4D Magnetism Solution

[bo198214]

Though the 4D worlds exists in our minds, so we can create arbitrary
physical laws in it, as long as they dont contradict, we are always
tempted to extend the laws we already know to the 4D/ND case.
The only difficulty I can see to do this (of course my physical
knowledge is restricted) are the elctro magnetic laws.
(Which was already discussed in the threads
4D magnetism and electricity, Magnetism and Light in 4D.)
Whereas in all other laws one usually only need to exchange the 3
above the sum sign into a 4 or an N, the magnetic fields are are
determined by being perpendicular to the movement direction and
the electrical field direction (and in 4D there is a whole bunch of
directions perpendicular to 2 axes).

Nonetheless the Maxwell-Equations are a bit unclear what regards the
relativity of the velocity. A simple paradox (that I couldn't get
answered in school) can illustrate the problem:
Assume two electrons parallely moving along an axis with the same
velocity. There are repelling electrical forces between them and by
the movement there are attractive magnetic forces between them. One
can compute that at the speed of light both forces would be cancel out
each other.
The Problem now is: The velocity is relative, so for an observer
sitting on one of the electrons there should be no magnetic field only
the electrical field. So which trajectory does the electron take, the
one computed by the still observer or the one computed by a moving
observer?

Or a similiar question: Every one learns in school that 2 parallel
wires having equally directed current attract each other. But the
electrons in the one wire has no relative velocity to the electrons in
the other wire, so how can they attract then each other?

The general answer is: Magnetism is only a relativistic effect. This is
especially astonishing regarding the velocity in the above examples
may be very small compared to the speed of light (and relativistic
effects are considered to happen only for velocities near speed of
light). Instead of explaining this myself I will refer you
to the following introduction. Or (if you have a bit more time)
to the books Craik and the classic text Purcell.
Summarizing magnetism is the effect of densing electrical fields by
relativistic length contraction. And: taking a five-dimensional (or
even n-dimensional) space-time-continuum instead of a four-dimensional
does not hurt in the relativistic equations.

Conclusions:
n-dimensional magnetism: Two wires with currents flowing in the same
direction attract each other with the same formula as in 3d (which
only depends on the distance, length and current). Especially this
magnetism is even possible in 2D.

The above 2 Electrons Experiment: For the still observer the forces
between the electrons are even more repelling then for still
electrons. But this is balanced by the relativistic mass increase. So
the repelling accelleration remains the same (as if they were still).

[PWrong]

Summarizing magnetism is the effect of densing electrical fields by
relativistic length contraction.

First of all I have to thank you for this. I knew that magnetism was a relativistic manifestation of electricity, but I've always wondered about the details. Somehow when I read that I instantly understood, or at least had a vague idea. After reading the link I almost understand it completely.

the magnetic fields are are
determined by being perpendicular to the movement direction and
the electrical field direction (and in 4D there is a whole bunch of
directions perpendicular to 2 axes).

This is the main problem with extending electromagnetism to 4D. At the moment we have two possible solutions. Either there are three perpendicular electromagnetic fields in 4D (electric, magnetic and tetric), or the magnetic field is not a vector field, but some kind of tensor field. That is, instead of having a magnetude and a "direction", it has a magnitude and a whole plane of directions. Instead of having a tiny 1D compass at every point in space, you'd have a tiny 2D disk-like compass that aligns with the magnetic field.

A magnet is essentially the same as a cylindrical coil of wire. I don't think you could have spherindrical coils or cubindrical coils, but it's possible you could you have a duocylindrical coil. Perhaps we should look at 4D coils in the geometry thread.

n-dimensional magnetism: Two wires with currents flowing in the same direction attract each other with the same formula as in 3d (which
only depends on the distance, length and current).

Not quite the same formula. Coulomb's law is a 1/r^3 law in 4D, like gravity. The force from a wire in 4D will fall off with 1/r^2, instead of 1/r. But length contraction is the same in any dimension so apart from the inverse square law, the formula will be exactly the same.

[bo198214]

At the moment we have two possible solutions. Either there are three perpendicular electromagnetic fields in 4D (electric, magnetic and tetric), or the magnetic field is not a vector field, but some kind of tensor field. That is, instead of having a magnetude and a "direction", it has a magnitude and a whole plane of directions. Instead of having a tiny 1D compass at every point in space, you'd have a tiny 2D disk-like compass that aligns with the magnetic field.

Yes I read this, but it seemed not very matured to me. The good news is we dont need all this pondering about the magnetic field in 4D. The magnetic field is only a helping tool to compute the forces that moving charges have to each other, right? But these forces can be computed with only the Columbs law and the relativistic laws.

But if I had to decide with respect to the relativistic explanation, I would tend to the kind of tensor magnetic field. We take one circle wire with current in it. Then the magnetic field "bundle" in the center of the circle would span the plane perpendicular to the plane of the circle. Then we take another circle wire parallel to the first circle wire (there are different declinations possible). By the relativistic considerations (or the-wires-attract-each-other-consideration) there would be a force in the direction perpendicular to both circle planes (this direction is determined, you have similar situation with two straight parallel lines in 3 dimensions.) Is this direction the cut of the both field "bundles"? (Not sure whether talking nonsense in this paragraph ...) Ok, thats only for the case, one is not satisfied regarding the moving charges, and must have a dedicated magnetic field, regardless how complicated everything becomes thereby.

A magnet is essentially the same as a cylindrical coil of wire. I don't think you could have spherindrical coils or cubindrical coils, but it's possible you could you have a duocylindrical coil. Perhaps we should look at 4D coils in the geometry thread.

As said above I am not sure, whether the notion of magnetism and magnets helps in any way in 4D (and by the way I dont know what spher-, cub- or duocyl-indrical coils are )

Not quite the same formula. Coulomb's law is a 1/r^3 law in 4D, like gravity. The force from a wire in 4D will fall off with 1/r^2, instead of 1/r. But length contraction is the same in any dimension so apart from the inverse square law, the formula will be exactly the same.

Yes you are right, the densitity of a field should be proportional to the volume of the surface (of the sphere) and that is proportional to r^n-1 in n dimensions (so we see, not only the number above the sums has to be increased by 1 but sometimes also the numbers above the products ).

Interestingly this yields, that in 2 dimensions there is a constant field intensity around an axis charge (or mass). Or for magnetics:
No matter how distant two straight parallel wires with currents in it are, their attraction/repulsion is not dependent on their distance.

[jinydu]

Interestingly this yields, that in 2 dimensions there is a constant field intensity around an axis charge (or mass). Or for magnetics:
No matter how distant two straight parallel wires with currents in it are, their attraction/repulsion is not dependent on their distance.

Well, perhaps that is not too surprising in light of the fact that in 3D, the electric field due to an infinite plane of uniform charge does not depend on the distance from the plane.

The solution to these counter-intuitive results is that they are based on physically unrealistic situations (infinitely long wires and infinite planes respectively).

[PWrong]

As said above I am not sure, whether the notion of magnetism and magnets helps in any way in 4D (and by the way I dont know what spher-, cub- or duocyl-indrical coils are )

Have a look at Alkaline's site. [url]tetraspace.alkaline.org[/url]
It's a great introduction to the 4th dimension and the kinds of cylinders that are possible.

Although I like the idea of using relativity to derive magnetism, I'm still used to doing it the ordinary way. I'm still reading through the notes at your link, and doing the exercises. By the way, how advanced are you with this sort of thing? Are you still in school? I've just finished my first year at uni. I've studied both EM and relativity seperately, but not together like this.

Interestingly this yields, that in 2 dimensions there is a constant field intensity around an axis charge (or mass). Or for magnetics:
No matter how distant two straight parallel wires with currents in it are, their attraction/repulsion is not dependent on their distance.

That would be very useful for 2D people like Fred (see Alkaline's site). In fact I think it's possible to construct a very effective public transport system in 2D.

The first step is to make a long wire underground and run current through it. This creates a uniform magnetic field directed "outwards" everywhere above the ground. Next, build a fixed U-shaped conductor (the train tracks), above the ground. Then place a conducting bar (the train) between the rails, so that it can slide freely. Here's a rough diagram.

Code: Select all

    /--<--<--<--<
   |      ||  >>>
    \-->-->-->-->
oooooooooooooo
-->-->-->-->-->
oooooooooooooo

ooo = ground
-->--> = current
>>> = force
||  = sliding bar


Current flows through the U-shape and up the bar to complete the circuit. From section 2.2 and 2.3 of your link, this should cause a force on the bar to the right. All that's needed is a little trapdoor to allow the passengers to get in. When Fred sits in front of the bar, he is accelerated until he reaches his destination.

[bo198214]

Nice!!!

Though I hope that these duonians (and not bionian! Hey folk, "bionian" is two errors in one, 1. its misleading to have something to do with biology, 2. either you count greek (mononian, duonian, trionian, tetronian, pentonian, ...) or you count latin (unionian, binonian, trionian, quadronian, quintonian, ...) but not both mixed!) have no gravity, or can jump very good, otherwise its like living in an interval restricted by your next neighbours, poor duonians ...

Are you still in school?

Not by far. I finished my mathematics study several years ago. Intensive physics only in school.
But the described problem remained always in my mind, till some years ago I got a hint but could not find any references, and now the paradox is resolved for me. And I think the explanation is not that difficult to understand.

And its interesting that in a good electrodynamics text book, for this example (the parallel moving electrons) the forces are ((how I now know) wrongly) computed but there is no remark, relative to what (to the ether??!!) the velocity should be regarded. And interesting too was that some school friends, who studied physics, denied the bare possibility that magnetics could be only a relativistic effect (at a time where I only had this hint, but no explicit explanation).

Currently looking at those xyz-linders ....

[PWrong]

(and not bionian! Hey folk, "bionian" is two errors in one, 1. its misleading to have something to do with biology, 2. either you count greek (mononian, duonian, trionian, tetronian, pentonian, ...) or you count latin (unionian, binonian, trionian, quadronian, quintonian, ...) but not both mixed!)

That's a good point, duonian is a much better word.

Though I hope that these duonians have no gravity, or can jump very good, otherwise its like living in an interval restricted by your next neighbours, poor duonians ...

The duonians I'm talking about do have gravity. The same device would work just as well in Flatland, where there isn't any gravity. Now that I think about it, it would be better to have the whole thing underground.

[bo198214]

underground/overground without pilars/fastening would simply fall down.
One pilar/fastening would allow only one entry to the underground/overground (and maybe instable for longer distances).
So maybe the best to constitute some electrical field instead of pilars
so that all platforms floating over each other :?
Then the duonians can live in some kind of skyscrapers ....
I would imagine it like this:

Code: Select all

Apartments
###### Roof
# D |                                                Work Place
#######                                                #####
# C  |                                                 |   #
#########                                            #########
# B    |                    overground               |       #
###########         #########################      #############
# A      |             ^^^^ Transport ^^^^         |           #
################## ########################### #######################
                 #     ^^^^ underground ^^^^   #
                 ###############################


# means solid material
| means door
^ means electrical field
A,B,C,D are the persons. We assume that they can well jump or climb.

[PWrong]

If you hold up the roof with an electric field, it might interfere with the current running the train.

To make life a little easier for duonians, what if we suppose the existence of a hypothetical substance, that can hold things up, but still allow things to pass through sideways? You'd want it to occur naturally, because it would take a while to invent.

The obvious solution would be something that opens on one side, closes to trap you inside, then opens on the other side. Maybe there would be some kind of tree that breathes on one side, and exhales on the other side, allowing people and animals to pass through. A tree like this might have an evolutionary advantage over one that just acts like a wall.

Or there might be a material with "orientational permeability". That is, you can pass through it at one angle as if you were moving through water, but it feels solid if you push it vertically. Sort of like the way a polariser only lets vertically polarised light through.

[jinydu]

I'm taking a physics course right now that covers electromagnetism (and relativity later). Gradually, I think I'm starting to get a better understanding of the questions raised about magnetism raised in this forum.

Basically, electromagnetic interactions are given by the Lorentz Force Law:

F = qE + q(v x B)

Quoting from my professor "What is perceived in one frame of reference as an electric field can be perceived as a magnetic field in another frame of reference". He explained to me that in a frame of reference where a charge is moving, the electric field generated by that charge is given not by Coulomb's Law, but by some more general transformation law whose calculation involves tensors. When I pressed him for more details, he said I would have to wait for an upper division electromagnetism course.

Still, I've been able to glean some details. For instance, from Faraday's Law, I found that some electric fields are not conservative.

I also brought up the question of whether magnetism can work in spaces with dimension not equal to three. I commented that Newton's Laws clearly work in any number of dimensions, but that the cross products that appear so often in magnetism are defined only for 3D vectors. His response was that he hadn't thought about whether could work in more than three dimensions; but he did say that the cross product is only a special case of matrix multiplication, which can work in any number of dimensions. His opinion was that magnetism in more than three dimensions "should work".

[bo198214]

So, Jinydu, now I am asking you back: Did you understand this article?

If so, you dont need to ask authorities, because then its clear to yourself.
There is no magnetic field at all. Magnetic forces are simply the result of relativity and electric forces. And I think the idea behind it is simple: If charge b moves with respect to a, then its lengthes in moving direction are contracted, meaning that the field to the sides becomes stronger (the field lines to the sides (i.e. perpendicular to the moving direction) become denser). Thats the whole trick. (If you need formulas, then do it with formulas, they are all contained in the article. Though they may not essentially raise understanding in this case.)

The for me most surprising fact was, as I know found out, that Einstein already derived the Maxwell equations from his relativity principle in his (first) revolutionizing paper "Zur Elektrodynamik bewegter Körper" (on the electrodynamics of moving bodies) in 1905. And now 100 years later still the Maxwell theory with its axiomatic (contrary to derived) magnetic field is mainly tought at the universities (or am I wrong?).

F = qE + q(v x B)

And exactly this formula is a problem, because you dont know what v is relative to (if you had an ether that wouldnt be the problem of course). As I already explained in the paradox with the two electrons.
And this was/is a really serious problem, as we can see from: In a good (german) physics book on electrodynamics, exactly this example (that I had thought of in my schooldays, and brought my physics teacher in difficulties with) was explained and computed, but wrongly - as I now know -, by assuming the velocity to be absolute to the reference frame!
So Einstein solved those blunts with the Maxwell equations.

He explained to me that in a frame of reference where a charge is moving, the electric field generated by that charge is given not by Coulomb's Law, but by some more general transformation law whose calculation involves tensors.

Yes, of course one can hide the main ideas behind tensors.

but he did say that the cross product is only a special case of matrix multiplication, which can work in any number of dimensions.

Cross product is simply the nth vector that is perpendicular to some linearly independent n-1 vectors (in n-dim space). With length being the product of the lenghtes and sign according to some orientation considering.

[houserichichi]

An exact analogue of the binary cross (vector) product that we use in three dimensions works in 7 dimensions and no others. There exist similar operations (wedges) which can be defined by matrices and operations on them but the results are no longer vectors.

For proof of the above take a look at this paper.

On the other comment, tensors are much more descriptive than standard "variable play" that they teach in lower level algebra and physics. To understand "modern" physics one needs to learn, understand, and be able to manipulate tensors and even more abstract structures. Relativity and QM can only be described so far by vectors, scalars, and basic entities along those lines.

[bo198214]

An exact analogue of the binary cross (vector) product that we use in three dimensions works in 7 dimensions and no others.

Cool! Unfortunately it has nothing to do with magnetism (by relativity) in n dimensions (which is even possible in 2 dimensions). What is an "exact analogue" btw?

On the other comment, tensors are much more descriptive than standard "variable play" that they teach in lower level algebra and physics.

I didnt say anything against tensors (though for my taste they have too much indices ).

To understand "modern" physics one needs to learn, understand, and be able to manipulate tensors and even more abstract structures. Relativity and QM can only be described so far by vectors, scalars, and basic entities along those lines.

But please note (also @Jinydu), that Einstein in his above mentioned paper didnt use any vectors, tensors or cross-products, he used the "standard 'variable play' that they teach in lower level algebra and physics" which didnt hinder him not only to understand but also to invent. So please take this into account when rating the importance of calculus. (And the "only" in your above quotation can be erased.)

[houserichichi]

Didn't mean to come off cold like that...I re-read what I wrote. Was only trying to add in corrections as a side note (the vector product thing). The tensor bit I was off on a tangent again. Special relativity can be described with high school mathematics, a detail that stunned me the first time I caught glimpse of the derivations. The point of my tangent was (and I fully acknowledge that it had no place in this conversation, so I apologize) that "modern" physics required much more than basic arithmetic and calculus. I misinterpreted when you said "Yes, of course one can hide the main ideas behind tensors."

Sorry about that...carry on

[bo198214]

I found the comment about the two binary crossproducts, one from the quaternions in 3d, and one from the octonions in 7d, really interesting.
And for me it seems there is anyway a teaching flow of advanced maths from university to high school. Ok, but enough of off topic talk.

[jinydu]

For proof of the above take a look at this paper.

I took a quick glance through the paper and it seems that by "generalization of the cross product", the author meant any binary vector multiplication that satisfies the following axioms:

1) A x A = 0

2) (A x B) dot A = (A x B) dot B = 0

3) |A x B| = |A||B| if A dot B = 0

They went on to show over the first four pages of the paper, using basic application of those axioms, that an any vector binary operation satisfying those axioms must have have a dimension, n, that satisfies:

n(n - 1)(n - 3)(n - 7) = 0

The case n = 0 could be rejected because this doesn't correspond to an actual vector space; n = 1 was rejected because it was trivial. Thus, the only remaining possibilities were n = 3 (the familiar 3D cross product) and n = 7.

Page 5, the final page of the paper, was considerably more advanced; but if I understood correctly, the author gave a precise definition of the 7D cross product and stated that the existence of cross products in only 3 and 7 dimensions was related to the existence of "composition algebras" (I think that means higher-dimensional generalizations of complex numbers whose multiplication satisfies some special properties) in only 4 and 8 dimensions.

Please correct me if I am mistaken.

Also, the link that bo198214 gave looks interesting. If I understood correctly, Schroeder is claiming that all experimental predictions of electromagnetism can be recovered using only Coulomb's Law and special relativity, is that correct.

In addition, regarding your comment on the formula

F = qE + q(v x B)

My professor taught me that v is relatve to any inertial frame of reference, so long as you consistently use that frame of reference. He claimed that different frames of reference really do measure different values of v, and that E and B change across different frames of reference to accomodate the change in v, so that F is the same for all frames of reference.

Nevertheless, the idea of regarding magnetism and non-electrostatic electricity as a consequence of special relativity and electrostatics looks promising. It certainly looks a lot easier to generalize to more than three dimensions than Maxwell's formulation of electromagnetism. Still, I have two concerns:

1) Can it recover all empirical predictions of Maxwellian electromagnetism? The link showed how to recover the formula for attraction between a moving charge and a parallel wire, but can it recover all the other many effects covered in my class?

2) Special relativity depends on the Principle of Constancy: The statement that the speed of electromagnetic waves is the same in all inertial frames of reference. Although I have not seen a detailed derivation (hopefully, my class will cover that within one month), I have read before that Maxwellian electromagnetism is able to prove this based on Maxwell's equations. Thus, in the conventional way of teaching physics, Maxwell's equations and the Lorentz force law are regarded as postulates while the Principle of Constancy is a theorem that can be proven from those postulates. But if we now regard magnetism as a consequence of special relativity, the Principle of Constancy must now be accepted as a postulate, which (I think) also means that we must postulate the existence of electromagnetic waves. But since we are no longer talking about magnetic fields, how should we define what an electromagnetic wave is?

In fact though, upon reading the webpage carefully, it seems that the experimental predictions of both approaches are in fact not exactly identical. In the first example of a charge moving parallel to a current-carrying wire, a binomial theorem approximation was used to approximate the force. But of a course, a binomial approximation is not exact for finite, nonzero values of v. Which means that if we carry out an experiment where the force is measured extremely accurately, it should be possible to refute Maxwell's formulation and establish the one presented on your link! Of course, such an experiment would be very difficult to carry out in practice, but with modern technology, who knows!

There is one part of that first example that I don't quite understand: the webpage states that the separation distance between the positive charges in the test-charge frame is:

l_+ = l / sqrt(1 - (v/c)^2)

But, just above the diagrams, the author stated that in the test charge frame of reference, the positive charges are stationary, and hence the distance between them is not affected by length contraction. So shouldn't the formula be:

l_+ = l

?

In any case, the idea sounds intriguing; I think I will ask my professor about it after the class covers special relativity.

[PWrong]

In fact though, upon reading the webpage carefully, it seems that the experimental predictions of both approaches are in fact not exactly identical. In the first example of a charge moving parallel to a current-carrying wire, a binomial theorem approximation was used to approximate the force. But of a course, a binomial approximation is not exact for finite, nonzero values of v. Which means that if we carry out an experiment where the force is measured extremely accurately, it should be possible to refute Maxwell's formulation and establish the one presented on your link! Of course, such an experiment would be very difficult to carry out in practice, but with modern technology, who knows!

Actually, the binomial approximation doesn't serve much purpose at all. It's easy to simplify lambda, and show that the force expression is out by a factor of gamma = 1/sqrt(1 - v^2/c^2). This isn't a surprise, since in special relativity nothing escapes being multiplied by gamma .
I wouldn't be surprised if the experiment you suggest has already been done.

But, just above the diagrams, the author stated that in the test charge frame of reference, the positive charges are stationary, and hence the distance between them is not affected by length contraction. So shouldn't the formula be:

l_+ = l

I didn't get this either at first. He mentions that in the test charge frame, the distance is "un-length-contracted". In the test charge frame, the +ve charges are stationary and l_+ apart. In the lab frame, they're moving, so they get contracted to l.

l_+ must be the rest length between the positive charges. It's weird, because if there was no current in the wire, the positive charges would be further apart anyway :?.

[jinydu]

l_+ must be the rest length between the positive charges. It's weird, because if there was no current in the wire, the positive charges would be further apart anyway :?.

That's what I'm confused about. I thought that l was defined as the rest length.

[bo198214]

No, 'l' was the length how the moving positive charges appear to the lab frame. Though there is indeed an error in this consideration. It is assumed that the (moving) positive charges have same distance as the resting negative charges in the lab frame. That doesnt seem reasonable to me.

I am currently developing a fixed version ...

[jinydu]

No, 'l' was the length how the moving positive charges appear to the lab frame. Though there is indeed an error in this consideration. It is assumed that the (moving) positive charges have same distance as the resting negative charges in the lab frame. That doesnt seem reasonable to me.

I am currently developing a fixed version ...

So the rest length is l multiplied by the Lorentz factor (to reverse the length contraction), right?

[bo198214]

So the rest length is l multiplied by the Lorentz factor (to reverse the length contraction), right?

Exactly.

After really desperate computations I came to the conclusion that there must be some law like:

The lengthes of B seen from A do not change by acceleration. (Or at least are not changed after acceleration. Or at least at least if A and B were resting in the beginning.)
Because I know relativistic only from secondary sources, I cannot answer this. But Jinydu can ask this his professor?

With this statement then the length between the negative resting charges are the same as the length between the positive moving charges (with respect to the lab frame), because initially when there is no current in the wire the distances are equal, so they are equal after the electrons are accelerated to some velecity (or in this example the positive charges take the role of the electrons).

My professor taught me that v is relatve to any inertial frame of reference, so long as you consistently use that frame of reference. He claimed that different frames of reference really do measure different values of v, and that E and B change across different frames of reference to accomodate the change in v, so that F is the same for all frames of reference.

No, thats nonsense, thatswhy I gave the example with the two electrons. If you use consistently the test frame that moves with the two electrons there are no magnetic forces at all.
If you use consistently the lab frame, then the one electron creates a magnetic field by its movement, and there is a magnetic force on the other moving electron. You can compute the magnetic force by F=q(vxB).
(of course vice versa for the other electron.)

So in the lab frame there is an additional magnetic force and in the test frame there is none. Contradiction.

Quite magicly these problems do not occur with parallel wires with current in the same direction because there is relative movement between the protons in the first wire and the electrons in the second wire, so between both conductors there is indeed a magnetic force that is computed by the maxwell equations as if there were no protons at all. Strange incidences.

[jinydu]

Come to think of it, if the source of electromagnetic foces is Coulomb's Law + Length Contraction, the webpage's claim that the wire is (net) uncharged in the lab frame now seems to be in doubt. In the lab frame, the positive charges are moving and the negative charges are stationary. If we say that the electric field from the positive charges exactly cancels the electric field from the negative charges outside the wire, it follows that if we stopped the current, the wire would gain a net negative charge, since the positive charges have been "un-length contracted", thus reducing their linear charge density. But that conclusion doesn't seem right; we expect the net charge on the wire to be zero when there is current, due to symmetry.

[bo198214]

Jinydu, sometimes I have the impression, that you dont listen to me (as it was already with the 2 electron example.)
The statement: "The lengthes of B seen from A do not change by acceleration." exactly would avoid your scenario. And I really thought you already asked your professor about this.

The statement of special relativity is not, that the lengthes become contracted, when one thing starts to move (because special relativity is not about accellerated movements). The statement is about length contraction when changing view from a resting observer to a moving observer.

[jinydu]

Actually, I wasn't going to ask my professor until after he had covered Special Relativity. Admittedly, the Special Relativity I learned was in high school, which tends to be less rigorous than university education, and it now seems that there are some subtleties to Special Relativity that I still don't fully understand now.

I think what I will do is hold off on commenting on this any further until after Feb. 22, when my professor will have finished covering Special Relativity.

[bo198214]

The lengthes of B seen from A do not change by acceleration.

As I think more about it, it seems quite obvious.
If I have two resting electrons (that I consistently observe from the lab frame) with distance l, and now I put the same force (and so accelleration) on both and waiting some time, then both should have moved by the same amount and same direction and hence still have the same distance.

So the presentation in the article of Schroeder has no flaws at all.

[jinydu]

My professor provided some further details. Apparently, in more advanced treatments of Maxwell's electromagnetism, one defines something called an "A-vector", a four-dimensional vector that contains all information about the electric and magnetic fields. But again, I have to wait for a more advanced physics course...

Also, it now seems, based on my rather limited understanding of an advanced calculus book, that Maxwell's equations (at least in the differential form), or the Lorentz Force Law for that matter, cannot be generalized to more than 3 spatial dimensions without introducing qualitatively new concepts. Apparently, the "vector field derivatives": gradient, curl and divergence, are special cases of a more general operation called an "exterior derivative" and vector fields are special cases of a more general kind of function called a "form fields", which is a function (meeting several special properties) that assigns a form (itself a function that assigns a number to an ordered list of vectors) to every point in R^n. So if you specify a point in space and a list of vectors (you need to have the correct number of vectors; a k-form is a function of k vectors), a form field gives you a (signed) number. Supposedly, there is a general mathematical theory about forms that, in the special case of three dimensions, reduces down to standard 3D vector calculus. (I'll admit that I don't really see how a vector field is a special case of a form form, or if that is what the author meant at all).

Here are some relevant quotes:

"The real difficulty with forms is imagining what they are ... However, in R^3, it really is possible to visualize all forms and form fields, because they can be described in terms of functions and vector fields. There are four kinds of forms on R^3: 0-forms, 1-forms, 2-forms and 3-forms. Each has its own personality."

"In R^3 and in any R^n, a 0-form is simply a number, and a 0-form field is simply a [scalar] function" (I added the word scalar)

"Let F be a vector field on R^n. We can then associate to F a 1-form field, W_F, which we call the work form field... if F is a force field, its work form field associates to a little line segment the work that the force field does along the line segment ... The unit of a force field such as the gravitation field is energy per length" (I edited it to make it look easier. But I guess this does sort of make sense. If I have, say, a gravitational field in space, and I specify a point in that field, and then I specify a vector, I can calculate the infinitessimal work done by moving a particle, originally located at the point, by an infinitessimal distance in the direction of the vector).

Continuing onwards, the book then goes on to discuss 2-forms. I tried to think about it like this: Suppose we have an electric field in R^3. Now, let us select a particular point in this field. If we choose two linearly independent vectors and place them at that point, they form a plane passing through that point. The value of the 2-form field (called the flux form field) is then the infinitessimal amount of flux passing through the part of the plane near the point per infinitessimal area. Of course, this interpretation is not valid for dimensions higher than 3, since a 2D surface then fails to enclose any "hypervolume".

Now, suppose we have a 3-form field in R^3. We start off by specifying a point in that field. If we then specify three linearly independent vectors, we can then form a solid near the point. The value of the form field at that point for those three vectors then gives the infinitessimal "amount of vector-stuff" passing through an infinitessimal volume located near the point, divided by that infinitessimal volume (the result is a density). But since three linearly independent vectors span R^3, the use of vectors is redundant; any three linearly independent vectors would give the same result. Thus, we might as well give only the "magnitude" of the form (which we associate with density), since we know that the "directions" of the form span all of R^3. The same argument applies to n-forms in R^n, since n linearly independent vectors span R^n.

So here is another quote (I've added by own words in square brackets):

"In all dimensions,

(1) 0-form fields are [scalar] functions.
(2) Every 1-form field is the work form field of a vector function.
(3) Every (n-1) form field is the flux form field of a vector function.
(4) Every n-form field is the density form field of a [scalar] function.

...

"In dimensions higher than R^3, some form fields cannot be expressed in terms of vector fields and [scalar-valued fields]: in particular, 2-forms on R^4, which are of great interest in physics, since the electromagnetic field is such a 2-form on spacetime. The language of vector calculus is not suited to describing integrands over surfaces in higher dimensions, while the language of forms is."

...

"The correspondences between form fields, [scalar] functions and vectors ... explain why vector calculus works in R^3 - and why it doesn't work in higher dimensions than 3. For k-forms on R^n, when k is anything other than 0, 1, n-1, or n, there is no interpretation of form fields in terms of [scalar] functions or vector fields.

A particularly important example is the electromagnetic field, which is a 6-component object, and thus cannot be represented either as a function (a 1-component object) or a vector field (in R^4, a 4-component object)."

Later in the chapter, the book goes on to talk about gradient, curl and divergence as special cases of something called the exterior derivative (which I, admittedly, don't fully understand; although I know that it turns a k-form into a (k+1)-form).

Suppose we have a scalar function defined on R^3. This function will be a 0-form field. If we take the exterior derivative, we get a 1-form field. In vector calculus, this is equivalent to taking the gradient of a scalar function and getting a vector field, that we then associate with work. Furthermore, this always works in R^n because the gradient (which is easy to generalize to n dimensions) of a scalar function gives a vector for every point in space, exactly what a 1-form field needs.

Now, if we take the exterior derivative of a 1-form field, we get a 2-form field. In vector calculus, this is equivalent to taking the curl of a vector field and getting another vector field. But there is a tension here: A 2-form fields needs to act on 2 vectors (for any point in space) while a vector field only provides 1 vector (for any point in space). Fortunately, we're lucky in this case; given any two linearly independent vectors (and most pairs of vectors in R^3 are linearly independent), there is an almost unique (up to multiplication by a constant) vector that is perpendicular to both vectors. We can then associate this normal vector with the single vector provided by the vector field and interpret the new vector field (obtained after taking the curl) as a function that tells you how much "vector stuff" flows through any point, per unit cross-sectional area, and the direction of this flow. This trick works because the result of the exterior derivative is a 2-form, and 2 = 3 (the dimension of space) - 1. It is the same idea as being able to (almost) uniquely determine a plane using only one vector (its normal vector), even though a plane is 2 dimensional. In general, the vector field interpretation of a form field works whenever we have a (n-1)-form in R^n, since there is only one vector (up to multiplication by a constant) in R^n that is perpendicular to each of (n-1) linearly independent vectors. So for instance, we can associate a vector to a 3-form field in R^4 or a 999-form field in R^1000.

If we then take the exterior derivative of a 2-form, we get a 3-form. The vector calculus analogue of this is taking the divergence of vector field (remember that we have chosen to associate 2-forms with vector fields). The result is a scalar field, for the reason explained in my paragraph about 3-forms in R^3. Furthermore, this can be generalized to any number of dimensions.

To summarize the last few paragraphs, gradient and divergence can be generalized to any number of dimensions greater than or equal to 3, but curl can't (as you might have guessed naively based on their defining formulas). Due to an anticommutative property of forms, it can be shown that in odd dimensions (3, 5, 7, etc.), all the terms in the divergence are positive (which is what we're used to); but in even dimensions (2, 4, 6, etc.), the terms in the divergence alternate in signs.

In 2D, divergence and curl are in fact the same thing.

To summarize my entire post (except the first paragraph):

I do not think that it is possible to generalize Maxwell's equations to four or more spatial dimensions using only scalars and vectors. The reason is that Maxwell's equation are based on vector calculus, which itself fails to generalize to four or more spatial dimensions. This, in turn, is because it is not possible, in general, to describe an infinitessimal surface using only one vector.

Of course, even if my interpretation of the book is correct (and that is not certain), it doesn't mean that trying to generalize Maxwell's equations to four or more dimensions is entirely futile; it might still be possible to write them in terms of other things. In fact, since a form relies on a list of vectors, and a matrix can be thought of as a list of vectors, I suspect that Maxwell's equations in four or more dimensions can be written using matrices.

[bo198214]

Jinydu, your ignorance is really unbeleavable.
You write a long article only to state that you cant find a generalization with your selfimposed limits of form theory, where on the other hand the magnetism long time ago was shown to be a mere relativistic effect (and the relativistic equations easily generalize to n dimensions). It was shown within your scientific community! And in this thread computations are placed directly in front of your nose which warrant those results.
And you turn your head, take some theories and assert its not possible (remember my citation of Planck?)

[PWrong]

Of course, even if my interpretation of the book is correct (and that is not certain), it doesn't mean that trying to generalize Maxwell's equations to four or more dimensions is entirely futile; it might still be possible to write them in terms of other things. In fact, since a form relies on a list of vectors, and a matrix can be thought of as a list of vectors, I suspect that Maxwell's equations in four or more dimensions can be written using matrices.

Well, the electromagnetic tensor is a 4*4 matrix. We should be able to extend this to a 5*5 matrix (Each B will have two subscripts e.g. B_x,y). Then we just need to find out what it actually does.

I think the A-vector would work pretty well, because it doesn't need the magnetic field, which isn't a real field anyway.

And you turn your head, take some theories and assert its not possible (remember my citation of Planck?)

I don't think Jinydu said that anywhere. All he said was that we can't do it without tensors, which we might not learn for over a year. Still, it seems that EM isn't much different in 4D. All our usual electric machines, given a bit more tridth, should work ok in 4D.

[bo198214]

I don't think Jinydu said that anywhere. All he said was that we can't do it without tensors, which we might not learn for over a year.

Ok, I was a bit overreacting, sorry for that.

We regarded here of course only the case of constant velocities. We easily can compute the magnetic forces between two constantly moving charges in any dimension with only using vectors and scalars.

As I already previously stated, the magnetic field is only helping field for computation. And its not necessary to generalize it to n dimensions. What we can compute in any dimension is the interaction of moving charges and there is no more to electromagnetism. So especially generalizing the magnetic field as in the Maxwell equations is an unnecessary obstacle (even if it would be possible with tensors).

So maybe Jinydu is literally taken right that the Maxwell-Equations - especially the magnetic field - cannot be generalized to arbitrary dimensions with vectors and scalars. But this statement is nearly irrelevant when regarding the relativistic possibilty to generalize electromagnetism (as we do in this thread).

I am not exactly sure what is needed for computing forces of arbitrary (accelerated, curved) movement of charges (as to explain electrical induction for example), perhaps that also involves tensors, who knows. At least Jinydus explanations give not hint to this.

[jinydu]

Jinydu, your ignorance is really unbeleavable.

I can't deny that I am nowhere near mastering Maxwell's theory; but hopefully the situation will improve with time. University has a steep learning curve.

All he said was that we can't do it without tensors, which we might not learn for over a year.

Actually, I meant matrices (which, according to Mathworld, is actually a special case of a tensor). I have to admit that I haven't been taught anything about tensors yet, although I am taking a Linear Algebra that, of course, covers matrices. So until I get to tensors, I try to think about things in terms of concepts I've already learned :wink: .

where on the other hand the magnetism long time ago was shown to be a mere relativistic effect (and the relativistic equations easily generalize to n dimensions). It was shown within your scientific community!

My professor did mention some kind of connection with special relativity, although he emphasized that I would have to wait for an upper division course in electromagnetism to see the full details. Please be patient, my education is still a work in progress :wink: . For now, I'm just experimenting.