Special Relativity Paradox

[M S La Moreaux]

Consider a current carrying circuit with a straight side.  Because of length contraction, in the straight side the moving electrons will appear closer together than they would be if there were no current.  This gives a greater charge density, so the straight side should have a negative charge.  But this is not observed.  How come?

[Markus Hanke]

3 hours ago, M S La Moreaux said:

This gives a greater charge density, so the straight side should have a negative charge.

I don’t really understand what you are getting at. First of all, it isn’t just one side that gets length-contracted, but the entire circuit in the direction of motion. Secondly, both the circuit itself and the classical electron radius in the direction of motion are shortened by the same factor, thus the total amount of charge (ie number of electrons in the wire) in any section of wire remains conserved. Particle number is a conserved quantity under Lorentz transformations.

I don’t see an issue here?

Quote

Special Relativity Paradox

SR is just the geometry of Minkowski spacetime; specifically, in the special case of relationships between inertial frames, it concerns hyperbolic rotations of the coordinate system. Let A be some coordinate-dependent description of an inertial frame, and L be a Lorentz transformation parametrised by rotational angle (=rapidity, a measure of relative velocity). The action of L on A is thus simply

\[A’=L(\omega)A\]

Furthermore, we know that, from definition, the matrix L is a square matrix and must preserve the metric:

\[L^{-1}gL=g\]

which implies that \(|det(L)|=1\). From linear algebra we know that all square matrices with determinant other than zero are invertible, and thus:

\[L^{-1}(\omega)L(\omega)A=A\]

So it is mathematically and logically impossible to construct physical paradoxes using just the axioms of SR, no matter what kind of physical scenario you try to concoct. Any purported SR “paradox” is by the above automatically an erroneous conclusion based on some misapplication of SR. 
 

Edited Thursday at 06:48 AM by Markus Hanke

[M S La Moreaux]

13 hours ago, Markus Hanke said:

I don’t really understand what you are getting at. First of all, it isn’t just one side that gets length-contracted, but the entire circuit in the direction of motion. Secondly, both the circuit itself and the classical electron radius in the direction of motion are shortened by the same factor, thus the total amount of charge (ie number of electrons in the wire) in any section of wire remains conserved. Particle number is a conserved quantity under Lorentz transformations.

I don’t see an issue here?

SR is just the geometry of Minkowski spacetime; specifically, in the special case of relationships between inertial frames, it concerns hyperbolic rotations of the coordinate system. Let A be some coordinate-dependent description of an inertial frame, and L be a Lorentz transformation parametrised by rotational angle (=rapidity, a measure of relative velocity). The action of L on A is thus simply

 

A′=L(ω)A

 

Furthermore, we know that, from definition, the matrix L is a square matrix and must preserve the metric:

 

L−1gL=g

 

which implies that |det(L)|=1 . From linear algebra we know that all square matrices with determinant other than zero are invertible, and thus:

 

L−1(ω)L(ω)A=A

 

So it is mathematically and logically impossible to construct physical paradoxes using just the axioms of SR, no matter what kind of physical scenario you try to concoct. Any purported SR “paradox” is by the above automatically an erroneous conclusion based on some misapplication of SR. 
 

 

[studiot]

Just now, M S La Moreaux said:

 

I'm sorry, what did you want to say ?

11 hours ago, M S La Moreaux said:

Consider a current carrying circuit with a straight side.  Because of length contraction, in the straight side the moving electrons will appear closer together than they would be if there were no current.  This gives a greater charge density, so the straight side should have a negative charge.  But this is not observed.  How come?

An explanatory diagram would help greatly her.

Observed ?

Observed by whom, in what frame ?

greater charge density ?

greater than what ?

length contraction again observed by whom in what frame ?

[swansont]

19 hours ago, M S La Moreaux said:

Consider a current carrying circuit with a straight side.  Because of length contraction, in the straight side the moving electrons will appear closer together than they would be if there were no current.  This gives a greater charge density, so the straight side should have a negative charge.  But this is not observed.  How come?

What happens to two parallel wires with current flow?

[M S La Moreaux]

This really should not be this difficult.  I assume a straight side to avoid acceleration so special relativity will apply.  The circuit is stationary.  The only things moving are the free electrons.  Their charge density appears greater while the charge density of the protons remains unchanged, thus resulting in a net negative charge for the segment of the circuit under discussion.  This should be observed by someone in the frame of reference of the circuit, but is not.

[Markus Hanke]

52 minutes ago, M S La Moreaux said:

Their charge density appears greater

No. Charge as well as particle number are conserved quantities under Lorentz transformations, as I have pointed out above. The ratio of electrons to protons doesn’t change just because something is in relative motion. How could it? Inertial motion doesn’t magically create or annihilate particles in a wire.

Consider, as a very primitive and purely classical analogy, a length of transparent plexiglass tube filled with - say - 100 ping pong balls, initially at rest. The balls are all perfectly spherical, and their number is constant and doesn’t change. Let’s say it’s a random mix of red and yellow balls.

Now you put the tube into motion at relativistic speeds, relative to an observer. What happens? For that observer, the tube becomes length-contracted in the direction of motion, and so does the radius of each ball. The balls no longer appear spherical to him, but instead look like flattened disks; however, both their total number and the ration between red to yellow balls remains the same.

It’s similar with the particles in the wire, with the caveat that those aren’t classical objects, so for a precise description you need to use relativistic quantum mechanics, which means fields of bispinors. But it all works out. As I said above, you cannot construct physically paradoxes with the axioms of SR.

[exchemist]

2 hours ago, Markus Hanke said:

No. Charge as well as particle number are conserved quantities under Lorentz transformations, as I have pointed out above. The ratio of electrons to protons doesn’t change just because something is in relative motion. How could it? Inertial motion doesn’t magically create or annihilate particles in a wire.

Consider, as a very primitive and purely classical analogy, a length of transparent plexiglass tube filled with - say - 100 ping pong balls, initially at rest. The balls are all perfectly spherical, and their number is constant and doesn’t change. Let’s say it’s a random mix of red and yellow balls.

Now you put the tube into motion at relativistic speeds, relative to an observer. What happens? For that observer, the tube becomes length-contracted in the direction of motion, and so does the radius of each ball. The balls no longer appear spherical to him, but instead look like flattened disks; however, both their total number and the ration between red to yellow balls remains the same.

It’s similar with the particles in the wire, with the caveat that those aren’t classical objects, so for a precise description you need to use relativistic quantum mechanics, which means fields of bispinors. But it all works out. As I said above, you cannot construct physically paradoxes with the axioms of SR.

This isn't my field at all, but I wonder if the conceptual difficulty is that, while the electrons are in motion relative to the observer, the wire in which they travel (i.e. the array of atoms of which it is made) is not. So it seems to me the external dimensions of the wire are unchanged to the observer, even though the contents, i.e. the moving electrons, are foreshortened in the direction of travel as you have explained. 

I'm not sure myself how I should picture this. In terms of your analogy, the balls inside the plexiglas tube are in motion, but the tube itself is not. 

[KJW]

11 hours ago, swansont said:

What happens to two parallel wires with current flow?

There is a force acting on the wires due to the magnetic field. Magnetic field??? Where did THAT come from???

 

 

[Genady]

5 hours ago, M S La Moreaux said:

to avoid acceleration so special relativity will apply

Special relativity applies with acceleration as well. Albeit the application is more involved.

[Markus Hanke]

3 hours ago, exchemist said:

This isn't my field at all, but I wonder if the conceptual difficulty is that, while the electrons are in motion relative to the observer, the wire in which they travel (i.e. the array of atoms of which it is made) is not.

I think you might be right, I misunderstood, see below.

7 hours ago, M S La Moreaux said:

The circuit is stationary. 

Ok, I just saw this while re-reading the thread. Apologies, I initially misunderstood your scenario. 

7 hours ago, M S La Moreaux said:

The only things moving are the free electrons.

So the observer is at rest in the circuit hardware frame.

7 hours ago, M S La Moreaux said:

Their charge density appears greater while the charge density of the protons remains unchanged

The drift velocity of electrons in a DC circuit is typically nowhere near relativistic, so you wouldn’t see any relativistic effects.

If that velocity somehow was made relativistic, the resolution is still the same - in the circuit frame, the classical radius of the electrons is length-contracted in the direction of motion; in the electron rest frame, it is the wire length and the classical radius of the protons that is length-contracted by the same factor. Crucially, in both cases, you furthermore have to consider relativity of simultaneity when figuring out how many particles are actually there in total, since the events “reached beginning” and “reached end” of wire are no longer simultaneous in both frames - the adjustment happens by the same factor, thus the ratio between the number of electrons and the number of protons is the same in both frames.

This is really just a variation of the classical ladder paradox, and is resolved similarly.

As an aside - an experimental example of where you’d see a type of length-contraction of a roughly spherical object into something resembling more of a flattened disk, is the collision of gold ions at the Relativistic Heavy Iron Collider.

Edited Friday at 11:57 AM by Markus Hanke
Ladder paradox mentioned

[studiot]

1 hour ago, M S La Moreaux said:

This really should not be this difficult.  I assume a straight side to avoid acceleration so special relativity will apply.  The circuit is stationary.  The only things moving are the free electrons.  Their charge density appears greater while the charge density of the protons remains unchanged, thus resulting in a net negative charge for the segment of the circuit under discussion.  This should be observed by someone in the frame of reference of the circuit, but is not.

Thank you for responding to my query about frames, even if the reply was short.

When I first read this I wondered if you are confusing the relativistic origin of magnetism .....

But We still have no diagram.

[exchemist]

42 minutes ago, Markus Hanke said:

I think you might be right, I misunderstood, see below.

Ok, I just saw this while re-reading the thread. Apologies, I initially misunderstood your scenario. 

So the observer is at rest in the circuit hardware frame.

The drift velocity of electrons in a DC circuit is typically nowhere near relativistic, so you wouldn’t see any relativistic effects.

If that velocity somehow was made relativistic, the resolution is still the same - in the circuit frame, the classical radius of the electrons is length-contracted in the direction of motion; in the electron rest frame, it is the wire length and the classical radius of the protons that is length-contracted by the same factor. Crucially, in both cases, you furthermore have to consider relativity of simultaneity when figuring out how many particles are actually there in total, since the events “reached beginning” and “reached end” of wire are no longer simultaneous in both frames - the adjustment happens by the same factor, thus the ratio between the number of electrons and the number of protons is the same in both frames.

This is really just a variation of the classical ladder paradox, and is resolved similarly.

As an aside - an experimental example of where you’d see a type of length-contraction of a roughly spherical object into something resembling more of a flattened disk, is the collision of gold ions at the Relativistic Heavy Iron Collider.

Aha. So the resolution of the seeming paradox is  actually quite subtle. To be honest, the ladder paradox almost does my head in. I'm not surprised @M S La Moreaux hadn't thought of it that way. 

[swansont]

6 hours ago, KJW said:

There is a force acting on the wires due to the magnetic field. Magnetic field??? Where did THAT come from???

You can look at it solely from electrostatics and relativity, as Markus has detailed.

[Markus Hanke]

17 hours ago, exchemist said:

Aha. So the resolution of the seeming paradox is  actually quite subtle. To be honest, the ladder paradox almost does my head in. I'm not surprised @M S La Moreaux hadn't thought of it that way. 

You can think of the number of electrons in the wire to correspond to the number of steps on the ladder. This number does not change as you switch between frames (so no excess of either electrons or protons), and despite the circuit being length-contracted, the “electron-ladder” still fits in the wire, due to relativity of simultaneity.