Is the asymmetry mentioned in 1905 SR paper fully removed?

[GregAshmore]

In his 1905 paper introducing special relativity, Einstein calls attention to the asymmetry in the classical treatment of the relative motion of a magnet and a conductor:

For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case.

In a previous post, it was explained to me that, when the conductor is moving, the electromotive force has "in itself ... no corresponding energy" because the integral of work along the path of the moving conductor cannot always be resolved to a scalar quantity. The integral of the work done along the path of the moving magnet, on the other hand, does resolve to a scalar in all cases. Hence the asymmetry. (I don't understand the math well enough to understand why the asymmetry arises. If in my ignorance I've butchered the explanation somewhat, my apologies to BCrowell.)

Einstein claims to have eliminated this asymmetry by transforming the event to the frame of the moving conductor. In that frame, the force on the charge is due to the electric field, which, as noted, has "a certain definite energy."

It seems to me that the asymmetry has been only partially removed. I explain in the following what I mean by "partially removed."

Here is Einstein on the removal of the asymmetry:

If a unit electric point charge is in motion in an electromagnetic field, the force acting upon it is equal to the electric force which is present at the locality of the charge, and which we ascertain by transformation of the field to a system of co-ordinates at rest relatively to the electrical charge. (New manner of expression.)

The analogy holds with “magnetomotive forces.” We see that electromotive force plays in the developed theory merely the part of an auxiliary concept, which owes its introduction to the circumstance that electric and magnetic forces do not exist independently of the state of motion of the system of co-ordinates.

Furthermore it is clear that the asymmetry mentioned in the introduction as arising when we consider the currents produced by the relative motion of a magnet and a conductor, now disappears.

In essence, the asymmetry is removed by always considering the charge to be at rest. This way, the force is always due to the electric field, and there is consequently always "a certain definite energy" associated with the force.

First, the success of the theory in establishing that the force can be legitimately considered as due to an electric field must be recognized. Prior to Einstein, it was not possible to consider the moving electric charge to be at rest, because it was the movement relative to the ether which generated all electromagnetic effects. Movement relative to the ether is an absolute fact; it cannot be transformed away. With the elimination of the ether, only the relative motion of magnet and charge come into play, and it is legitimate to consider the conductor to be at rest.

It is the rest of the claim that I am having trouble accepting. It is not at all clear to me (as Einstein says it should be) that the asymmetry has been removed. Rather, it seems to me that the theory establishes the rest frame of the conductor as the preferred frame.

According to the principle of relativity, as declared by Einstein himself, the laws of nature should be of the same form for all observers. Yet, as I understand the text above, the event as seen by the charge at rest is of a different form than the event as seen by the magnet at rest. The need for the concept of electromotive force is driven by this difference in form.

Further, Einstein says that the view from the rest frame of the charge is the true view, while the view from the rest frame of the magnet is affected by the motion of that frame relative to the frame of the charge.

Comments?

 

[Dale]

In essence, the asymmetry is removed by always considering the charge to be at rest.

No. The asymmetry is removed by the derived transformation equation for the fields. You don't have to consider the charge to be at rest, although you certainly can use the frame where it is at rest if you wish.

It is the rest of the claim that I am having trouble accepting. It is not at all clear to me (as Einstein says it should be) that the asymmetry has been removed. Rather, it seems to me that the theory establishes the rest frame of the conductor as the preferred frame.

That seems like a rather untenable interpretation. What would you do if you had two identical conductors moving relative to each other? Which would you designate as preferred?

According to the principle of relativity, as declared by Einstein himself, the laws of nature should be of the same form for all observers. Yet, as I understand the text above, the event as seen by the charge at rest is of a different form than the event as seen by the magnet at rest. The need for the concept of electromotive force is driven by this difference in form.

The laws of nature are the differential equations governing the behavior of a physical system. Here we are specifically talking about Maxwell's equations. In what way do you think that the form of Maxwell's equations is different in the frame of the charge or in the frame of the magnet?

Further, Einstein says that the view from the rest frame of the charge is the true view

Nonsense. He said no such thing.

 

[GregAshmore]

No. The asymmetry is removed by the derived transformation equation for the fields. You don't have to consider the charge to be at rest, although you certainly can use the frame where it is at rest if you wish.

That's not what Einstein says. He says that the force is ascertained by transforming to the frame in which the charge is at rest. "...the force acting upon it is equal to the electric force which is present at the locality of the charge, and which we ascertain by transformation of the field to a system of co-ordinates at rest relatively to the electrical charge."

That seems like a rather untenable interpretation. What would you do if you had two identical conductors moving relative to each other? Which would you designate as preferred?

To follow the instructions from Einstein, one would consider each conductor separately, transforming the field to the rest frame of each conductor.

The laws of nature are the differential equations governing the behavior of a physical system. Here we are specifically talking about Maxwell's equations. In what way do you think that the form of Maxwell's equations is different in the frame of the charge or in the frame of the magnet?

In the general form, Maxwell's equations include a term which indicates the electromotive force on a moving charge. When that term is non-zero, there exists the asymmetry which troubled Einstein. The transformation to the frame of the charge makes that term go to zero. The upshot of Einstein's argument is that the electromotive force term is not quite valid. The valid form of the equation is without the electromotive force term.

Nonsense. He said no such thing.

Not in those words, no. If I were speaking to him directly, I would not have been so free with my words. But I would suggest to him that what he says amounts to the same thing as saying that the rest frame is the true frame, insofar as ascertaining the electrical force on the charge is concerned. His meaning is clear: if one wants to know the electrical force on the charge, one must transform the field to the rest frame of the charge. In any other frame, one must introduce the notion of electromotive force, which puts one in precisely the situation which Einstein complained about in the introduction of the paper.

 

[Dale]

To follow the instructions from Einstein, one would consider each conductor separately, transforming the field to the rest frame of each conductor.

If you do the same thing for both frames then neither frame is prefered.

In the general form, Maxwell's equations include a term which indicates the electromotive force on a moving charge. When that term is non-zero, there exists the asymmetry which troubled Einstein. The transformation to the frame of the charge makes that term go to zero.

None of that represents a change in the laws of physics between frames.

The upshot of Einstein's argument is that the electromotive force term is not quite valid. The valid form of the equation is without the electromotive force term.

This is certainly not the upshot of his argument. It is not even a remote logical consequence of his argument. In fact, the implication of his argument is almost exactly opposed to your assertion here. The implication of his argment is that the standard form of Maxwell's equations, including the EMF term, is valid in every frame, provided the frames are related by the derived transforms.

Not in those words, no.

Then don't pretend that he did. Appeal to authority is not a valid form of argument anyway, and when you try to use it by completely misquoting the authority it destroys any vestige of credibility that you may have. Not only are you presenting an illogical argument, you are not being factual in the presentation.

 

[GregAshmore]

If you do the same thing for both frames then neither frame is prefered.

I see that I misread your question. I took it to mean that you were asking about two conductors moving in the vicinity of the magnet. My response was directed to that case.

To respond to your actual question, I am not asserting that Einstein asserts that one or the other of the conductors is truly at rest, and the other moving. I am asserting that Einstein asserts that for each conductor, and in general for any charge, there is only one frame in which the force on the charge can be determined without appeal to a force "to which in itself there is no corresponding energy". That one frame is the rest frame of the charge. That is why Einstein says that the force on the charge is to be determined by transforming the field to the rest frame of the charge. In that limited sense, I (not Einstein) assert that the rest frame of a charge is the preferred frame for determining the force on the charge.

None of that represents a change in the laws of physics between frames.

Yes, it does, in the limited sense of considering whether the force on a charge is one "to which in itself there is no corresponding energy". In one frame only, the equation for the force on the charge has just one term, the term for the electric field. In all other frames, there is another term, for the electromotive force.

This is certainly not the upshot of his argument. It is not even a remote logical consequence of his argument. In fact, the implication of his argument is almost exactly opposed to your assertion here. The implication of his argment is that the standard form of Maxwell's equations, including the EMF term, is valid in every frame, provided the frames are related by the derived transforms.

Of course Maxwell's equations are valid in all frames; that is not merely implied by Einstein, it is explicitly stated. In all frames, the equations correctly predict the current which rises in the conductor. But in all frames except one, those equations include a term for a force to which in itself there is no corresponding energy. For that reason, the force is not quite valid. If it were without fault, Einstein would not have objected to it in the introduction, nor would he call it an auxiliary concept in the body of the paper.

Then don't pretend that he did.

I am sorry to have let myself be undisciplined, and put words in someone else's mouth.

Appeal to authority is not a valid form of argument anyway, and when you try to use it by completely misquoting the authority it destroys any vestige of credibility that you may have. Not only are you presenting an illogical argument, you are not being factual in the presentation.

Perhaps you believe that I am claiming that the rest frame of the charge is preferred in the sense of being absolutely at rest. In that case, my argument would be illogical and completely at odds with the text of Einstein's paper. I am not making that case. I am asserting that, by declaring that the force on a charge is to be determined by transforming the field to the rest frame of the charge, and by further declaring that in all other frames there is present in Maxwell's equations a term for the auxiliary concept of electromotive force, Einstein is in effect establishing the rest frame of the charge as the preferred frame, for the limited purpose of establishing the force on the charge. I believe that this is a logical argument, soundly based on the text.

 

[Dale]

I see that I misread your question. I took it to mean that you were asking about two conductors moving in the vicinity of the magnet. My response was directed to that case.

Actually, the point I was making is a little more basic than that. For a frame to be "preferred" in the physical sense means that the laws of physics must be different in that frame than in other frames. Furthermore, those laws of physics must be able to describe all objects in that one preferred frame. So if the preferred frame is the one where one conductor is at rest then in the preferred frame the other conductor is moving. So you still need your laws of physics to be able to deal with moving conductors because you cannot have a preferred frame where all possible conductors are at rest.

I am not asserting that Einstein asserts that one or the other of the conductors is truly at rest, and the other moving. I am asserting that Einstein asserts that for each conductor, and in general for any charge, there is only one frame in which the force on the charge can be determined without appeal to a force "to which in itself there is no corresponding energy". That one frame is the rest frame of the charge.

This is correct.

That is why Einstein says that the force on the charge is to be determined by transforming the field to the rest frame of the charge. In that limited sense, I (not Einstein) assert that the rest frame of a charge is the preferred frame for determining the force on the charge.

This does not follow, unless by "preferred" you simply mean "I personally prefer it". It is not "preferred" in any physical sense. There is one force law, and that same force law can be used in all frames to correctly determine the force on the charge.

Yes, it does, in the limited sense of considering whether the force on a charge is one "to which in itself there is no corresponding energy". In one frame only, the equation for the force on the charge has just one term, the term for the electric field. In all other frames, there is another term, for the electromotive force.

The term is there in all frames.

Consider an example with a magnet and two charges, A and B, all moving relative to each other. There is one law which governs the force on both charges in all frames, the Lorentz force law. In the magnet's frame they are both moving, so you need the v term in the magnet's frame. In A's frame B is moving, so you still need the v term in A's frame. In B's frame A is moving, so you still need the v term in B's frame.

There is no frame in which the correct force law lacks a v term.

But in all frames except one, those equations include a term for a force to which in itself there is no corresponding energy. For that reason, the force is not quite valid.

Nonsense. It is still a perfectly valid force. There are lots of valid forces without a corresponding energy. Friction, for example.

I am asserting that, by declaring that the force on a charge is to be determined by transforming the field to the rest frame of the charge, and by further declaring that in all other frames there is present in Maxwell's equations a term for the auxiliary concept of electromotive force, Einstein is in effect establishing the rest frame of the charge as the preferred frame, for the limited purpose of establishing the force on the charge. I believe that this is a logical argument, soundly based on the text.

The term is there in all frames, including the frame where a given charge is at rest. See the example above.

 

[GregAshmore]

Actually, the point I was making is a little more basic than that. For a frame to be "preferred" in the physical sense means that the laws of physics must be different in that frame than in other frames. Furthermore, those laws of physics must be able to describe all objects in that one preferred frame. So if the preferred frame is the one where one conductor is at rest then in the preferred frame the other conductor is moving. So you still need your laws of physics to be able to deal with moving conductors because you cannot have a preferred frame where all possible conductors are at rest.

Okay. I see that "preferred frame" is a technical term, and that I have misused it.

This does not follow, unless by "preferred" you simply mean "I personally prefer it". It is not "preferred" in any physical sense. There is one force law, and that same force law can be used in all frames to correctly determine the force on the charge.

The force on the charge can be calculated in all frames using the Maxwell/Lorentz equations. The practical effect on the charge indicated by the calculation will be the same in all frames. But the force is not of the same character in all frames. In every frame except the rest frame of the charge, the force is one "to which there is no corresponding energy". Only in the rest frame of the charge does the force have "a certain definite energy". Einstein considers this distinction to be of importance. And, to use your term of expression, he prefers the force in the rest frame.

The term is there in all frames.

It would not be there if the equation were developed from observations made by an observer on the charge. That observer would rightly object to its introduction into the equation.

Consider an example with a magnet and two charges, A and B, all moving relative to each other. There is one law which governs the force on both charges in all frames, the Lorentz force law. In the magnet's frame they are both moving, so you need the v term in the magnet's frame. In A's frame B is moving, so you still need the v term in A's frame. In B's frame A is moving, so you still need the v term in B's frame.

True. But according to Einstein, "the force acting upon it [the charge] is equal to the electric force which is present at the locality of the charge, and which we ascertain by transformation of the field to a system of co-ordinates at rest relatively to the electrical charge." To correctly (that is, according to the instructions given by Einstein) calculate the force on each charge in the system, one must transform the field to the rest frame of the charge.

There is no frame in which the correct force law lacks a v term.

See above.

Nonsense. It is still a perfectly valid force. There are lots of valid forces without a corresponding energy. Friction, for example.

Granted that whether Einstein would use the word "valid" or not is a matter of speculation. The clear sense of the text is that he considers the force which has "a certain definite energy" to be somehow better than the force "to which there is no corresponding energy". He instructs us to calculate the force based on the field in the rest frame. He calls the force in every other frame an auxiliary concept.

 

[Dale]

Einstein considers this distinction to be of importance. And, to use your term of expression, he prefers the force in the rest frame.

OK, no problem. You have a personal preference which coincides with Einstein's personal preference. Neither personal preference represents a preferred frame in the standard physics sense.

It would not be there if the equation were developed from observations made by an observer on the charge. That observer would rightly object to its introduction into the equation.

No, it would still be there for such an observer. They would require that term to explain the forces they observe acting on other charges which are moving wrt them.

To correctly (that is, according to the instructions given by Einstein) calculate the force on each charge in the system, one must transform the field to the rest frame of the charge.

No. One "may" transform the field to the rest frame of the charge. The word "must" is incorrect. That is essentially the point of the first postulate of relativity. You are free to use any frame you like.

Granted that whether Einstein would use the word "valid" or not is a matter of speculation. The clear sense of the text is that he considers the force which has "a certain definite energy" to be somehow better than the force "to which there is no corresponding energy". He instructs us to calculate the force based on the field in the rest frame. He calls the force in every other frame an auxiliary concept.

Sure, but again, this is not "prefered" in the physics sense, nor is it invalid.

 

[pervect]

The total symmetry is, I think, manifest if one writes the equations using tensor notation.

I'll give a brief exposition and encourage the OP to look into the tensor formulation (background permitting) as it should answer his question. For a fuller explanation, any E&M textbook that covers tensors should suffice, I'd suggest Griffiths.

I apologize in advance if the material is too advanced, but I thought I should at least mention it.

Proceeding, one can, then write the usual non-relativistic force law

f = dp/dt = q(e+v x B)

using tensors as

$f\prime^a = dp/d\tau = F^{a}{}_{b} u^b$

here $f\prime$ is the f-fource, the 4-vector equivalent of the 3-fource f. u is the 4-velocity, the 4-vector equivalent of the 3-velocity v. $F^{a}{}_{b}$ is the Faraday tensor, which combines both the electric and magnetic fields into a single unified geometric entity.

F can be thought of more or less as a 4x4 matrix, though in tensor terms it would be called a rank 2 tensor.

There's a brief description of F in the wiki at http://en.wikipedia.org/w/index.php?title=Electromagnetic_tensor&oldid=520946298

As this is a tensor equation, summation over the repeated index is implied, i.e. the above equation is actually 4 equations, one for each of the 4 values of a, giving the 4 components of f', and each equation is the sum of four terms , the sum of b=0,1,2,3.

All of the entites, f', u, and F, transform as tensors should, the equation is symmetrical, and neither E nor B is singled out for special treatment, being combined into the single gemoetric entity, the Faraday tensor.

 

[Dale]

Greg, one quick comment before I call it a night:

I think you are getting too caught up in small details of the wording of one seminal paper. Much more important than that is the clear and unambiguous math. It is mathematically clear that Maxwells equations are not invariant under the Galilean transform. It is also mathematically clear that they are invariant under the Lorentz transform. Those two clear mathematical facts should resolve any confusion or ambiguity introduced by the English.

 

[GregAshmore]

Greg, one quick comment before I call it a night:

I think you are getting too caught up in small details of the wording of one seminal paper. Much more important than that is the clear and unambiguous math. It is mathematically clear that Maxwells equations are not invariant under the Galilean transform. It is also mathematically clear that they are invariant under the Lorentz transform. Those two clear mathematical facts should resolve any confusion or ambiguity introduced by the English.

I do see that the Maxwell equations are invariant under the Lorentz translation. I have sort-of-known that for while; meaning that I have read about it, and what I read made sense to me. Reading the Maxwell paper and stumbling for a bit as I tried to work out the meaning of the equations, until it dawned on me that the equations as he wrote them are all with reference to the ether, solidified what I had sort-of-known.

I agree that the issue of whether the emf on a moving charge is an "asymmetry" or an "auxiliary concept" is a small detail in the large scheme of special relativity. The reason I have pursued it is that there have been many times, in my journey toward understanding special relativity, that I have not followed up on a "small detail", only to find out later that it in skipping over the detail I failed to uncover a major misconception on my part. Relativity is not at all intuitive; I can't trust my normal reasoning processes to get me to the correct answer.

Then, too, discussing these issues with people who know what they are doing gives me a chance to learn some things that are not going to be covered in a popular treatment of relativity. And it gives me a chance to learn (sometimes the hard way) to be more disciplined in my thinking.

My biggest problem re relativity at this point is that I do not have the mathematical tools to process the technically oriented books. Unfortunately, I have to prioritize my effort, and learning the math is a couple of levels down in priority. Understanding relativity is part of a larger project, which is mostly metaphysical. Where the project touches on physics, I must have my facts straight. But, much as I think I would enjoy the exercise, mastering the math is not top priority. (I figure it would take me, including GR, two years as a full-time student, and perhaps triple that in my spare time. I bought some books 20 months ago to start; then I broke my hip in a cycling accident and lost eight months.)

 

[GregAshmore]

The total symmetry is, I think, manifest if one writes the equations using tensor notation.

I'll give a brief exposition and encourage the OP to look into the tensor formulation (background permitting) as it should answer his question. For a fuller explanation, any E&M textbook that covers tensors should suffice, I'd suggest Griffiths.

I apologize in advance if the material is too advanced, but I thought I should at least mention it.

Proceeding, one can, then write the usual non-relativistic force law

f = dp/dt = q(e+v x B)

using tensors as

$f\prime^a = dp/d\tau = F^{a}{}_{b} u^b$

here $f\prime$ is the f-fource, the 4-vector equivalent of the 3-fource f. u is the 4-velocity, the 4-vector equivalent of the 3-velocity v. $F^{a}{}_{b}$ is the Faraday tensor, which combines both the electric and magnetic fields into a single unified geometric entity.

F can be thought of more or less as a 4x4 matrix, though in tensor terms it would be called a rank 2 tensor.

There's a brief description of F in the wiki at http://en.wikipedia.org/w/index.php?title=Electromagnetic_tensor&oldid=520946298

As this is a tensor equation, summation over the repeated index is implied, i.e. the above equation is actually 4 equations, one for each of the 4 values of a, giving the 4 components of f', and each equation is the sum of four terms , the sum of b=0,1,2,3.

All of the entites, f', u, and F, transform as tensors should, the equation is symmetrical, and neither E nor B is singled out for special treatment, being combined into the single gemoetric entity, the Faraday tensor.

Well, it is advanced for me because I do not know tensors. I do know linear algebra and differential equations, so I have some sense of what is going on. And I have read that tensors are independent of coordinate system, in the sense that the value of the tensor is the same in all coordinate systems. So I think I get the gist.

I find it curious, then, that Einstein makes a distinction in the 1905 paper between the force on the charge in its rest frame and the force in other frames. The one is due to E only; the other is some combination of E and v × B. That (in more precise terms than I originally framed my thoughts) is why I decided to ask the question in the original post. Given that the Faraday tensor makes no geometric distinction between the E and B fields, why does Einstein say that the emf due to velocity in the B field is an auxiliary concept, appearing only because (paraphrasing) the electric and magnetic fields are not independent of the movement of the frames?

 

[Dale]

IMy biggest problem re relativity at this point is that I do not have the mathematical tools to process the technically oriented books. Unfortunately, I have to prioritize my effort, and learning the math is a couple of levels down in priority. Understanding relativity is part of a larger project, which is mostly metaphysical. Where the project touches on physics, I must have my facts straight. But, much as I think I would enjoy the exercise, mastering the math is not top priority. (I figure it would take me, including GR, two years as a full-time student, and perhaps triple that in my spare time. I bought some books 20 months ago to start; then I broke my hip in a cycling accident and lost eight months.)

Tensors are a bit challenging if you really need GR. But if you are only interested in SR then you can learn the math very easily. I would estimate less than a week if you already know algebra and matrices and vectors.

Personally, I never understood SR until I learned the mathematical framework of four-vectors. It is very simple, and once I learned it everything "clicked" almost immediately.

 

[Sugdub]

I think the question raised in this thread is important, however I don't share the views expressed so far. What does the "asymmetry" noticed by Einstein consist in? It relates to the fact that according to the former electric and magnetic theories, a change of the reference frame affects the description of the EM field whereas it leaves invariant the relative speed between the magnet and the conductor. Here lies the asymmetry. In other words, the description of the EM field provided by the former theories is dependent on something else than the relative speed between both items, and in this context the only candidates are the velocities of both items in respect to the reference frames. This infringes the principle of relativity of motion: the cause of the observed current should depend only on the relative motion of both items in respect to each other.
The only way to remove or resolve this asymmetry consists in proposing a new EM theory in which the change of reference frame leaves the expression of the EM field invariant, because it leaves invariant the relative motion between the magnet and the conductor... But this is not what Einstein proposes.
Indeed in his 1905 SR paper Einstein invokes his new SR theory to state that the EM field (E,B) corresponding to the reference frame where the conductor is at rest can be transformed, using the Lorentz transformation derived in the SR context, into (E',B') corresponding to the same EM field as described in the reference frame where the magnet is at rest. Both descriptions lead to the same prediction concerning the observed current. So Einstein's strategy consists in demonstrating that the descriptions of the EM field provided by the former theories are equivalent in spite of their apparent incompatibility... But the "asymmetry" is still there.

 

[Dale]

Here lies the asymmetry. In other words, the description of the EM field provided by the former theories is dependent on something else than the relative speed between both items, ... But the "asymmetry" is still there.

No, it isn't. What do you think still depends on something other than relative speed?

 

[GregAshmore]

Tensors are a bit challenging if you really need GR. But if you are only interested in SR then you can learn the math very easily. I would estimate less than a week if you already know algebra and matrices and vectors.

Personally, I never understood SR until I learned the mathematical framework of four-vectors. It is very simple, and once I learned it everything "clicked" almost immediately.

A bargain at twice the time. Can you recommend a book or on-line tutorial?

[later] I went back to a 2010 post and saw that pervect recommended Taylor and Wheeler's Spacetime Physics because it has a four-vector approach. I bought that book then, and was partially through it when I broke my hip. I didn't remember it as particularly math-intensive; didn't think of it for four-vectors. I'll get back into it now.

 

[Dale]

A bargain at twice the time. Can you recommend a book or on-line tutorial?

[later] I went back to a 2010 post and saw that pervect recommended Taylor and Wheeler's Spacetime Physics because it has a four-vector approach. I bought that book then, and was partially through it when I broke my hip. I didn't remember it as particularly math-intensive; didn't think of it for four-vectors. I'll get back into it now.

I don't have Taylor and Wheeler, but I have heard good things about it. I stumbled on 4-vectors at hyperphysics, and I still consider it a good intro:

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/vec4.html

Then, of course, there are a bunch of good Wikipedia pages:
http://en.wikipedia.org/wiki/Four-vector
http://en.wikipedia.org/wiki/Four-velocity
http://en.wikipedia.org/wiki/Four-momentum

You also need to know about spacetime intervals:
http://en.wikipedia.org/wiki/Spacetime#Basic_concepts
http://en.wikipedia.org/wiki/Minkowski_Space

And spacetime diagrams will help too:
http://en.wikipedia.org/wiki/Minkowski_diagram
http://en.wikipedia.org/wiki/World_line

There are also many good university pages, but many of them seem to have a tendency to jump into tensor notation which can be a little intimidating at first. So I personally tended to stick with less formal sources in the beginning.

 

[pervect]

Taylor has the first chapter of the older 1965 editio of "Spacetime Physics" available for dowload at his website:

http://www.eftaylor.com/special.html

Probably not enough to get the complete story, but enough to see if you like the book.

 

[GregAshmore]

Thank you both.

 

[Sugdub]

No, it isn't. What do you think still depends on something other than relative speed?

Your question pushed me into second thoughts and I must withdraw my previous statement. Indeed a change of reference frame is a purely mental act by a theoretician who describes the same experiment in two different ways: it cannot affect the physical quantities involved in the experimental setup. However, according to Einstein's paper, E and B should no longer be considered as genuine physical quantities, but somehow "components" of a physical entity at an higher level, the EM field. Thus, and contrary to my previous statement, it is no longer a contradiction that their value changes through the swap between two reference frames, provided the overall EM field itself remains invariant. Then it is correct to state that the "asymmetry" has been removed by establishing the equivalence of (E,B) and (E',B') .
But still there is one aspect which puzzles me: according to SR the impact of applying the Lorentz transformation to (E,B) is negligible for small values of v (in respect to c) envisaged here, so that (E',B') should be nearly identical to (E,B). How can one expect that the Lorentz transformation will justify the equivalence of both descriptions at stake? I hope you can explain this otherwise Einstein's strategy seems to fall apart.

 

[Dale]

However, according to Einstein's paper, E and B should no longer be considered as genuine physical quantities, but somehow "components" of a physical entity at an higher level, the EM field.

Correct. This "physical entity at an higher level" is called the Electromagnetic tensor or the Faraday tensor. For more details see:
http://en.wikipedia.org/wiki/Electromagnetic_tensor

But still there is one aspect which puzzles me: according to SR the impact of applying the Lorentz transformation to (E,B) is negligible for small values of v

I don't think this statement is true in general. In fact, I don't think I can come up with an example where it is true.

What you are probably thinking is that the difference between the Galilean transform and the Lorentz transform is small for small values of v. But since Maxwell's equations are not invariant under the Galilean transform that fact cannot be used for studying EM.

 

[Sugdub]

What you are probably thinking is that the difference between the Galilean transform and the Lorentz transform is small for small values of v.

Your answer convinced me. Thanks. A last question to close the loop if you don't mind. As far as I understand SR deals with "observers" performing analog measurements in different experimental conditions, thereby respectively obtaining different results for space and time quantities. The Lorentz transformation explains the divergence of their results.
But in the context of the moving magnet and conductor problem, one is dealing with one single experiment. The variation of the EM field components is purely mental, it cannot be traced to any effective measurement. The Lorentz transformation explains the equivalence of theoretical views, it does not relate to observers who actually perform measurements. So what is the rationale for invoking SR and its Lorentz transformation for resolving the problem at stake?

 

[Dale]

As far as I understand SR deals with "observers" performing analog measurements in different experimental conditions, ...
But in the context of the moving magnet and conductor problem, one is dealing with one single experiment.

Your second case is actually more common. It is the basis for all of the traditional problems/paradoxes and a huge volume of homework problems. They take the same scenario and describe it in two different reference frames.

The Lorentz transformation explains the equivalence of theoretical views, it does not relate to observers who actually perform measurements.

The first statement is correct, but the second is not correct and it certainly does not follow from the first. You can use the Lorentz transform to do the theoretical analysis in any reference frame that you choose, whether or not there is an actual "observer" at rest in that frame. However, if you do have an actual observer doing some measurement of some frame variant quantity, then that observer must select a frame to do the measurement with respect to, and the Lorentz transform would be able to relate the measurements of observers which measured the same quantity wrt different frames.

 

[GregAshmore]

I don't have Taylor and Wheeler, but I have heard good things about it. I stumbled on 4-vectors at hyperphysics, and I still consider it a good intro:

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/vec4.html

Then, of course, there are a bunch of good Wikipedia pages:
http://en.wikipedia.org/wiki/Four-vector
http://en.wikipedia.org/wiki/Four-velocity
http://en.wikipedia.org/wiki/Four-momentum

You also need to know about spacetime intervals:
http://en.wikipedia.org/wiki/Spacetime#Basic_concepts
http://en.wikipedia.org/wiki/Minkowski_Space

And spacetime diagrams will help too:
http://en.wikipedia.org/wiki/Minkowski_diagram
http://en.wikipedia.org/wiki/World_line

There are also many good university pages, but many of them seem to have a tendency to jump into tensor notation which can be a little intimidating at first. So I personally tended to stick with less formal sources in the beginning.

I worked through chapter 7 of Taylor Wheeler today, on the momentum-energy four-vector. This will be a good prep for frame-to-frame transformations using the four-vector. A lot of rust will shake loose then; I haven't had to use vectors in matrix form since college. I used force vectors regularly in machine design, but not in matrix form.

As a side note, I find Taylor-Wheeler to be extremely annoying in the presentation. I feel exactly the same way I do when I read a theological article that is doing more selling than logical exposition.

 

[Jano L.]

Greg,
do not worry about complicated math and tensors at first. They are not that important to understand the basic ideas of relativity. They can be learned later and more easily then. Actually, I found that knowing relativity first was a good prerequisite to learn the basic algebra and analysis of tensors.

First, you need to get some good books. There are many good books, but more bad ones. I recommend for example

R. Resnick, Introduction to Special Relativity, Wiley, 1968

R. Katz, Introduction to the Theory of Special Relativity, van Nostrand, 2000

D. Bohm, Special Theory of Relativity, Routledge, 1996

The first two books discuss, among other things, how to transform quantities like momentum or electric field from one frame to another, but beware, it is not the first thing to start with.

The third book presents concise exposition of the subject closer to history, with some interesting non-mathematical reflections. I think you may find them interesting and helpful, as you indicated you are interested in metaphysics as well.

And it gives me a chance to learn (sometimes the hard way) to be more disciplined in my thinking.

Indeed, this was my feeling too when learning relativity. For example, try to derive the Lorentz transformation from the simple length contraction and time dilatation; it is first very difficult and the goal slips away as the old thinking creeps in, but after one tries few times, it gets better and better until one gets it. It is a great exercise in reasoning.

 

[Sugdub]

Your second case is actually more common. It is the basis for all of the traditional problems/paradoxes and a huge volume of homework problems. They take the same scenario and describe it in two different reference frames.

I'm afraid you didn't get my point. I will develop it below. The magnet and conductor problem deals with one single experiment described alongside two reference frames. A change of reference frame leaves invariant the relative speed between any pair of physical objects. Typically the list of those physical items which are considered "at rest" has changed. It is a purely mental act. There is no way it will induce a change of any physical quantity and neither of the outcome of any measurement. The issue at stake is whether and why SR qualifies for being invoked to help demonstrating the equivalence of both descriptions of the EM field.
SR assumes that two "observers" obtain different outcomes when performing analogous measurements of space and/or time quantities. Here we are dealing with reproducible experiments, therefore one must assume that both observers actually do not operate in the same experimental conditions: it must be possible to trace the divergence of experimental outcomes back to a statement which holds for one observer about its experimental conditions but does not hold for the second observer. Due to its full symmetry, the "relative motion" between observers, which holds for both, does not fulfill the need. According to Einstein's SR paper in 1905, the experimental conditions differ insofar one observer is at rest in respect to the target object of the measurement, whereas the second observer is in motion in respect to it: their relative speed in respect to the target object is not the same (and this obviously implies a relative motion between observers). In SR the swap from one experimental context to the other one cannot be considered formally as a change of reference frame, since the relative speed between the observer and the target object changes from nil to v (a mental act cannot achieve that).
There appears to be a confusion between a change of reference frame (EM paradigm) and a change of experimental conditions which affects the relative speed between a pair of physical objects (SR paradigm). There appears to be a confusion between the explanation of the divergence of two theoretical descriptions for the same experiment (EM paradigm) and the explanation for the divergence of experimental results delivered by two experiments (SR paradigm). On the one hand two theoreticians producing different descriptions of the same thing and on the other hand two observers who perform different experiments. Indeed the confusion between both paradigms will pave the way to many paradoxes.
Assuming that in SR the Lorentz transformation deals with the difference of measurement outcomes in response to a change of experimental conditions which does not qualify as a change of reference frame (this is the SR paradigm), I'm challenging the view that it would be legitimate, without further justification, to invoke SR to deal with a change of a theoretical description (purely mental) in response to a genuine change of reference frame (this is the EM paradigm). Overall the conditions which prevailed for the derivation of the Lorentz transformation in the SR paradigm do not match at all the conditions which prevail in the EM paradigm.
Hence my question: what is the rationale for invoking SR and its Lorentz transformation for resolving the EM problem at stake?

 

[Dale]

I'm afraid you didn't get my point.

No, I got your point.

The magnet and conductor problem deals with one single experiment described alongside two reference frames. A change of reference frame leaves invariant the relative speed between any pair of physical objects. Typically the list of those physical items which are considered "at rest" has changed. It is a purely mental act. There is no way it will induce a change of any physical quantity and neither of the outcome of any measurement.

Most of this is fine, but you have to be careful about the statement "there is no way it will induce a change of any physical quantity". If you make that statement then you are essentially defining "physical quantity" in such a way that all physical quantities must be frame invariant. There is nothing wrong with that, but then you will find that some very useful quantites, such as energy, are not "physical quantities" as you have defined the term.

SR assumes that two "observers" obtain different outcomes when performing analogous measurements of space and/or time quantities.

No. SR assumes that the two postulates hold. That is it. Everything else is derived.

Here we are dealing with reproducible experiments, therefore one must assume that both observers actually do not operate in the same experimental conditions

I disagree. Why must one assume that?

As you point out elsewhere, a reference frame is a purely mental construct. So you can take any single physical experiment and analyze it as many times as you like using a different reference frame each time. You do not need to vary the experimental conditions or even actually run the experiment again; you simply re-analyze it.

In SR the swap from one experimental context to the other one cannot be considered formally as a change of reference frame, since the relative speed between the observer and the target object changes from nil to v (a mental act cannot achieve that).

I think that this is a reasonable objection that you bring. It is due to some highly unfortunate and (IMO) sloppy terminology.

In the usual SR thought experiments or homework problems it is common (but not universal) to use the term "observer" as shorthand for "reference frame where a hypothetical observer is at rest". Generally SR "observers" assign coordinates to events, but do not have any location, mass, fields, charge, energy, etc. The "observer" in SR is a purely mental construct that does not participate in the experiment at all (unlike an "observer" in QM). You can postulate as many such observers as you like, one at rest wrt the target object and another with relative speed v.

Hence my question: what is the rationale for invoking SR and its Lorentz transformation for resolving the EM problem at stake?

Hopefully the understanding of an SR "observer" as a non-participating mental construct answers that question in general.

However, in the specific scenario at hand, I would like to point out that Einstein's description of the magnet and the conductor problem is presented without reference to any observer and is clearly and unambiguously presented as a change in reference frame. Even if you take the term "observer" to be a real human, as is sometimes done, it is not a relevant objection to the specific example of the magnet and conductor.

 

[Sugdub]

Most of this is fine, but you have to be careful about the statement "there is no way it will induce a change of any physical quantity".

Agreed. My statement should read:"There is no way it will induce a change of the outcome of any measurement."

No. SR assumes that the two postulates hold. That is it. Everything else is derived. ...You do not need to vary the experimental conditions or even actually run the experiment again; you simply re-analyze it.

My wording was incorrect, however I keep thinking that the 1905 SR paper which is discussed in this thread refers to two different experimental contexts. First Einstein refers to two observers in relative motion in respect to each other and performing measurements. There is no indication that observers exchange any signal between themselves and therefore one can assume that they perform their measurements independently: there is no way the outcome of each measurement is dependent on the presence or absence of a second observer. So overall Einstein's presentation describes two experimental contexts.
However on the basis of a set of fully symmetrical hypotheses there is no way one could build up a theory demonstrating that the results respectively obtained by both observers may differ. The hypothesis of a "relative motion" between observers is certainly compatible with a divergence of outcomes but its symmetry prevents that this divergence could be logically derived in the absence of an additional non-symmetrical hypothesis. Hence the importance of the auxiliary hypothesis set by Einstein whereby one observer is at rest in respect to the target whereas the other one is not. This sets a tangible and objective difference in experimental conditions and therefore removes the blockage above. In addition, the nature of the difference in experimental conditions (a change in the relative speed between the observer and the target object) makes it impossible to invoke a change of reference frame. So waiting for further objections I maintain my statement whereby the conditions which prevailed for the derivation of the Lorentz transformation in the context of the 1905 SR paper do not match at all the conditions which prevail for resolving the moving magnet and conductor problem.

In the usual SR thought experiments or homework problems it is common ... The "observer" in SR is a purely mental construct that does not participate in the experiment at all (unlike an "observer" in QM).

This seems to indicate that the SR theory, or at least the way it can be presented, has evolved in the meantime, but it does not stand as such as a justification which would operate in the context of the 1905 SR paper.

However, in the specific scenario at hand, I would like to point out that Einstein's description of the magnet and the conductor problem is presented without reference to any observer and is clearly and unambiguously presented as a change in reference frame.

Should you read again my input you will no doubt convene that I fully agree. However, should one try to extend the metaphor of human involvement, the "observers" referred to by Einstein in the SR part of his 1905 SR paper, who are assumed to perform measurements as discussed above, would become "theoreticians" in the second part of the paper dealing with the magnet and conductor problem: with this I highlight again the incompatibility of the SR and EM paradigms and therefore the absence, in the 1905 SR paper, of a proper justification for invoking SR in demonstrating the equivalence of both descriptions of the EM field.

 

[Dale]

My wording was incorrect, however I keep thinking that the 1905 SR paper which is discussed in this thread refers to two different experimental contexts. First Einstein refers to two observers in relative motion in respect to each other and performing measurements.

Here is the entire first paragraph:

It is known that Maxwell's electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case.

There is not a single reference to any observers. Also, it is clear that the "two cases" are related to each other by a simple boost, and therefore the use of the Lorentz transform is clearly justified.

You are really "grasping at straws" with this objection. It makes you seem unreasonable to continue with it.

 

[Sugdub]

Your last reply won't convince those who followed our debate.

Here is the entire first paragraph: ... it is clear that the "two cases" are related to each other by a simple boost, and therefore the use of the Lorentz transform is clearly justified.

I've never challenged the fact that Maxwell's equations are invariant under a boost. I challenge the view that the transformation derived by Einstein in the "kinematics" part of his 1905 SR paper could actually account for a boost: the transformation he derived does not respond to a change of reference frame. It responds to a change of experimental conditions involving a change of the relative speed (from nil to v) between the measurement device (the "observer") and the target object. This cannot account for a change of reference frame and therefore the transformation at stake is not a boost.
Our recent exchanges show that you have supported the core of this analysis:
Sugdub:"In SR the swap from one experimental context to the other one cannot be considered formally as a change of reference frame, since the relative speed between the observer and the target object changes from nil to v (a mental act cannot achieve that).

I think that this is a reasonable objection that you bring. It is due to some highly unfortunate and (IMO) sloppy terminology.

However you now become aware of the unavoidable consequences of that discovery and you want to escape. You should feel free to retract your initial support to my analysis. No need to argue, it's much better to play it honestly.

 

[Dale]

Your last reply won't convince those who followed our debate.

I think that is making a pretty big assumption about the disposition of those following the debate.

I challenge the view that the transformation derived by Einstein in the "kinematics" part of his 1905 SR paper could actually account for a boost: the transformation he derived does not respond to a change of reference frame.

The transformation he derived is a single-parameter linear transform which maps lines of constant position in one frame to lines of constant velocity in another frame. That is a boost, by definition. Simply look at the equation at the end of section 3, it is clearly a boost.

However you now become aware of the unavoidable consequences of that discovery and you want to escape. You should feel free to retract your initial support to my analysis. No need to argue, it's much better to play it honestly.

The thing I was supporting was your understandable confusion over the use of "observers" in SR as a shorthand for a reference frame. It isn't even relevant here since there are no observers mentioned at all in the scenario under discussion, as I already pointed out with a verbatim quote of the entire paragraph in question.

 

[Sugdub]

The thing I was supporting was your understandable confusion over the use of "observers" in SR as a shorthand for a reference frame.

We might be close to identifying the root cause of our divergence of views. Indeed Einstein believed he could derive the Lorentz transformation in response to a change of reference frame steered by the relative velocity between two description frames. It is therefore not surprising if he raises the equations you refer to. Having achieved this derivation he could then invoke SR and its Lorentz transformation to justify the equivalence of both descriptions of the EM field resulting from his analysis of the moving magnet and conductor problem. His strategy in the 1905 SR paper is clear.
However, as already pointed out, a change of reference frame can only account for a change of description for the same realm, it is a pure mental act which cannot trigger, predict or explain the change of any physical quantity directly observable such as space or time quantities. Let's assume with Einstein that the relative speed between two measurement devices be accountable for the change of value of directly observable physical quantities (please note that I'm no longer referring to "observers"). It would mean that the state of motion of these devices is objectively different, that one objectively moves faster than the other … and this would breech the principle of relativity of motion. Had SR been conceived as a theory only dealing with mental descriptions without any observable consequences, it could lead to a transformation steered by the relative speed between two description frames with the caveat that this transformation could not apply to any directly observable physical quantity. It could however be invoked in support to the EM paradigm since this only deals with mental descriptions of non-observable physical quantities.
Conversely, in the "kinematics" part of his 1905 SR paper Einstein clearly referred to a change of directly observable physical quantities, namely space and time quantities. Such changes must be traceable to a change in experimental conditions. As demonstrated above, the relative speed between two measurement devices cannot account for such a change, hence the need for an additional hypothesis which sets an objective difference in experimental conditions: one device is at rest in respect to the target object whereas the second is in relative motion in respect to it. This has two consequences which Einstein did not spot: first the transformation to be derived cannot respond any longer to a "change of reference frame" since the relative speed between the measurement device and the target object changes from nil to v; second the parameter which steers the transformation must be the one which accounts for the change in experimental conditions, namely the relative speed between the second measurement device and the target object and no longer the relative speed between both measurement devices. These consequences are unavoidable if one wants to develop a theory which affects space and time quantities.
The "kinematics" part of Einstein's paper is a mix of both approaches, but I'm afraid physicists cannot have it both ways: they are just incompatible.

 

[Dale]

Let's assume with Einstein that the relative speed between two measurement devices be accountable for the change of value of directly observable physical quantities (please note that I'm no longer referring to "observers"). It would mean that the state of motion of these devices is objectively different

The paragraph in question also made no reference to any measuring devices. The fallacy you are committing is called the "strawman fallacy". You are having trouble attacking Einstein's actual position, so you add things that he didn't put in there and attack those additions. It is rather transparent.

I agree that two situations, one where a given measuring device is at rest wrt a given object and one where the same measuring device is in motion wrt the same object are not related by a boost. I agree, but point out that it is completely irrelevant to the scenario under consideration. Please try to make a non-fallacious argument.

Furthermore, when talking about reference frames or observers in terms of a system of physical clocks and rods, each reference frame has its own set of clocks and rods. So a Lorentz transform does not change the relative velocity of any clock or rod to any other object, it simply changes which set of clocks and rods is considered stationary. That is a boost.

The mathematical FACTS are thus: Both a Galilean transform and a Lorentz transform are boosts (i.e. single parameter linear transforms mapping lines of constant position to lines of constant velocity). Maxwell's equations are not invariant under a Galilean boost, which is an asymmetry. Maxwell's equation are invariant under a Lorentz boost, which is a symmetry.

If you don't like Einstein's presentation, that is fine; you don't even need a logical reason to dislike it, which is good since you don't seem to have one. However, your dislike of the presentation doesn't alter the math of SR one bit, nor does it reflect on its validity at all.

 

[ghwellsjr]

Taylor has the first chapter of the older 1965 editio of "Spacetime Physics" available for dowload at his website:

http://www.eftaylor.com/special.html

Probably not enough to get the complete story, but enough to see if you like the book.

Potential buyers of Spacetime Physics should be aware that there is a huge difference between the first and second editions. They are really different books. So if you look at the first chapter of the first edition in the link above and you don't like it, that doesn't mean you won't like the second edition. Conversely, if you do like it, and you decide to save some money by buying a used copy of the first edition, you should not think you are merely buying an older version of the second edition. When most people are recommending Taylor and Wheeler, they are referring only to the second edition. Your best bet is to find a library that has both or at least the second edition if you want to evaluate it before you buy it. But whatever you do, if you see just a copy of the second edition and like it, don't buy the first edition or you'll be sadly disappointed.

 

[Sugdub]

The paragraph in question also made no reference to any measuring devices.

How long will it take until you accept reading that the "Kinematical" part of Einstein's paper deals with observable quantities whereas the "Electrodynamical" part exclusively relates to non-observable quantities?
Sugdub: "… in the "kinematics" part of his 1905 SR paper Einstein clearly referred to a change of directly observable physical quantities, namely space and time quantities."
Einstein's SR paper: I-Kinematical part; §1 Definition of simultaneity... "If at the point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous with these events. If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the immediate neighbourhood of B."
I-Kinematical part; §2 On the relativity of times and Lengths … "By means of stationary clocks set up in the stationary system and synchronizing in accordance with §1 the observer ascertains at what points of the stationary system the two ends of the rod to be measured are located at a definite time. The distance between these two points, measured by the measuring-rod already employed, which in this case is at rest, is also a length which may be designated “the length of the rod." … "We imagine further that with each clock there is a moving observer, and that these observers apply to both clocks the criterion established in § 1 for the synchronization of two clocks."
I-Kinematical part; §3 Theory of Transformation of Co-ordinates …. "We now imagine space to be measured from the stationary system K by means of the stationary measuring-rod, and also from the moving system k by means of the measuring-rod moving with it; ... Further, let the time t of the stationary system be determined for all points thereof at which there are clocks by means of light signals in the manner indicated in §1 …."

Obviously Einstein refers to "observers" and measuring devices ("rods" and "clocks") in the Kinematical part of his paper and he compares two different experimental scenarios leading to different values of observable quantities. The root cause of this argument is that you were so far unable to read / admit that the "Kinematical" part refers to observable quantities whereas the "Electrodynamical" part exclusively refers to non-observable quantities. The same transformation cannot fit to both paradigms since Electrodynamics must deal with a change of reference frame whereas Kinematics cannot do so.
Hence my question: what is the rationale for invoking SR and its Lorentz transformation for resolving the EM problem at stake?
II-Electrodynamical part; §6 Transformation of the Maxwell-Hertz Equations for Empty Space..."If we apply to these equations the transformation developed in §3, by referring the electromagnetic processes to the system of co-ordinates there introduced, moving with the velocity v, we obtain the equations..."
Capito?