4 is equal to 3. The error should be obvious by now: simply choosing to evaluate a function along some given path cannot imply any relation between the function's partial derivatives. The error is in failing to distinguish between the independent parameters of a function, and in naively treating the partial derivatives as fractional expressions when they aren't. Recall that the partial derivatives of f evaluated at \(\bigl( \tilde{x},\, \tilde{y} \bigr)\) are defined by:

\( \begin{align} \frac{\partial f}{\partial x}\bigl(\tilde{x},\,\tilde{y}\bigr) \,&=\, \lim_{h \rightarrow 0} \frac{f\bigl( \tilde{x} \,+\, h,\, \tilde{y} \bigr) \,-\, f\bigl( \tilde{x},\, \tilde{y} \bigr)}{h} \\ \frac{\partial f}{\partial y}\bigl(\tilde{x},\,\tilde{y}\bigr) \,&=\, \lim_{h \rightarrow 0} \frac{f\bigl( \tilde{x},\, \tilde{y} \,+\, h \bigr) \,-\, f\bigl( \tilde{x},\, \tilde{y} \bigr)}{h} \end{align} \)​

It should be clear there's no reason any relation between \(\tilde{x}\) and \(\tilde{y}\) should imply any relation between \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\).

Try this for the specific case of \(\tau(x,\,t) = \gamma \bigl( t - \frac{v}{c^2}x\bigr)\). You'll see that the LHS and RHS of equation 10 on your page are not equal to the corresponding LHS and RHS of equation 11, so equation 11 simply does not follow from equation 10. This is where your contradiction comes from.

 

No, but I would be interested why you wrongly think something is wrong.

Thomas

 

It should be stated explicitly, but it may as well not be. And it definitely doesn't follow that when nothing is stated explicitly, the velocity can't be a relative one. Of course, that alone logically allows only the possibility of the velocity being a relative one. A proof would strictly speaking require corresponding experimental verification (although as I said, I don't see how in case of physical force laws it could be anything else).

The theory of electrodynamics in general and the Lorentz force law in particular existed way be before Quantum Mechanics and Quantum Field Theory and does not need justification from the latter. It should be formulated conceptually consistently on its own. This includes a clear and explicit definition of the velocity in the Lorentz force law.

A velocity referred merely to some coordinate system has no physical relevance, only an abstract mathematical (geometrical) one. A physical force needs a physical reference point, not just an abstract one.

I haven't implied that there must be an alternative explanation. I only said that, contrary to the impression you are trying to give, the Stern-Gerlach effect can not be proven to exist for free electrons. All the experimental evidence for the electron spin is derived from electrons in atoms and ions (or rather in electric fields in general).

Again, you shouldn't try to justify Classical Electrodynamics by means of Quantum Field Theory. In this case, the locality requirement arises indeed from the same misunderstanding about 'locality' that you seem to apply to the equations of electrodynamics, i.e. you are using a circular argumentation here.

No, I am afraid you are in denial. The various derivations I have been quoting clearly assume an infinite wire, so no endpoints exist.

These are just hand-waving arguments and don't do my treatment any justice. As I suggested, you should give a strict mathematical proof of your claim that non-static effects would invalidate the result derived on my page http://www.physicsmyths.org.uk/lorentzforce.htm .

This has nothing to do with the relativistic case we have been considering earlier. I am referring here to the velocity of the electrons relative to the ions in the boosted frame, so the boost velocity has already been subtracted and we are in fact dealing with a stationary loop. In this case, different electron velocities in the top and bottom horizontal sections (which is what you claimed) violate the energy conservation law.

Thomas

 

'Limiting velocity' is the maximum velocity that v can take in the factor gamma = 1/sqrt(1-v^2/c^2) . Only the solution corresponding to the Lorentz transformation contains this factor, so accepting this solution means postulating the existence of a limiting velocity v=c.

The trajectory of the moving row of boys in your reference frame is irrelevant here. What defines the coordinate transformation of events related to the propagation of a light signal is how a separate row of boys at rest in your frame respond to the light signal. Assuming these are identical contradicts the invariance of c.

As you said yourself, the event is only defined here by the interaction of the e.m. field with a given detector. A detector in a different reference frame defines an altogether different event, and it doesn't make any sense to compare the location of the two.

As an example, consider Einstein's train-embankment thought experiment: if you have two light detectors, one on the embankment and one on the moving train, both at the same distance from the light source (assumed to be stationary on the embankment) when the light flash is emitted, then this means that both detectors will be triggered at the same time due to the invariance of c (as seen from the detector on the train, the light source is receding, but as the travel time of the light signal does only depend on the distance but not the velocity, it is the same as the travel time required for the light signal to reach the stationary detector. The important point here is, as mentioned above, that both detector events define actually different light signals, so it is not a contradiction when the location of the two events differ (which they do here as the train is moving on whilst the light signal is on its way). Einstein quite obviously did not realize this and assumed the two loactions must coincide (which implies indeed that the signal speed is not frame invariant, i.e. that we are dealing with a signal consisting of material objects).

On the contrary, ss I said already, it was Einstein who mad up up his own contraint here: this transformation does not strictly speaking result in a velocity independence of the speed of light. I have have explained it before and I just repeat it here:
If you have line of light detectors with synchronized clocks at the same distances from the origin going into opposite directions, and send out a light signal from the origin into both directions, then, according to the principle of the invariance of c, you should find that the time corresponding detectors (at the same distance) show for the arrival of the light signal is identical, irrespective of whether the light source was stationary or moving when the light signal was emitted. In other words, in any case, the positions of the two light signals must be symmetrical to the origin. Or if you want to formulate this in terms of two reference frames, if you flip the coordinate in one reference frame, it must also flip in the other (a constraint which the Lorentz transformation formula does not satisfy)

It's not a rule. It is self-evident. Assume you have a computer subroutine TRANSFORM(x,y,x,t,v) that takes as arguments the unprimed frame coordinates x,y,z,t and transforms it into the primed frame coordinates x',y',z',t' in dependence of v. If you plug in for x any coordinate not belonging to the unprimed systen, then this will result only in nonsense. And x-vt (instead of x) is not a coordinate in the unprimed reference frame.

Again, you are using an inappropriate (and in this case incorrect) generalization: the differentiation in Einstein's paper is not with regard to x but x'=x-vt. If the chain rule is applied correctly, this results in an equation (Eq.(10) on my page http://www.physicsmyths.org.uk/lorentz3.htm ) that can not be satisfied unless v=0.

This exactly shows that a correct differentiation with regard to x' can not determine gamma. Only the incorrect one applied by Einstein appears to fix gamma to the known expression.

Effectively Einstein did (try to) differentiate. Using the Taylor expansion is just a convoluted way of doing it. Essentially he is trying to re-invent the wheel where he could have applied just fundamental rules of calculus.
Note: Einstein himself remarked once that he has his problems with mathematics; he apparently just didn't realize how big his problems were.

The invariance of c obviously exists as a constraint before he starts the derivation. As I said before, he just tacitly changes this constraint so that it is consistent with a velocity dependent transformation formule. The unchanged constraint x=ct<=>x'=ct' and x=-ct<=>x'=-ct implies however that the tranformation can not be velocity dependent (i.e. it has to be the identity transformation).


If (I) is supposed to be an analogue to the identity constraint or a simultaneous x = +ct and x = -ct constraint then I've already told you where you've gone wrong.

You are merely claiming that they are defined for all x,t. But since they can't be consistently defined even on the light cone, this is evidently a false claim. The point is that, contrary to what you are saying, f = ct + x =0 and g = ct - x =0 must hold simultaneously if you are sending a light signal simultaneously in the positive and negative x-direction (like Einstein implies for his 'algebraic derivation' (see http://www.bartleby.com/173/a1.html )).

But you can never know that a priori. Otherwise you could as well claim that you have certain 'powers' if you happen to win the lottery; even if you would win the lottery ten times in succession, you could not claim to have any powers in this respect unless you know why you won it ten times.
There is a saying 'Knowledge is Power'; well, the reverse is also true 'Power is Knowledge'. The point is a theory is just that and it can not be taken for granted that it represents the truth.

I am not making up constraints. I am just applying the constraints that should be applied instead of the made-up constraint that Einstein applied in his Relativity.

Questioning things that you take for granted without having sufficiently reflected on them is always a good thing. In this case you seem to forget that collision cross sections are in general energy dependent. If you make wrong assumptions regarding the particle energy, you will associate this also with an incorrect cross section.

As I said, it should not be relevant for the GPS system in practice. Besides, as I am sure you know, the GPS system is essentially a military system, so I think it would be naive to assume that all the relevant information about it is in the public domain. In other words, I can't accept any information from military sources as reliable due to the obvious conflict of interest here between publication and military secrecy.

Thomas

 

Your example is missing the point: first of all, note that both here and in Einstein's example (as discussed on my page http://www.physicsmyths.org.uk/lorentz3.htm ) we don't have two truly independent variables, but both can be expressed linearly in terms of each other. What the substitution of the corresponding differentials does is simply to reduce the function to one variable only. You will then of course get either twice the derivative for the first variable or the second variable (in this case 2 or 4). The point is that on my web page I am not comparing one-variable derivative to the actual two-variable derivative (which is unknown at this point) but I am comparing the one-variable derivatives of the two paths implied by the two sides of Einsteins 'master equation'. And it should now be obvious that the these two paths can in general not be equal i.e. there can not be function tau that could fulfil the equation 1/2*tau(x2,t2)=tau(x1,t1).

Thomas