How To Consistently Explain Electromagnetism With Relativity

[A.T.]

But what about the opposite currents scenario?

DrGreg's diagramm could help here:
https://www.physicsforums.com/threa...agnetism-with-relativity.932270/#post-5886984

 

[jartsa]

I wasn't sure how to use your formula so I did my analysis without it. I wonder if you can confirm whether the magnetic repulsion between a proton moving left at 87% c and an electron moving left at double that speed is 1.5 times the magnetic attraction between two electrons moving left at 87% c?

I can only calculate magnetic attractions between charges moving side by side at the same velocity. Because that is very simple.

There's one thing I should mention: All your pictures are kind of unrealistic, as the two electrons are always perfectly lined up. It's not unphysical, it's just not correct for electrons in wires. I mean, according to an electron the other electrons should disappear into the distance as the current increases. That effect seems to be missing in all of the calculations - I just noticed that.There is also an easy way to calculate these things. That involves going into electron's frame, calculating electric forces on the electron there, there are no magnetic forces, which was the point of the frame change. And then that force can be easily transformed to any frame, by using relativity's force transformation formulas.

 

[Geocentricist]

DrGreg's diagramm could help here:
https://www.physicsforums.com/threa...agnetism-with-relativity.932270/#post-5886984

Not sure how that's supposed to help figure out the force between a moving electron and proton.

I can only calculate magnetic attractions between charges moving side by side at the same velocity. Because that is very simple.

Okay, I hope someone else can help me out here then.

There's one thing I should mention: All your pictures are kind of unrealistic, as the two electrons are always perfectly lined up. It's not unphysical, it's just not correct for electrons in wires. I mean, according to an electron the other electrons should disappear into the distance as the current increases. That effect seems to be missing in all of the calculations - I just noticed that.

Why should the separation between electrons increase?

There is also an easy way to calculate these things. That involves going into electron's frame, calculating electric forces on the electron there, there are no magnetic forces, which was the point of the frame change. And then that force can be easily transformed to any frame, by using relativity's force transformation formulas.

I prefer doing it this way, in the way I can understand.

 

[jartsa]

Okay, I hope someone else can help me out here then.

Hey, why don't I use my wonderful easy calculation method myself.

So, an electron moves at speed 0.87 in a wire, there is another wire where electrons move at the opposite direction at speed 0.87c.

Let's say electron sees the electron formation on the other wire contracted to 1/4 of the normal. That is not the correct number.

So four times more charges produces four time more electric force.

Now the transformation to the lab frame. That is division by two in this case.

Total force in lab frame = 4 * electric_force_in_ lab_frame / 2 = 2 * electric_force_in_ lab_frame

Total force doubled, so magnetic force in lab frame must be same as the electric force in the lab frame. (Not the actual correct result because of the wrong length contraction)This is the force transformation formula: F'=F/gamma

 

[Geocentricist]

(Not the actual correct result because of the wrong length contraction)

But why should the electron spacing increase in their own frame?

 

[Ibix]

Why should the separation between electrons increase?

Because they are accelerating, and their notion of distance is changing as this happens.

I prefer doing it this way, in the way I can understand.

Philosophical point: if you can't do the maths, do you really understand it? You will always need someone to tell you what the maths says.

Working in the lab frame, the magnetic field from an infinitely long current carrying wire is ##B=\mu_0 I/2\pi r##, always in the tangential direction. Defining ##\theta## as the angle in the y-z plane anticlockwise from the y axis, ##B_x=0##, ##B_y=-B\sin\theta##, ##B_z=B\cos\theta##. The electric field is everywhere zero.

Then you can write down the electromagnetic field strength tensor, F$$F=\left (\begin {array}{cccc}
0&-E_x/c&-E_y/c&-E_z/c\\
E_x/c&0&-B_z&B_y\\
E_y/c&B_z&0&-B_x\\
E_z/c&-B_y&B_x&0
\end {array}\right)$$Then you carry out a Lorentz transform to the electron rest frame. This is to a velocity v in the +x direction (v will turn out to be negative when we calculate it later, due to conventional current having its sign defined in an unfortunate way). That's way too much work to typeset for the general case. The result is that in the electron frame there is an electric field whose components are ##E'_x=0##, ##E'_y=-\gamma v B_z##, ##E'_z=\gamma v B_y## (Edit: removed erroneous factor of c) and a magnetic field whose components are ##B'_x=0##, ##B'_y=\gamma B_y##, ##B'_z=\gamma B_z##, where ##\gamma=1/\sqrt {1-v^2/c^2}##.

Now you need to determine the electron drift velocity, v. Wikipedia has the maths (although it uses u instead of v) and even works out a number for you using 1A through a copper wire. Note that they use a slightly inconsistent sign convention - the answer needs to be multiplied by -1 to get the negative velocity I mentioned above. https://en.m.wikipedia.org/wiki/Drift_velocity
Dale already gave you the formula to calculate the force on a charge from the field components.

 

[A.T.]

But why should the electron spacing increase in their own frame?

https://www.physicsforums.com/threa...sm-with-relativity.932270/page-3#post-5888171

 

[Geocentricist]

https://www.physicsforums.com/threa...sm-with-relativity.932270/page-3#post-5888171

The Wikipedia article says spaceships A and B disagree they both accelerated at the same time. But since in S they did accelerate at the same time, and did so with the same acceleration, it seems to me A and B share a frame and definition of simultaneity throughout their acceleration. In which case both spaceships agree they accelerated at the same time.

 

[Dale]

The Wikipedia article says spaceships A and B disagree they both accelerated at the same time.

Actually it doesn’t say that. What it says is subtly different.

In the article neither S nor S’ represent A or B’s perspective. S and S’ are inertial frames and the article is talking about their perspective, not the perspective of A or B which would be non inertial. A and B are only momentarily at rest in S or S’.

 

[Geocentricist]

Actually it doesn’t say that. What it says is subtly different.

In the article neither S nor S’ represent A or B’s perspective. S and S’ are inertial frames and the article is talking about their perspective, not the perspective of A or B which would be non inertial. A and B are only momentarily at rest in S or S’.

Okay, but I still don't see why A and B should disagree they accelerated at the same time, since they always share the same frame and therefore, time.

 

[PAllen]

Okay, but I still don't see why A and B should disagree they accelerated at the same time, since they always share the same frame and therefore, time.

Think about this. In S you have two separated simultaneous events: A started accelerating, B started accelerating. S’ is moving relative to S. It is then impossible for these acceleration events to be simulataneous in S’. Two relatively moving frames can never agree on simultaneity of events separated in the direction of relative motion.

 

[Dale]

Okay, but I still don't see why A and B should disagree they accelerated at the same time, since they always share the same frame and therefore, time.

It isn’t a disagreement between A and B. It is a disagreement between S and S’

 

[PAllen]

Consider a related question. In some S’, there is an event when A is stationary and an event when B is stationary. These necessarily correspond to events described in S as A has speed v and B has speed v. These events are simultaneous by construction in S. Thus the corresponding events in S’ are not simultaneous. Thus, once acceleration has begun, there are no inertial frames where A and B are simultaneously at rest.

 

[Geocentricist]

It isn’t a disagreement between A and B. It is a disagreement between S and S’

Consider a related question. In some S’, there is an event when A is stationary and an event when B is stationary. These necessarily correspond to events described in S as A has speed v and B has speed v. These events are simultaneous by construction in S. Thus the corresponding events in S’ are not simultaneous. Thus, once acceleration has begun, there are no inertial frames where A and B are simultaneously at rest.

Any S' equidistant to A and B will consider the events simultaneous. So I don't see how a possible disagreement between S and some hypothetical S' proves A and B will move relative to one another.

 

[PAllen]

Any S' equidistant to A and B will consider the events simultaneous. So I don't see how a possible disagreement between S and some hypothetical S' proves A and B will move relative to one another.

S’ is a reference frame, not an observer, so equidistant is a nonsequiter. There is no reference frame in which A and B are both at rest at the same time except the original reference frame for the times before acceleration starts. If you can’t see this, then you need to review the basics of relativity of simultaneity. You are making statements comparable to 1 + 1 = 3 and insisting they are right.

 

[Geocentricist]

S’ is a reference frame, not an observer, so equidistant is a nonsequiter. There is no reference frame in which A and B are both at rest at the same time except the original reference frame for the times before acceleration starts. If you can’t see this, then you need to review the basics of relativity of simultaneity. You are making statements comparable to 1 + 1 = 3 and insisting they are right.

I didn't say A and B were both at rest in S' and I'm not aware anyone else did either. I thought everyone said S' is a frame moving inertially relative to S, which would of course mean A and B are moving inertially relative to S' when they are at rest in S.

I still maintain it seems A and B share a frame at all times and this means they also agree on what is simultaneous. What part of this sentence is wrong?

 

[PAllen]

I didn't say A and B were both at rest in S' and I'm not aware anyone else did either. I thought everyone said S' is a frame moving inertially relative to S, which would of course mean A and B are moving inertially relative to S' when they are at rest in S.

I still maintain it seems A and B share a frame at all times and this means they also agree on what is simultaneous. What part of this sentence is wrong?

Do you understand that in any frame S’, moving with respect to S, in which A is momentarily at rest at some time t, then B will not be at rest at that time t; and further, A and B will have started accelerating at different times in this frame?

As to your second question, all of it is wrong. That is what I am trying to get you to see by statements about S’. What do you think sharing a frame means? It isn’t standard usage, but I am guessing you think it means there is a frame in which they are both at rest at some given time. There is no such frame after acceleration begins.

 

[Geocentricist]

Do you understand that in any frame S’, moving with respect to S, in which A is momentarily at rest at some time t, then B will not be at rest at that time t; and further, A and B will have started accelerating at different times in this frame?

No.

As to your second question, all of it is wrong. That is what I am trying to get you to see by statements about S’. What do you think sharing a frame means? It isn’t standard usage, but I am guessing you think it means there is a frame in which they are both at rest at some given time. There is no such frame after acceleration begins.

I know A and B aren't at rest while they're accelerating, but they are still sharing the same accelerating frame.

 

[Ibix]

What's going on is the relativity of simultaneity. In S, the initial rest frame, the ships always have the same velocity at the same time, and that velocity is always changing. But any other frame, S', does not share the same definition of "at the same time". So in S' the ships are always moving at different speeds once at least one of them is accelerating, because what it calls "at the same time" is what S calls "at different times".

 

[pervect]

Ibix is right. To understand how special relativity consistently explains electromagnetism, one needs to understand special relativity. And how special relativity is consistent. This requires undestanding the whole of the theory, which includes understanding the relativity of simultaneity. There are a lot of threads on the topic, either under the name relativity of simlutaneity or "Einsteins train".

 

[A.T.]

since they always share the same frame

There is no frame (inertial or non-inertial) where all the rockets remain at rest, throughout the acceleration.

- There is a series of inertial frames, where all the rockets are instantaneously at rest, but along that series the distances between the rockets are increasing.

- There are non-inertial frames, where one of the rockets remains at rest throughout the acceleration, but the acceleration of the others is affected by position dependent time-dilation:
https://en.wikipedia.org/wiki/Rindler_coordinates

 

[Ibix]

There is a series of inertial frames, where all the rockets are instantaneously at rest, but along that series the distances between the rockets are increasing.

Perhaps I'm misunderstanding you, but I don't think that's right. There is a series of frames in which one or other of the rockets is always at rest, but the other one is always moving. That's why (or, at least, one way of conceptualising why) the distance is changing.

The other way of putting that is that the spacelike axes of the family of frames are not parallel, and it's only parallel to the spacelike axis of S that the separation is constant.

 

[A.T.]

Perhaps I'm misunderstanding you, but I don't think that's right. There is a series of frames in which one or other of the rockets is always at rest, but the other one is always moving. That's why (or, at least, one way of conceptualising why) the distance is changing.

I conceptualize as each inertial frame in that series seeing a different distance.

But either way, even if you construct a single frame where one rocket remains at rest during the acceleration, the other rocket does not remain at rest in that frame. And if you insist to know why they increase separation according to that frame, you have to take the position dependent time dilation in that frame into account.

 

[Geocentricist]

But either way, even if you construct a single frame where one rocket remains at rest during the acceleration, the other rocket does not remain at rest in that frame. And if you insist to know why they increase separation according to that frame, you have to take the position dependent time dilation in that frame into account.

I still don't see why the rockets aren't at rest with respect to each other throughout the acceleration. Everyone keeps repeating they aren't without really explaining the cause.

 

[PAllen]

I conceptualize as each inertial frame in that series seeing a different distance.

But either way, even if you construct a single frame where one rocket remains at rest during the acceleration, the other rocket does not remain at rest in that frame. And if you insist to know why they increase separation according to that frame, you have to take the position dependent time dilation in that frame into account.

I think this is wrong. Consider the events A has speed .7c, B has speed .7c as described in the original rest frame. These events are simultaneous in this frame. Consider ther frame moving at .7c relative to the original rest frame. These events are not simultaneous in this frame. Thus, when A is at rest in this frame, B is moving, and when B is at rest in this frame, A is moving.

We are not dealing here with any non inertial frames. The statement is that no inertial frame has both rockets at rest at the same time except the original inertial frame. Also, in SR, it is normal practice that frame refers to inertial frame unless one specifically discusses noninertial (and then, as you know, there are multiple valid ways to construct noninertial frames.)

 

[Ibix]

I still don't see why the rockets aren't at rest with respect to each other throughout the acceleration. Everyone keeps repeating they aren't without really explaining the cause.

I told you why in #89. They are the same distance apart in S because in that frame they are always traveling at the same speed at the same time. But no other frame has the same definition of "at the same time", so in any other frame the rockets are traveling at different speeds at the same time (by their definition of "the same time") so are always changing their separation.

 

[A.T.]

I think this is wrong. Consider the events A has speed .7c, B has speed .7c as described in the original rest frame. These events are simultaneous in this frame. Consider ther frame moving at .7c relative to the original rest frame. These events are not simultaneous in this frame. Thus, when A is at rest in this frame, B is moving, and when B is at rest in this frame, A is moving.

Yes, you and Ibix are right. There are no other inertial frames, where they are both at rest, even instantaneously.

We are not dealing here with any non inertial frames.

The OP want's to know what happens in the frame of a rocket throughout acceleration. That frame is non-inertial and has position dependent time dilation.

 

[PAllen]

The OP want's to know what happens in the frame of a rocket throughout acceleration. That frame is non-inertial and has position dependent time dilation.

Somehow, I think th OP is ill equipped to deal with this. Imagine trying to explain why a non inertial frame for rocket A can always include B, but for a noninertial frame for B, A cannot be included after some time (it falls below the Rindler horizon for B).

 

[bob012345]

Yes, you and Ibix are right. There are no other inertial frames, where they are both at rest, even instantaneously.The OP want's to know what happens in the frame of a rocket throughout acceleration. That frame is non-inertial and has position dependent time dilation.

Let me jump in and ask if m getting the point correctly. Since there is no size requirement on what constitutes a valid reference frame, that's defined by velocity, so distance separated events along the direction of relative motion within one reference frame, S, can be seen as non simultaneous in another frame, S', even if they are always moving at the same velocity within frame S?

 

[PAllen]

Let me jump in and ask if m getting the point correctly. Since there is no size requirement on what constitutes a valid reference frame, that's defined by velocity, so distance separated events along the direction of relative motion within one reference frame, S, can be seen as non simultaneous in another frame, S', even if they are always moving at the same velocity within frame S?

Correct.

 

[bob012345]

Correct.

Thanks. So if the only separation were 90 degrees to the direction of motion, simultaneous events in S could be simultaneous in S'?

 

[PAllen]

Thanks. So if the only separation were 90 degrees to the direction of motion, simultaneous events in S could be simultaneous in S'?

Correct.

 

[jartsa]

I still don't see why the rockets aren't at rest with respect to each other throughout the acceleration. Everyone keeps repeating they aren't without really explaining the cause.

Well one thing is that the motor of the trailing rocket burns fuel at slowed down rate. And why does that happen? The vicinity of the Rindler-horizon causes that kind of things. Rindler-horizon is like event horizon of a black hole, in every way, for an accelerating observer, for inertial observers it does not exist.

What do you think is the situation for the observers in the rockets after a long time, if both rockets run out fuel after a quite short time?

 

[A.T.]

I still don't see why the rockets aren't at rest with respect to each other throughout the acceleration. Everyone keeps repeating they aren't without really explaining the cause.

It was explained to you pages ago for the electrons:

If you actually attach a frame to an electron, during it's acceleration, you have time running at different rates along the wire in that frame. This effectively delays the electrons in the back, and speeds up the electrons in the front, so their distances increase.

 

[Geocentricist]

Yes, you and Ibix are right. There are no other inertial frames, where they are both at rest, even instantaneously.The OP want's to know what happens in the frame of a rocket throughout acceleration. That frame is non-inertial and has position dependent time dilation.

A and B share a frame at all times. There may be time dilation in their frame relative to some other frame but they cannot experience time dilation relative to their own frame, so there is still no cause for their separation to increase.