## Paul Ehrenfest on Physics and Dimensions, 1918

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.

### Paul Ehrenfest on Physics and Dimensions, 1918

https://dwc.knaw.nl/DL/publications/PU00012213.pdf
P. Ehrenfest, "In that way does it become manifest in the fundamental laws of physics that space has three dimensions?"
KNAW, Proceedings, 20 I, 1918, Amsterdam, 1918, pp. 200-209

I have seen these sorts of arguments before but it is very nice to get them in such a polished and authoritative format.

I feel fairly certain though that he isn't looking at magnetism correctly. He says having six planes of rotation causes trouble. Well, here in 3D we have three planes of rotation and seem to do all right. In 3D all these planes always sum to one plane. In 4D they sum to two perpendicular planes. Another way to look at it is that according to Purcell magnetism is a child of special relativity. This is essentially a 2+1 theory so extra dimensions are irrelevant and magnetism should be pretty much the same no matter how many extra dimensions one might be pleased to add. But I haven't looked at it for long so maybe I just don't understand what Ehrenfest has to say.
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### Re: Paul Ehrenfest on Physics and Dimensions, 1918

Having looked more closely at what Ehrenfest has to say about magnetism. He points out that 3D is the only dimension where the number of basis vectors is equal to the number of basis bivectors. Well...so, what? I think what he is indirectly getting at is that 3D is the only situation in which the mathematics of the day -- cross products or quaterions -- is applicable. Though this doesn't really have anything to do with the number of basis vectors being equal to the number of basis bivectors. If that were somehow true in some other number of dimensions it wouldn't matter. What matters is that two dimensions of 2D plane plus one of vector equals three dimensions. We have a convenient thing that the dual of a plane is a vector. Vectors are easy to deal with. So we convert each plane to its dual, calculate there, then convert them back to planes.

Magnetism is essentially a 2D force. If there were a (N-1)D force then cross products would be useful in ND. Who knows? Maybe there is and they are. :-)
Last edited by PatrickPowers on Thu Feb 02, 2023 11:25 am, edited 1 time in total.
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### Re: Paul Ehrenfest on Physics and Dimensions, 1918

There are two kinds of generalizations of the cross product to higher dimensions: the perpendicular product and the bivector product.

The bivector product can be rationalized using the geometric product, given any arbitrary pair of vectors in N dimensions, it produces a bivector which is essentially an oriented 2D plane with associated magnitude.

The perpendicular product generalizes the (pseudo-)determinant definition of the cross product, i.e., as the "determinant" of the following pseudo matrix:
Code: Select all
[ x1 y1 z1 ][ x2 y2 z2 ][  X  Y  Z ]

corresponding to the cross product of <x1,y1,z1> with <x2,y2,z2>. X, Y, Z are the basis vectors of 3D space; when you expand this determinant and group the terms by X, Y, Z, you get the components of the cross product of the two vectors. The N dimensional analogue, therefore, is an NxN matrix with the basis vectors of N-space as the bottom row, and the components of (N-1) vectors filling the rest of the rows. So it's a product not of 2 vectors, which only occurs in 3D, but of (N-1) vectors. So a 4D perpendicular product is a ternary operator taking 3 vectors, a 5D perpendicular product is a quatenary operator taking 4 vectors, etc..

For a generalization of electromagnetism, I'd hazard to guess that the bivector generalization is probably more useful, since I wouldn't know how to rewrite the Maxwell equations for 4D if I had to add an extra vector into every cross product (where would that vector come from?). It also "makes more sense" physically-speaking: if you have 2 interacting charged particles, it would be strange if a 3rd vector appeared out of nowhere just to satisfy the requirements of the mathematical perpendicular product. That said, having an extra dimension probably has some fundamental consequences as to how electromagnetism would behave in 4D. In 3D a circulating electric current induces a directed magnetic force parallel to the "extra" dimension; in 4D there would be two dimensions leftover, so what would the magnetic field be? A circulating field in the orthogonal plane? In 3D, a linearly-travelling electric charge induces a circulating magnetic field around the line; in 4D there would be 3 dimensions left over. What shape would the magnetic field take? Whatever the answers, it will certainly produce some fundamentally different behaviours than in 3D.
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### Re: Paul Ehrenfest on Physics and Dimensions, 1918

According the authoritative Purcell magnetism is a consequence of special relativity. Special relativity is a 2+1 dimensional theory. Extra dimensions are irrelevant. So magnetism works pretty much the same in 4D. Use geometric algebra. Instead of Faraday's arrows that are the result of the baneful cross product, think of a sequence of spinning discs each perpendicular to that arrow. Sort of like Chinese coins on a string.

In 4D it is possible to have magnetic fields just like ours, with a 2D planar force at every point and no force in all other directions. In 4D nature though that would be unlikely. Instead you get 4D fields with an ellipsoidal force with a maximal plane and a perpendicular minimal plane. Think of two spinning perpendicular discs, which in 4D intersect only at a point.

Once I got used to it I think it's in some ways better than the trad method even in 3D. Consider two ordinary 3D magnets. Put them side to side with their poles facing the same way. Shouldn't they reinforce one another? They don't. One of the magnets will do everything it can to flip over and point the opposite way. I always thought that was weird.

Instead think of each of those magnets as a rotating wheel. Put them side to side like that and the wheels grind together in a nasty way. Flip one over and everything goes smoothly. It's like gears.

"In 3D, a linearly-travelling electric charge induces a circulating magnetic field around the line; in 4D there would be 3 dimensions left over. What shape would the magnetic field take?"

Well, I believe I can't do the integral to derive the shape in 3D so I can't do it in 4D either.
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### Re: Paul Ehrenfest on Physics and Dimensions, 1918

Aha, for the magnetic field around a straight wire with a current I need to use cylindrical coordinates. Never used them before.

In 3D there are length, radius, and azimuth, with azimuth being an angle. 4D is the same with two azimuths. I don't need two angles so I'll call that the az plane. It is a tangent plane and is used only very locally so its orientation differs from point to point.

In this case the magnetic field is at every point a single plane and that plane is the length-radius plane, just as it is in our 3D. There is no magnetic force in the az plane. Adding more az dimensions makes no difference so you may have as many as you please.

As to the magnitude of the force I'd have to do the integration to trust the answer. My guess is that it is the same as in 3D.

It is true that you would need the azimuth angles to identify the location in space you are interested in, and the orientation of the magnetic field at that point. The magnetic field is perpendicular to the az plane at that point. But the values of those angles aren't relevant to the strength of the field.
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### Re: Paul Ehrenfest on Physics and Dimensions, 1918

This is something I have been playing with for a long while. It's not easy.

Instead of using Maxwell's equations, we use variously the definitions and Jefimenko's equation. Jefimenko's equations are the cause of electromagnetism. Also, we use the four fields, rather than the constants $$\epsilon$$ and $$\mu$$.

The four fields are E, H, D and B. H and D arise from charge and moving charge, as

D = Q a and H = Qv × a. Q is chargc, v = velocity, and a is the radiant field, eg $$a=1/\gamma r$$.

If one were to integrate 'a' over a sphere, then we get $$\int a\ ds = 4\pi/\gamma$$ and $$\int v\times a\ ds = 2\pi/\gamma$$ In Si units, $$\gamma=4\pi$$, in CGS units $$\gamma=1$$.

The second pair produce a force on the charge, as

F = Q E = Qv × B Note that if these are added, you get the Lorentz field for moving charges.

The common relation, which we suppose to be the case if a photon is to remain continious, is

E = zH = zcD = cB

In a photon, all four fields exist at all times, since this is observed in things like defraction.

In the style of Dirac, we write electric + i magnetic, where i is an unknown value, we get

z (cD+iH) = (E+icB).

This is the conversion of the generating fields to the recieved fields, electricity goes from a bi-vector to a vector, and magnetic goes from a vector to a bi-vector.

If circulation is driving these fields, then we just substitute 'a' for the four-dimensional variant. If one were to integrate 'a' over a sphere, then we get $$\int a\ ds = 2\pi^2/\gamma$$ and $$\int v\times a\ ds = 4\pi/\gamma$$ In Si units, $$\gamma=2\pi^2$$, in CGS units $$\gamma=1$$.

As in 3d, the vector turns into the a-vector, (ie n-1 vector). z changes units, but the snell relation E=cB=zH=zcD is unchanged. The importance of this is that 'c' is purely kinematic, and nothing to do with EM theory. There is however a gaping hole between the vector and the trivector.

c plays a role in both retarded potentials and special relativity, but the former is much more clearly explained.

The underlying cause is an equation $$\nabla \cdot J + \tau \rho = 0$$. Here $$\tau = d/dt$$ .This is the equation of continuity, and from this José Haras derived Jefimenko's equations. These equations are the causitive solution to Maxwell's equations (as reformulated by Heaviside).

[this needs a bit of development, but it deals with retarded potentials.]
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### Re: Paul Ehrenfest on Physics and Dimensions, 1918

I was looking for pictures of the magnetic bivector field around a wire in 3D. I didn't find exactly what I wanted, but I did find these pages on John Denker's site.
Maybe I'll make my own picture.
The one on the second page is good, except that it doesn't show how the field decreases with respect to distance from the wire, or how its orientation changes with respect to angle around the wire (though that's easily gotten by rotational symmetry).

ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
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### Re: Paul Ehrenfest on Physics and Dimensions, 1918

quickfur wrote:The perpendicular product generalizes the (pseudo-)determinant definition of the cross product, i.e., as the "determinant" of the following pseudo matrix:
Code: Select all
[ x1 y1 z1 ][ x2 y2 z2 ][  X  Y  Z ]

corresponding to the cross product of <x1,y1,z1> with <x2,y2,z2>. X, Y, Z are the basis vectors of 3D space; when you expand this determinant and group the terms by X, Y, Z, you get the components of the cross product of the two vectors. The N dimensional analogue, therefore, is an NxN matrix with the basis vectors of N-space as the bottom row, and the components of (N-1) vectors filling the rest of the rows. So it's a product not of 2 vectors, which only occurs in 3D, but of (N-1) vectors. So a 4D perpendicular product is a ternary operator taking 3 vectors, a 5D perpendicular product is a quatenary operator taking 4 vectors, etc..

This is also easy with the geometric product. It is the Hodge dual of the wedge product of the N-1 vectors. That is, multiply all of the vectors with the geometric product, take only the highest-grade part (that's grade N-1), and then multiply by the N-vector. The result is a vector perpendicular to the others.

vN = ± (v1v2v3 ∧ ... ∧ vN-1) (e1e2e3...eN)
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
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### Re: Paul Ehrenfest on Physics and Dimensions, 1918

This is about how magnetism is a result of special relativity. The Microscopic Origins of the Magnetic Field - John Denker https://www.av8n.com/physics/magnet-relativity.htm

The way I see this is
1) assume (wrongly) that electrical charges attract and repel one another instantly. The resulting predictions will be incorrect.
2) call the error "magnetism."
Last edited by PatrickPowers on Sat Feb 04, 2023 10:12 am, edited 1 time in total.
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### Re: Paul Ehrenfest on Physics and Dimensions, 1918

From Wikipedia's Jefimenko equations article

There is a widespread interpretation of Maxwell's equations indicating that spatially varying electric and magnetic fields can cause each other to change in time, thus giving rise to a propagating electromagnetic wave[6] (electromagnetism).

I never did believe that. I thought, "time does not pass for a photon. Without time, it can't oscillate."

However, Jefimenko's equations show an alternative point of view.[7] Jefimenko says, "...neither Maxwell's equations nor their solutions indicate an existence of causal links between electric and magnetic fields. Therefore, we must conclude that an electromagnetic field is a dual entity always having an electric and a magnetic component simultaneously created by their common sources: time-variable electric charges and currents."[8]
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### Re: Paul Ehrenfest on Physics and Dimensions, 1918

If you look at Jefimenko's equations, both exist simultaneously, including H=v×D. But Jefimenko's equations can be largely set to something like a definition for H and D, as D = Qa and H = Qv×a. An extra term is added in for retarded fields.

If you are looking for some reading matter on it, you could look for Theirry de Mees and José Heras. These provide an excellent alternative to relativity, You can also find Oleg Jefimenko's books as PDF as well, the one on the causality thing in electrics and magnetics is quite good.

Just as Bohr worked with the quantum notion of e and h, and that h has the same units as angular momentum, my study derives from trying to unravel the rationalisation and linkage problems between CGS and SI. The rationalisation thing is about shuffling $$4\pi$$ around in equations. This is down to a fine art now: one can attack it at reading speed.

The linkage problem is shuffling the factor $$c$$ around. It's not as well advanced, but it turns out to be quite useful when one is reading books on the current model of gravity.

The form of the Maxwell equation on the Lorentz-Heaviside page on the wikipedia are the ones that I use. You can even see this on the talk page.
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### Re: Paul Ehrenfest on Physics and Dimensions, 1918

At the moment, i am not dealing with any particular algorithms, but trying to devise some solution that works with polytopes as well. The Jefimenko thing tells us that we can treat c as a property of space-time, and any weightless particle moves at that speed. For electomagnetism, the properties are handled by two variables n and z. In the photon continuity equations, you insert n if c is not present. z is already given, but some stuffing around is telling me that z itself ought be a square factor.

It goes without saying that the GEM process does not involve special or general relativity, and that approach is not looked at. Note that once you start playing with quarterions, it is pretty much 3+1, but radiant flux and retarded field is not tied to any specific geometry.
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### Re: Paul Ehrenfest on Physics and Dimensions, 1918

quickfur wrote:In 3D a circulating electric current induces a directed magnetic force parallel to the "extra" dimension; in 4D there would be two dimensions leftover, so what would the magnetic field be? A circulating field in the orthogonal plane? In 3D, a linearly-travelling electric charge induces a circulating magnetic field around the line; in 4D there would be 3 dimensions left over. What shape would the magnetic field take?

The magnetic vector b is not the direction of any force. The force f on a particle with charge q and velocity v is
f = q v×b
= q v•B.
The magnetic bivector B is perpendicular to b but has the same magnitude.
(What is the dot product of a vector and a bivector? It is the projection of v onto the B plane, rotated 90° in the B plane, and scaled by the magnitude of B.)

There is no force parallel to the extra dimension.

Note that two parallel wires in a plane, with electric current flowing in the same direction, are attracted directly toward each other.

Here's that picture I was thinking of. The current is considered to flow downward (meaning electrons are moving upward in the wire). The magnetic bivector is the blue disk; its magnitude decreases in inverse proportion to the distance from the wire, going off to the right; and it is always parallel to the wire. The bivector is also shown rotating around the wire in increments of 45°.

Wire magnetic field.png (23.31 KiB) Viewed 244 times

In N dimensions, the magnetic bivector produced by a wire with direction a, at the point x in space, is
B = k (xa) / ||xa||N-1
where k is a constant scalar, proportional to the current in the wire.
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### Re: Paul Ehrenfest on Physics and Dimensions, 1918

That's precisely the picture I had in mind. It will be true in any number of dimensions.
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