Electromagnetism in 4D

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

Electromagnetism in 4D

Postby PatrickPowers » Thu Jan 07, 2016 7:08 pm

http://physics.weber.edu/schroeder/mrr/MRRtalk.html

This talk describes how an electromagnetic field is the direct result of electric charge plus special relativity. (Permanent magnets are much harder to explain.) This shows that electromagnetism is essentially two-dimensional. It works the same in all Minkowski spaces with dimension of 2+1 or more.

To calculate the magnetic force between a current flowing in a straight wire and a free charge in standard 3+1 space, do the following. Determine the plane defined by the line of the straight wire and the location of the free charge. Project the velocity vector of the free charge onto that plane. Rotate 90 degrees, depending on the direction of motion and the signs of the charges involved. The resulting vector shows the acceleration on the free charge due to the magnetic field.

The magnitude of the magnetic field vector is proportional to the dot product of free charge and the velocity vector of the charge in the wire.

All this has been known for decades. It seems that the most elegant way to calculate these things is with tensors, which I may or may not someday understand.
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Re: Electromagnetism in 4D

Postby PatrickPowers » Thu Jan 14, 2016 8:23 pm

Here is a more sophisticated and detailed version. The basic idea is that the total force, electrical + magnetic, is a bivector in Minkowski space.

http://www.av8n.com/physics/magnet-relativity.htm#bib-lienard-wiechert
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Re: Electromagnetism in 4D

Postby quickfur » Thu Jan 28, 2016 2:07 am

Oooh... This looks like a very promising approach to deriving a workable model of electromagnetism in 4D! Thanks for posting this... I've been trying to find an approach to electromagnetism that isn't tied to a specific dimension of space. If we assume the obvious analogies to special relativity and basic electric charges, we could probably calculate the effect of a current of charged particles to a test charge, and thereby derive 4D-specific equations for magnetism.
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Re: Electromagnetism in 4D

Postby PatrickPowers » Thu Jan 28, 2016 2:32 am

quickfur wrote:Oooh... This looks like a very promising approach to deriving a workable model of electromagnetism in 4D! Thanks for posting this... I've been trying to find an approach to electromagnetism that isn't tied to a specific dimension of space. If we assume the obvious analogies to special relativity and basic electric charges, we could probably calculate the effect of a current of charged particles to a test charge, and thereby derive 4D-specific equations for magnetism.


The equations may be seen in the Magnetism in N D thread.
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Re: Electromagnetism in 4D

Postby PatrickPowers » Mon Feb 01, 2016 4:58 am

There is a very nice article about Maxwell's Equations and Electromagnetism at John Denker's web site.

http://www.av8n.com/physics/maxwell-ga.htm

I got some warnings about security but these were easy to get around.

So..it's a solved problem. He even has the relativistic version. A PDF is at http://www.av8n.com/physics/maxwell-ga.pdf but it is quite different and at first glance I prefer the web page.
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Re: Electromagnetism in 4D

Postby quickfur » Mon Feb 01, 2016 6:19 am

Very nice stuff. I got scared off bivectors by the wikipedia article, which probably does not do it justice... but seeing them in action has convinced me that they are worth looking into. I suppose 4D rotational mechanics would benefit from using a geometric algebra representation too. I agree with the note on the website that cross products are trouble. They are specific to 3D, and are generally quite fragile and must be handled with care, etc..

Now your next challenge is to come up with a workable model of 4D atomic physics. :- :lol: The problem is, the last time somebody tried to solve the 4D Schroedinger's equation, it had no local minima except at r=0, which implies that the direct equivalent of a 4D atom would simply collapse and disintegrate.

There's also the issue of wave propagation through a 4D medium, which some time ago somebody linked an article about, that indicates that 3D is the only dimensionality of space where a signal transmitted from a point source can be recovered faithfully at a receiver at some distance away. In even dimensions, in particular, the wavefront would induce back-propagation which causes the signal to interfere with itself in a complex way, that makes it difficult to recover the original signal at the receiving end.

These things seem to be suggesting that a workable model of 4D physics may turn out to be far more alien than what one might expect based on a naive generalization from 3D by dimensional analogy.
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Re: Electromagnetism in 4D

Postby PatrickPowers » Mon Feb 01, 2016 6:34 am

quickfur wrote:Very nice stuff. I got scared off bivectors by the wikipedia article, which probably does not do it justice... but seeing them in action has convinced me that they are worth looking into. I suppose 4D rotational mechanics would benefit from using a geometric algebra representation too. I agree with the note on the website that cross products are trouble. They are specific to 3D, and are generally quite fragile and must be handled with care, etc..


See the Magnetism in N D thread. Cross products need love too. It's not that big of a deal.

Wikipedia math is nice for a reference but I find it too terse for learning.

Frankly, the math of physics seems like a big kludge. Things are kind of slapped together for historical reasons, when people were feeling their way and didn't yet quite understand what was happening. But the system has so much inertia it might never change.

I tried and failed to learn vector calculus in college. I felt it didn't make sense. Now I know it didn't make sense.

quickfur wrote:Now your next challenge is to come up with a workable model of 4D atomic physics. :- :lol: The problem is, the last time somebody tried to solve the 4D Schroedinger's equation, it had no local minima except at r=0, which implies that the direct equivalent of a 4D atom would simply collapse and disintegrate.


Very interesting! Maybe one could use vorticies in superfluid superconductors instead. But that math is way too hard.

quickfur wrote:There's also the issue of wave propagation through a 4D medium, which some time ago somebody linked an article about, that indicates that 3D is the only dimensionality of space where a signal transmitted from a point source can be recovered faithfully at a receiver at some distance away. In even dimensions, in particular, the wavefront would induce back-propagation which causes the signal to interfere with itself in a complex way, that makes it difficult to recover the original signal at the receiving end.

These things seem to be suggesting that a workable model of 4D physics may turn out to be far more alien than what one might expect based on a naive generalization from 3D by dimensional analogy.


Indeed they do. I'll stick with easier things for now. Like precession of a rotating sphere. That seems doable (though not exxactly a breeze.) Do 4D spheres precess? Inquiring minds want to know...
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Re: Electromagnetism in 4D

Postby PatrickPowers » Mon Feb 01, 2016 2:18 pm

A useful article about the mystery of quantum spin.

http://people.westminstercollege.edu/faculty/ccline/courses/phys425/AJP_54%286%29_p500.pdf

What I have been discovering is that those diagrams of light waves are bogus. What is really going on is that EM waves are helical. They usually have roughly equal amounts of helices of both chirality superimposed on one another. The phase is undefined. You may think of it as an infinite number of helices superimposed. When the EM wave hits something, the phase becomes defined. This is called polarized light. Now we have only two counterrotating helices. The cancellations give us a sine wave.

It is possible to filter out one of the helices and get circularly polarized light. That's the most basic thing. Instead of a wave we see a helix. So the "wave" doesn't have those weird nodes where it is equal to zero. Instead it is of constant norm and has cylindrical symmetry. But for some reason it isn't taught this way.

And the electric and magnetic fields are not orthogonal. According to MacDonald they are both manifestations of the same thing: a bivector field in 3+1D spacetime.

That quantum spin is a 4D phenomenon is no big surprise: quantum spin is orientable, so it must be in an even-dimensional space. Quantum spin in 2+1 or 4+1 space would not be orientable, so basic physics would be very different.
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Re: Electromagnetism in 4D

Postby quickfur » Mon Feb 01, 2016 4:25 pm

The problem with generalizing precession to 4D is that precession assumes a rotational axis. However, rotational axes don't exist in 4D. Rotation being a 2D phenomenon, if you were to place a spinning object on a 3D hypersurface, the rotation would mainly occupy only 2 out of the 3 horizontal directions, so I wouldn't expect rotational inertia to hold it upright. It would probably just fall sideways along the direction perpendicular to the rotational plane. But then again, I've never managed to work out how to generalize rotational inertia to 4D, so perhaps I'm wrong. But there will definitely be an inhomogenous situation in the horizontal hyperplane, unlike rotation in 3D.

Now, precession in free-floating objects in 4D space is a different story altogether. According to wendy, if such objects have any rotation, they should eventually settle into one of the two chiral Clifford rotations, in which there are an infinite number of stationary (2D) planes. I don't know how you'd even apply the concept of precession to that!
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Re: Electromagnetism in 4D

Postby PatrickPowers » Mon Feb 01, 2016 11:22 pm

quickfur wrote:The problem with generalizing precession to 4D is that precession assumes a rotational axis. However, rotational axes don't exist in 4D. Rotation being a 2D phenomenon, if you were to place a spinning object on a 3D hypersurface, the rotation would mainly occupy only 2 out of the 3 horizontal directions, so I wouldn't expect rotational inertia to hold it upright. It would probably just fall sideways along the direction perpendicular to the rotational plane. But then again, I've never managed to work out how to generalize rotational inertia to 4D, so perhaps I'm wrong. But there will definitely be an inhomogenous situation in the horizontal hyperplane, unlike rotation in 3D.

Now, precession in free-floating objects in 4D space is a different story altogether. According to wendy, if such objects have any rotation, they should eventually settle into one of the two chiral Clifford rotations, in which there are an infinite number of stationary (2D) planes. I don't know how you'd even apply the concept of precession to that!


Yep, I just don't know. Unlike 3D physics, one cannot resort to experiment.

Precession can make sense without an axis. The plane of rotation can change.

The math of precession of tops is not that easy even in 3D. I had to get an upper-division text to show me how it was done. You have to use the Lagrangian or Hamiltonian, and to get a specific number you have to solve a cubic with an elliptic integral. I wouldn't be confident of the result if the answer weren't already known.

As to the precession of 4D planets, start with a planet rotating only in one plane. The planet is oblate. In an inhomogenous gravitational field this will act as a lever arm, which will induce a perpendicular rotation on the planet. This will reduce the oblateness (oblasity?). However, if the decrease is exponential then this will take an infinitely long time. I once calculated the distance an object would travel if its speed decreased exponentially. Infinite distance! Bit of a surprise, that.

It's also worth noting that it is expected that 4D planets can't orbit stars. They would be traveling solo. The asymmetry in the gravitational field would be tiny.

If the plane of rotation is perpendicular to the gravitational field then the planet's rotation is metastable and there is no precession. Uranus is sort of like that.

As one last blast, if the plane of rotation is coplanar with that of gravity then there is no precession at all.

Hmm, could there be galaxies in 4D? I would guess yes. You don't need orbits to have a galaxy. The gravity has to confine most of the stars, doesn't it?
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Re: Electromagnetism in 4D

Postby quickfur » Tue Feb 02, 2016 12:35 am

It has been well known around these parts that 4D planetary systems cannot exist, at least not for a meaningful amount of time. At least, that's the case if we assume the flux theory of force propagation, which dictates that the strength of a force emanating from a point source should diminish in proportion to the area of an n-dimensional sphere centered on the source. Thus, in 3D the strength of gravity would diminish in accordance with an inverse square law, whereas in 4D, it would diminish in accordance with an inverse cube law.

The inverse cube law, unfortunately, is what wrecks everything, because it causes the force to increase too quickly when an object is approaching the source, and to diminish too quickly when the object is moving away from the source. Due to the mismatch in the shape of the gravity well vs. how the object's momentum varies, the only possible stable "orbiting" motion is the perfect circle, which is virtually unheard of in real-life, because even the slightest of perturbations will throw it off the orbit into an unstable path. There are 5 possible paths under the inverse cube law of gravity:

1) The momentum of the prospective planet is too high, and it flies off into space after being somewhat deflected by the gravity of the star. This is analogous to a hyperbolic or parabolic path in 3D.

2) The momentum of the prospective planet is too low, and it quickly crashes into the star. This is analogous to the 3D case of an object with low momentum that falls along a collision path into the star.

3) The momentum of the prospective planet is approximately balancing the pull of gravity, but somewhat errs on the high side. The resulting path is a spiral, in which each revolution around the star will cause the orbital radius to increase by a constant amount. So the planet somewhat remains in orbit for a while, but every orbital year it flies farther away from the star. Eventually, it will be so far away that the influence of the star will be negligible, and it pretty much becomes an unbound planet or it will get captured by another star.

4) The momentum of the prospective planet is approximately balancing the pull of gravity, but somewhat errs on the low side. The resulting path is also a spiralling path, but this time an inward spiral, where every orbital year its distance from the star decreases by a constant amount. Eventually, it will crash into the star.

5) The momentum of the prospective planet exactly balances out the pull of gravity. The resulting path is a perfect circle. In practice, of course, this is impossible, since the slightest perturbation will send it along either (3) or (4). (In 3D, perfectly circular orbits are also pretty much unheard of -- they are virtually all elliptical, just with a small eccentricity. Unlike 3D, however, there are no analogues of stable elliptical orbits in 4D. The closest equivalents are (3) and (4), which do not last in the long term.

I haven't done any further rigorous study on this, but from what little I did do, it appears that the cause of this inherent instability of 4D orbital motion is related to momentum being proportional to the square of velocity. When the gravitational well obeys an inverse square law, it seems that the reciprocal square radius in the equation neatly balances out the momentum, giving rise to various quadratic relationships, that happen to include stable solutions in circular and elliptical orbits. When the gravitational well obeys an inverse cube law, however, its interaction with momentum produces various cubic relationships, and unfortunately only under exceptional circumstances will a path defined by a cubic equation produce a periodic path. Most of the paths will be aperiodic, due to the general nature of cubic curves.

Because of this, my suspicion is that larger conglomerations of orbiting bodies will also be unstable; so galaxies and other such familiar structures would be unlikely to exist in 4D.

Unless, of course, we declare by fiat that gravity continues to obey an inverse square law in 4D. Then the familiar elliptical orbits return, except with a much larger scope of possibilities granted by the extra dimension of space. Sometimes this is a convenient fancy to explore, as it yields many interesting results that are sufficiently similar to our own 3D universe that we understand it more easily, and gain insights into the behaviour of 4D space. However, it's also a rather unrealistic model of how an actual 4D universe might look like.
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Re: Electromagnetism in 4D

Postby quickfur » Tue Feb 02, 2016 12:38 am

P.S. My prediction for 4D galaxies is that the imbalance of momentum and the gravity well will cause any such structure to quickly fling off high-momentum masses into outer space, and the residual low-momentum masses will quickly collapse into a giant black hole where the center of the galaxy is.

("Quickly", of course, on an astronomical timescale. But it will definitely be much shorter than the lifetimes of the typical 3D galaxy.)
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Re: Electromagnetism in 4D

Postby PatrickPowers » Tue Feb 02, 2016 3:22 am

quickfur wrote:Unless, of course, we declare by fiat that gravity continues to obey an inverse square law in 4D. Then the familiar elliptical orbits return, except with a much larger scope of possibilities granted by the extra dimension of space. Sometimes this is a convenient fancy to explore, as it yields many interesting results that are sufficiently similar to our own 3D universe that we understand it more easily, and gain insights into the behaviour of 4D space. However, it's also a rather unrealistic model of how an actual 4D universe might look like.


At it's root physics is based on the 3+1 dimensions and on symmetries. (Then there are all the particles, which aren't understood yet, but that's another story.) If you give up on the symmetries then physics collapses completely -- energy isn't conserved and so forth -- so I'm not going to do that. But as you say this is all fancy, so it's a matter of taste.
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Re: Electromagnetism in 4D

Postby PatrickPowers » Tue Feb 02, 2016 3:46 am

This space intentionally not left blank.
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Re: Electromagnetism in 4D

Postby quickfur » Tue Feb 02, 2016 4:34 am

Well, by generalizing physics to 4+1 dimensions instead of 3+1, we're already "invalidating" it, strictly speaking. :P

I don't see why symmetries can't continue to hold in 4D space, though. That has nothing to do with why point force sources ought to attenuate according to an inverse cube law. In fact, symmetry turns out to be an excellent way of generalizing aspects of physics that initially may seem inextricably bound to 3D space. E.g., the idea of deriving electromagnetism just from charged particles and special relativity is an awesome approach that can apply to all dimensions. (Special relativity may be considered a fundamental kind of symmetry, the symmetry of physical laws under a change of inertial frame of reference.) In fact, it makes me wonder if one can also derive quantum chromodynamics (or at least the "classical" model thereof, whatever that means) from "first principles" by assuming a 3-fold color charge and applying special relativity.
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Re: Electromagnetism in 4D

Postby wendy » Wed Feb 03, 2016 9:46 am

I've been playing with a different set of equations to maxwell's. This is basically by carefully filtering the hidden stuff out of cgs units, one gets these.

These equations are to be regarded as definitions, of the valuse in single quotes, and when the symbol in double quotes is included, the relations are as described.

'D' = Q S 'H' = P×S Radiant field equation (Ampere relations)

'z'cD = E "=" zH = cB Photon continuity equation (Snell's law)

F = Q.'E' "+" P×'B' Electron force equation (A H Lorentz) (Coulomb relations).

S is the radiant vector. c is a property of space-time, and affects the nature of S.

P is the momentum of charge, × is an unspecified operator corresponding to vector-product.

z is a specific leakage of fluxkins to feildkins. Without z, there is no field.

Maxwell's law are

del. D = rho. Q del × H = rho P + tau D
del. B = 0 del × E = -tau B

rho = space density / z.
tau = time variance / k [k = EM linkage constant]

See zb http://www.os2fan2.com/tmp/physics.pdf
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Re: Electromagnetism in 4D

Postby PatrickPowers » Wed Feb 03, 2016 4:00 pm

You write that light is a scalar field, but this is not so. Light carries momentum, which is a vector quantity. Light may even carry angular momentum, which is in my opinion a bivector quantity. All this comes from Einstein in 1905. His original papers are very readable, more so than any "explanation" that I've come across.
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Re: Electromagnetism in 4D

Postby wendy » Wed Feb 03, 2016 11:47 pm

On the other hand, light as measured in lumens and foot-candles, is a scalar.

It is true that p = h/l but this is not smoething that we take account of in the sunshine. On the other hand, the number of foot-candles is.

The solar constant is 2 pyrons, or 2 g.cal/cm² min. Poking in the values gives 1386 W/m². We divide this by c to get 4.5 µPa. This means that 1 sq km of solar sail will give a momentum of 33 ft lb / s. in one second. You can reduce the sail down to 50 ft square, but it would then take an hour to get this momentum. Certainly it's much less than the scalar effect that powers the roof-donkeys (solar panels) .
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Re: Electromagnetism in 4D

Postby PatrickPowers » Thu Feb 04, 2016 5:18 am

Aha. Thanks for the clarification.
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Re: Electromagnetism in 4D

Postby PatrickPowers » Mon Feb 08, 2016 4:25 pm

In geometric algebra, Maxwell's equation in 3+1 D is ultra-simple.

grad( F ) = J

where F is the force and J is what the charge is doing. Needless to say there is a great deal packed in there, so much one might fear an explosion when opened. Let's see whether we can disarm the device. Many clues are contained in http://www.av8n.com/physics/conservation-continuity.htm.
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Re: Electromagnetism in 4D

Postby PatrickPowers » Sun Feb 21, 2016 10:03 pm

I'm looking for help. I'm working out of the Feynman lectures. Maxwell's equations and their solutions are there about a third of the way down. http://www.feynmanlectures.caltech.edu/ ... n-EqII2113

Note: That is actually the full set (or at least a fuller set) of Maxwell's original equations. What are known today as Maxwell's equations are actually due to Oliver Heaviside. He didn't like potentials so he left them out.

The sticking point is this. Consider the current density j. In 3D this is a vector at any moment of time. But it would be possible to define it as a bivector. This is because there isn't really any such thing as the current flow through a point. Instead we define a 2D surface with that point in the center, get the flow through that surface, then take the limit as the surface shrinks to zero. In 3D we can define that surface with either a bivector or its dual, which is a vector.

So in ND, should j still be defined this way? It could be. The surface would be N-1 D instead of 2. Or should we use current through 2D manifolds instead? Let's call these the (N-1) solution and the 2D solution. One might choose either or both. It could be that both definitions are useful.

?
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Re: Electromagnetism in 4D

Postby quickfur » Mon Feb 22, 2016 2:23 am

I'm pretty sure it should be an (N-1)-dimensional multivector. The key in deciding between 2D and (N-1)D lies in the fact that we are considering flow per unit area of a cross-section. Cross-sections in n-dimensional space are made by intersection with a (n-1)-dimensional subspace, because anything lower-dimensional would not divide n-dimensional space, and therefore the result wouldn't be a cross-section.

While it's certainly possible to derive a 2D solution in 4D, for example, such a concept would not be very useful, because current through a 2D subspace doesn't tell us very much in 4D. A conductor like a wire, for instance, would be basically a thin rod in 4D, which generally speaking would form a sphere-like cross section perpendicular to the direction of the current. (Well, technically a ball, not just the spherical surface.) A 2D subspace of the wire, while certainly computable, wouldn't tell you very much about the current flowing through the wire. Lots of current could be flowing outside the 2D subspace, and it wouldn't be included in the result. So the resulting 2D current density measurement would be rather useless.
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Re: Electromagnetism in 4D

Postby PatrickPowers » Mon Feb 22, 2016 1:43 pm

Here are Alan Macdonald's GA version of N D Maxwell's equations from his Vector and Geometric Calculus.

With h is just about anything, the directional derivative of f at x is

dhf = lim t->0 (f(x+th)-f(x))/t

With ei an orthonormal basis vector this can be condensed to

dif = lim t->0 (f(x+tei)-f(x))/t

Define the "grad" operator as eidif using the Einstein summation convention. If an index appears twice then sum over that index. Then with | the inner product "dot" operator

eidif = ei|dif + ei^dif

ei|dif is "divergence"

ei^dif is "curl."

Note that the divergence of F is one grade lower than F. The curl is one grade higher than F.

The electric field E is a vector. The magnetic field B is the curl of a vector potential, so it is a bivector.

Maxwell's equations without charges come out as

ei|diE = 0
ei^diE = -dtB

ei^diB = 0
ei|diB = -dtE

-----------------

This can be made into a single equation (dt + eidi)(E+B) = 0. Multiply it out and segregate the different grades and one gets these Maxwell's equations without charges.

Macdonald leaves adding in the charges as an exercise for the reader. Grrr.

ei|diE = rho/eps0,

but is it

ei|diB = j/eps0 - dtE

or

ei|diB = -j/eps0 - dtE

?
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Re: Electromagnetism in 4D

Postby PatrickPowers » Mon Feb 22, 2016 2:31 pm

The Lagrangian is

(eidif)2 = (eidif)(eidif) = di2f/dei2

Then using Maxwell's,

(dt2 - eidi2)(E+B)=0
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Re: Electromagnetism in 4D

Postby PatrickPowers » Sun Feb 28, 2016 10:28 am

Summing up, let E1 be the electric field monovector. E2 is the electric field bivector. A static charge generates a purely E1 field. A permanent magnet generates a purely E2 field. The subscript indicates the grade.

Assign grades to zero as well. This clarifies what is going on with the grades. Div lowers by one grade, curl raises by one.

With no charges present,

E = 00 + E1 + E2 + 03 + 04 .... 0N
div E1 = 00
curl E1 = -dtE2

div E2 = -dtE1
curl E2 = 03

Energy is conserved. Only curl can do "work."

div E1 is static electricity. In the absence of charges it is zero.

curl E = curl( E1 + curl E2) is the electromotive force. curl E2 must be zero, otherwise a permanent magnet would do perpetual work.
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Re: Electromagnetism in 4D

Postby Teragon » Sun Feb 28, 2016 9:27 pm

PatrickPowers wrote:Macdonald leaves adding in the charges as an exercise for the reader. Grrr.

ei|diE = rho/eps0,

but is it

ei|diB = j/eps0 - dtE

or

ei|diB = -j/eps0 - dtE

?


Normally I don't mind signs much and just look them up, but this one turned out to be a really good question.

E points from positive to negative charge.
J is defined in the flow direction of positive charges - also from positive to negative.
Any J can be understood as a change in E (more generally D) and vice versa, independent of the presence of actual charges.
As J acts in a way to equate regions of opposite charges it's corresponding to a decrease in magnitude of E, which means that dE/dt will always be of opposite sign as E and therefore J.
Everything seems consistent, but Maxwell's equations say just the opposite. Can someone explain that?
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Re: Electromagnetism in 4D

Postby Teragon » Fri Jul 01, 2016 9:19 pm

I'm wondering about how a magnetic field created by a double rotating charge distribution would look like.
A current going around in a loop the magnetic field lines will be distorted spheres linked to the loop. Each of the will be extented in one dimension of the loop and two dimensions perpendicular to the loop.
If we have a double rotating charge distribution looking like a Clifford torus it gets really confusing for me.
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Re: Electromagnetism in 4D

Postby The Shadow » Wed Nov 02, 2016 5:39 pm

Geometric algebra is definitely the way to go for EM. I recommend the book _Geometric Algebra for Physicists_ by Doran and Lasenby. I've learned a lot from it! (Down with the cross product!)

In an n+1 spacetime, the electromagnetic bivector will have n(n+1)/2 components. n of these include the time direction, and thus produce Lorentz boosts - this is the electric part.

The remaining n(n-1)/2 are purely spatial, and thus make charges rotate. This is the magnetic part.

So you can treat the electric field as a vector in any number of dimensions if you really want to; but treating the magnetic field as a vector only works in 3d. In 4d the magnetic field has 6 components.
The Shadow
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