Relativity fails with Magnetic Force

BenTheMan,
I have already admited that I was wrong in the proposition that the relativistic prediction differs from the classical prediction. They are the same. But another inconsistence or absurd appears:

I have realized that I was wrong in one point: actually the relativistic prediction do agree with the classical prediction. The relativistic electrodynamics with lenght and charge contraction with Lorentz Transform actually predict changing Electric and Magnetic Fields so that the the Force obtained is the same as in Classical Physics.
But the inconsistence rises then from the other point of view: Relativity states there are no privileged frames of reference and so in the presented problem there's a change in the real movement of the electron just because a change in the frame of observation (the observer)!
This is unsustainable...
What do Relativity defenders would argue now?

And I have already posted:
"Sorry przyk but I cannot "swallow" that. Do you realize you are admiting that a change in the selected frame of observation of a "system" would alter its dynamics!
No, no way...
This is absurd. "

 

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What does ``real movement'' mean?

What do indeed.

 

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Sorry BenTheMan but I was editing the post when you wrote a new post. Please re-read it.

Note:
By "real movement" I mean the relative movement of the objects of some system being analyzed. In this case the relative movement of the isolated electron in relation to the beam of the electrons.

 

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To avoid confusions now I have already rewrote the problem in the page of the site.

 

I really don't see what's bothering you about this - what would you have expected? If you take any system and view it in a reference frame in which the system is moving at high velocity, relativity predicts that this frame's local description of the dynamics of the system will be different. Specifically, relativity predicts that the dynamics will be length contracted, time dilated, and exhibit the relativity of simultaneity effect. On its own there isn't anything particularly remarkable about this. If you perform the following coordinate substitution on all the trajectories that make up your system:

\( \begin{align} t^{\prime} \,&=\, \gamma \bigl( t - \frac{v}{c^{2}}x \bigr) \\ \\ \\ \\ x^{\prime} \,&=\, \gamma \bigl( x - v t \bigr) \\ \\ \\ \\ y^{\prime} \,&=\, y \\ \\ \\ \\ z^{\prime} \,&=\, z \end{align} [/indent]\)
then anything defined in terms of the coordinates will change - there's no reason for \(\bar{a}^{\prime} \,=\, \frac{\partial^{2}\,\bar{x}^{\prime}}{\partial t^{\prime}\,^{2}}\) to be numerically equal to \(\bar{a} \,=\, \frac{\partial^{2} \bar{x}}{\partial t ^{2}}\), and in general it won't be. This is true of arbitrary coordinate substitutions and isn't a feature in any way specific to relativity.

In relativity, the interesting question isn't how you can distort the description of dynamics by playing around with coordinate changes - that's just fiddling with math. The interesting question is why inertial observers have a natural tendency to define cartesian coordinate systems that are related to one another by Lorentz (or in general Poincaré) transformations.​

 

przyk,
The point is that a change in the reference frame (the observer) cannot alter the behavior of the system ot phenomenon being analyzed. In other words the reality of the system/phenomenon cannot be changed and that is what is happening in the presented problem. By reality I mean all the characteristics and properties of the system/phenomenon itself.
If you change the reference frame you would get different expressions for many things as velocity, acceleration, the Force, etc but for example, as in our case, the relatrive movement between the electron and the beam of must be the same, but if you observe carefully the relativistic transformation of the frames you will find that they determine two different movements, two different phenomenons. This cannot happen.

In other words, in the problem the relativistic prediction is the same as the classical prediction, fine, but the classical prediction is that the isolated electron will behave differently in the two presented cases of the problem. Now, Classical Physics determines two different behaviors because it considers that if a new absolute velocity is given to the isolated electron and the beam then this actually is a new phenomenon, a new system whith justified different behavior.
Now how Relativity consider and justify the two cases? The relative initial velocity of the two cases is the same so they would represent the same phenomenon, the same system for Relativity, then, how two different behaviors (to match with the classical prediction) are justified? Remember that just a change in the freference frame have been done. Then how?
I don't understand. I see a big problem here.

 

Veliskyov is warned for trolling in physics.

As an aside, your posting style is oddly reminiscent of a previous, banned member, even though your IPs don't match.

 

Sure not me. For a long time I only posted times to time just with the aim to receive possible honest criticism on my theory which I present in the site (can be accessed through the "HOME" link at the top of the posted page).

 

Magnetic forces (in one frame)
=
Electric forces (in the other frame)

 

In defining ``reality'', you have implicitly assumed a specific reference frame.

A simple example, which you've surely encountered in your extensive studies of special relativity, is the idea of simultaneity. That is ``simultaneous'' events are only simultaneous in a specific reference frame. For example, suppose I turn on a switch that operates two lights, A and B:


A------------------------>---------------------- ME ------------------------<----------------------B


I measure photons from light A and light B arriving at exactly the same time: I say that the lights turn on simultaneously. A boosted observer, however, claims that either light A or light B turns on first, depending on the direction which he has been boosted.

His ``reality'' is different than my ``reality'', which is exactly the conclusion of Special Relativity.

 

I have not been confusing anything. On the contrary, I pointed out that if one does not define the velocity v in F=q*vxB and the velocity associated with the current (that determines B) separately, the problem is insufficiently defined (and thus will get confused). In mortillo's example there is only one velocity, namely the velocity of the test charge with regard to the row of the 'beam electrons'. So the situation is ambiguous here, as you can interpret this velocity as being either solely due to v or solely due to the current (with v=0), or any other combination of the two. That's why a test charge moving relatively to a single beam can not possibly create a magnetic force. You need two beams (like e.g. in a wire) to unambiguously (and frame-independently) define the current and thus the magnetic field, and v is then (also unambiguously) the velocity of the test charge with regard to the center of mass of the two beams (e.g. the wire).

As indicated above, you need two beams of (interacting) charges to create a magnetic field. Otherwise you could not define the current and the velocity v in the Lorentz force unambiguously.
Besides, a configuration like suggested by mortillo would not be stable anyway, as the charges would fly apart due to the electrostatic repulsion. So it is not a valid scenario to discuss the Lorentz force.

If the charge density is higher everywhere by a constant factor, then this means that for an infinite wire you have charge conservation violated by an infinite amount. I wouldn't exactly call this negligible.

If you consider a finite wire, you can of course enforce charge conservation by this. But the point is that then the field produced by this finite wire (with different charge densities for positive and negatve charges) is inconsistent with Maxwell's equations: the far field would decrease like 1/r^4, but according to Maxwell's equations the magnetic field of a current in a finite wire should decrease like 1/r^2 in the far field (I have shown this on my page Magnetic Fields and Lorentz Force).

Thomas

 

They *are* seperately defined: the \(\bar{v}\) in the Lorentz force equation is the velocity of the charge the force is acting upon in the particular frame you're considering. In Maxwell's equations and electrodynamics in general, the current \(\bar{I}\) is usually defined as the the flux of electric charge passing through a stationary surface perpendicular to the flow per unit time (\(I = \frac{\text{d}Q}{\text{d}t}\)). The current density is equal to the charge density multiplied by the local velocity of the charged medium (\(\bar{j} = \rho \bar{v}\)).

All these quantities are defined in a frame-dependent manner in electromagnetism. It's not obvious just looking at the equations, but the resulting theory is known (and rigorously proven) to be relativistically covariant, and it's the theory everyone has been using since the nineteenth century with no problems.

No, the situation wasn't ambiguous. martillo defined "case 1" as the isolated electron's rest frame (where \(v_{\text{e}} = 0\)) and called the electron beam's velocity \(v_{1}\) in this frame. In my first post in this thread, I explicitly stated I was going to calculate the forces for the particular case where \(v_{e} = v_{1} = 0\) in the rest frame, so both have the same velocity \(v\) in an arbitrary frame. As far as electrodyamics is concerned, the problem is completely defined.

The nice thing about thought experiments is that you don't have to worry about these things. You can assume that the electrons in the beam are being held in place by some other force that has nothing to do with electromagnetism and leave it at that. Since you're trying to isolate and study certain theoretical properties of electromagnetism you don't need to care what this "force" is or even whether such a force exists in the real world.

(Alternatively, if for some reason you don't accept this, you can still study the setup at the instant in time it's configured the way martillo described, before it blows apart).

Take an infinitely long wire of some given charge density, compress the infinitely long wire by a factor of two- squashing all the charges together without adding any new charges anywhere, and you end up with a wire with double the charge density, but you haven't had to add any charges to obtain this configuration so you haven't violated "charge conservation".

Even better: wipe away half the charges and repeat the above procedure, only compressing by a factor of 4. Now you doubled the charge density and lost charge in the process. My point? You're comparing infinities and you could "argue" this to get any result you want.

This is a moot point since what's meant by "charge conservation" is that the charge and current obey the continuity equation I posted earlier (which basically requires that the net accumulation of charge in any volume is due to the net flow of charge into that volume. As far as I know, it just so happens that the total charge is the same in a given region in different reference frames (if you're careful about the volumes you define in each frame), but this is a meaningless comparison if the charges are infinite and isn't what the continuity equation is concerned with anyway.

I can see at least two errors on that page.

The first is that the quadrupole moment is nonexistent if the net charge is zero and the charge is linearly distributed (which you obviously assumed when calculating the integral). Look at the equation for the electric field you derived: if \(Q\) is zero then \(E\) is zero.

The second is that neither Coulomb's law nor Biot-Savart are valid in dynamical situations. Your radio and television wouldn't work if they were.

 

Qualitatively the same thing is happening in both frames: in case 1, the electron accelerates away from the beam. In case 2, the electron accelerates away from the beam - just at a reduced rate consistent with time dilation of the entire system.

 

With time dilation, space contraction and whatever else the intrisic dynamics (the relative movements of objects) of a system of objects cannot be altered by changing the selected frame of observation.
No way.

 

So what would then be the magnetic force on the test electron in the rest frame of the beam (where the current would be zero)?

If you offset some electromagnetic forces by other forces, then, as far as electrodynamics is concerned, the whole theoretical consideration becomes inconsistent.

You could only strictly speaking study this scenario if the beam had a zero charge (such as to have no electrostatic field in the first place).

It was not my idea to use infinities here, but this is what all 'derivations' of the relativistic Lorentz force use. The point is that those authors don't seem to understand that the concept of 'infinity' merely means 'applying to any finite value'. Now if you have a finite wire and increase the charge density of the electrons, there are two options: either you keep the charge density constant over the whole wire (which could only be achieved by adding charges out of nowhere (i.e. by violating charge conservation); or you end up with an inhomogeneous charge distribution that would lead to an electromagnetic field contradicting Maxwell's equations (see below).

The continuity equation is irrelevant here. We are not concerned about the time dependence of the total charge here, but about the fact that in one reference frame the total (net) charge of a wire is zero, but in another reference frame not (which is what the relativistic interpretation of the magnetic field implies). The total charge should be independent of the reference frame.

No, there is no error. Q is the total line charge, and this results in an electric field E=Q*(1/r^2-L^2/2r^4) (+higher order terms in 1/r).
If you have a second line with the opposite charge -Q (like in a wire) but with a different length L (due to the length contraction), then the first term (proportional to 1/r^2) vanishes and only the quadrupole term proportional to 1/r^4 remains . This contradicts Maxwell's equations as the magnetic field of a short current element should be proportional to 1/r^2.

We are considering only electro- and magnetostatics here.

Thomas

 

If there's just the beam, then zero, regardless of the velocity of the test charge (although if the test charge is moving it will create a magnetic field that will pull on the wire).

In the case of a neutral wire, a magnetic field is created by the motion of the ions in the current-carrying electrons' rest frame.

If you really want to be pedantic about this then the logic would go something like this: assume some relativistic theory supplies a force that holds the electron beam together and in place, as in martillo's setup ("relativistic" meaning that it will cause the beam to length contract, etc.). Calculate the acceleration of the test charge. If you get a result inconsistent with relativity, you know it's electromagnetism's fault because you were careful that the fictitious theory you added was relativistic. If you get a result consistent with STR, you have circumstantial evidence that the combination of electromagnetism + fictitious theory is relativistic, which translates to circumstantial evidence that electromagnetism is relativistic.

If you were setting out to disprove STR (as martillo was) then the idea is that you think STR seems that little bit more plausible (it worked where you didn't expect it to), and you think you should maybe study this "relativistically covariant formulation of electromagnetism" that all these pro-STR people are so excited about and would consider their strongest argument (and because as long as you don't address their general proof, they'll think you're as likely to prove electromagnetism is inconsistent with STR as you are to square the circle).

You're remembering that the matter composing the wire would also be subject to length contraction effects, which would affect the charge density of the ions, right?

Note also that if you want to propose a relativity thought experiment about a finite wire, you should take care to specify what happens to the current-carrying charges at the ends of the wire (eg. "the charges drain from one end and accumulate at the other") - the relativity of simultaneity effect makes this relevant if you really want to calculate the fields in different frames.

In addition to what I said earlier about the wire material itself contracting, who says it isn't? You found a \(\frac{1}{r^{4}}\) for the electric field in the reference frame you were considering. Why should it have the same asymptotic behaviour as the magnetic field? In general you wouldn't expect it to - a feature of electromagnetism is the absence of magnetic monopoles, for instance.

 

If a fast moving clock is time-dilated, are you saying you'd expect the internals of the clock to tick along at the normal rest rate?

I don't know what you mean by "changing the dynamics", since in changing the coordinate system - which is really just variable substitution - what you get is a different description of the dynamics. In your "case 2" this is the same electron moving away from the beam as in case 1. The only difference is that an observer at rest in the "case 2" frame will measure a smaller acceleration because he doesn't measure time (or space) the same way an observer in "case 1" does. And how different observers will measure out space/time and define coordinate systems isn't something "obvious" that you can just guess about: observers are themselves physical systems that obey physical laws, and ultimately it is the symmetry properties of those laws that will determine their choice of coordinate systems. If a fast moving observer is length contracted and time dilated (as anything held together by electromagnetic forces will be), this will affect the way they view and measure the world.

 

przyk,

As I said: the intrinsic dynamics of a system is determined by the relative movement of its objects, I mean their relative trajectory.
I agree that the velocity, acceleration, etc could be frame dependent (it also can happen in Classical Physics) but not the relative trajectory.

By the way, the acceleration of the MagneticForce is in the direction perpendicular to the initial velocity of both the isolated electron and the beam and Lorenta Transform predicts time dilation and length contraction happen in the direction of the velocity only, not in the other perpendicular directions. There's no time dilation in the direction of the acceleration and so your argument that the difference in the dynamics could be accounted by the time dilation is not valid.

 

If you apply the Lorentz transformation to a system, it's a mathematical inevitability that the "relative trajectories" (if you mean the difference between trajectories calculated in a particular reference frame) will also Lorentz-transform.

Direct from the Lorentz transformation I posted a while back:

In case 1 the transverse momentum is \(p_{y} \,=\, m \frac{\partial y}{\partial t}\) (the electron is at rest in this frame so \(\gamma = 1\)). Since \(t = \frac{t^{\prime}}{\gamma}\) along the electron's path, and \(y = y^{\prime} \), you have:

\(p_{y} \;=\; m \, \frac{\partial y}{\partial t} \;=\; \gamma \, m \, \frac{\partial y^{\prime} }{\partial t^{\prime} } \;=\; p_{y}^{\phantom{y}\prime}\)​

ie. the transverse momentum is invariant.

If the acceleration is completely transversal, then:

\(F \;=\; \frac{\partial p_{y}}{\partial t} \;=\; \gamma \, \frac{\partial p_{y}^{\phantom{y}\prime} }{\partial t^{\prime} } \;=\; \gamma \, F^{\prime}\)​

Or \(F^{\prime} = \frac{1}{\gamma} F\), simply as a result of the coordinate change.

You're the first person I've ever seen try to associate an orientation with time dilation.

 

You are right in the second part of your post. For a moment I forgot that in Relativity transversal force could appear. Time dilation actually could create transversal forces.

I disagree with the first part:

It is a "mathematical inevitability" within Relativity Theory.
In the problem Lorentz Transform does not affect the distance in the direction perpendicular to the beam and the velocities (y = y') and the point is that if in the two cases differences forces are applied to the isolated electron (relativistic prediction is the same as the classical one) then it will follow two different trajectories (different coordinates in the "y" direction) and so actually they are not obtained from each other by the Lorentz Transform which states y=y'.
Then it is right to say that in Relativity Theory just a change of the reference frame (the observer) could produce a change in the relative movement (relative trajectories) of a system (particularly this one) what is inconsistent or just wrong or absurd.