Maxwell Could Have Done it All By Himself!

I posted an example involving \(x\to -x\) and how you get at least two possible simple transforms for the E-M fields which preserve Maxwell's equations, but that's not a continuous symmetry, and if I required that it be consistent with the composition of a set of continuous symmetries leading to this final coordinate transform, maybe then I do get a restriction indeed. Need to think about this more...

Edit: Actually I don't think this one can be built from repeated infinitesimal transformations, so I'd better consider rotations instead...

 

Log in or Sign up to hide all adverts.

What you're asking is whether or not the Lie group is connected. The Lorentz group is SO(3,1) is not, as you get the issue with determinant signs and \(\Lambda_{0}^{0}\) sign. Hence why we generally consider the orthochronous orientated Lorentz group, \(SO(3,1)^{\uparrow}\).

If you want to make the element responsible for \(x \to -x\) be connected to the identity then you need to consider the universal covering of SO(3,1), which is \(SL(2,\mathbb{C})\), ie the whole supersymmetry-like representation of things.

 

Log in or Sign up to hide all adverts.

Ah, so I could maybe move around in the complex plane to get from \(x\) to \(-x\) and avoid hitting a fixed point at \(x=0\), say?

 

Log in or Sign up to hide all adverts.

No! The Lie algebraic approach includes non-local transformations viz. generalized symmetries. The very paper I gave computes 1st order generalized symmetries of Maxwell's equations! The subject area of Lie symmetry analysis includes generalized symmetries (unfortunate naming convention "Lie type symmetries" only form a subset of the symmetries considered in "Lie symmetry analysis"). And yes, by non-local I mean that the transformed solution need not depend only on the original function u and derivatives thereof at the point x.

There isn't a "vast category" of Lie type symmetries for Maxwell's equations! There is a finite number of them, and they can readily be computed using Lie symmetry analysis. I strongly suspect that wherever you read that, the author is referring to generalized symmetries (and if he isn't, he's wrong!). And none of these approaches employ any sort of physical postulates -- they simply use standard tools to classify all the Lie type symmetries of a given PDE (whether the PDE is related to electromagnetism or population dynamics, the methods are the same).

Erm, yes, local symmetries, meaning Lie type symmetries! To make sure this is clear:

1) Lie symmetry analysis involves different types of symmetry: the classical "Lie type" symmetries and "generalized" symmetries.
2) The full group of Lie type symmetries of Maxwell's equations is well known and can be derived in a straightforward manner using tools from Lie symmetry analysis.
3) Generalized symmetries of Maxwell's equations can also be computed using the methods of Lie symmetry analysis, but this is more complicated.

Whether what you're doing has any worth -- if you're enjoying it, then it's worth it!

Please Register or Log in to view the hidden image!

 

I'd have to think about that. I didn't enter this thread with the intention of making up my own derivation of relativity after all, but if I had, I'd prioritise postulates that could be justified via observation over ones that just subjectively seemed "reasonable" to me (because they might not seem so "reasonable" to someone else). I'd also make a point of deriving the full symmetry group of electrodynamics (or in practice I'd cite the work of someone else who did), just to see how many assumptions were really needed and how many were already included in electrodynamics.

Actually, before continuing, I think it would be a good idea for you to try to clarify some of the terminology you're using. What exactly do you mean by the phrase "inertial reference frame" for instance? This might sound silly and basic, but it's one of the things I find everyone discussing relativity is vague about, and in this case (a derivation of relativity that is supposed to be "mathematically airtight"), it's preventing me from getting a firm handle on a lot of what you're trying to say. Normally I avoid the term "frame" or just use it as a synonym for "coordinate system", but from the way you write about reference frames I don't get the impression you're using these terms synonymously, so what exactly is a "reference frame" to you?

 

I'm actually using your suggested Lie-type symmetry method to see if the EM fields can be shown to uniquely transform as spatial vectors under spatial rotations. That would be very useful, we wouldn't need to take it for granted and could apply the resulting symmetries if need be (I was originally going to take it for granted after first demonstrating/arguing that the field transformations between different coordinate systems and different frames are linear)... I'm also going to try and check it with the generalized Galilean transform of Section II and see if it forces \(\kappa=-1/c^2\).

 

Sort of, you are allowed to wander into different regions of the space of linear maps without violating the required properties. In the complex number case, you're allowed to move off the real line, provided you don't mind using complex numbers.

If you wanted to apply \(x \to -x\) then you're changing the orientation, which means the determinant must be -1, which isn't allowed if you're working in SO(3,1), but it is if you're in O(3,1). Nothing wrong with that in principle. You don't want to swap the sign on t though, so \(\Lambda_{0}^{0}\) must be positive. Thus at worst you should be using \(O(3,1)^{\uparrow}\).

In SL(2,C) you have the identity I but you also have -I, since det(-A) = \((-1)^{2}\)det(A) = det(A) for A a 2x2 matrix. Hence it contains the map which flips all the coordinates, which normally resides in \(SO(3,1)^{\downarrow}\).

To use your complex plane example (though this isn't really how universal covers work), consider \(S^{0}\) embedded in \(\mathbb{R}^{1}\). It's a pair of disjoint points. You can't move from one to the other without leaving the sphere. However, if you upgrade to \(\mathbb{C}\) and view \(S^{0}\) as the intersection of \(S^{1}\) and the real axis you can now move from +1 to -1 without leaving the set \(|z|=1\). Actually, in that case you have one of the reasons complex analysis and complex Hilbert space stuff is so much richer than the real versions.

 

Yeah I was thinking "maybe I can build this from continuous symmetries via the transformation \(x\to ax\), but I suspected that would hit a fixed point at \(x=0\). Instead I should have considered \(x\to xe^{i\theta}\) and converted it to infinitesimal form.

Anyhow I did a bunch of that "overdetermined system of PDE's, get an undergrad to solve them" work for rotations and the corresponding actions on EM fields a couple days ago, am getting close to a final answer which I hope will be unique, so far it matches what we'd want (that the EM fields must rotate like vectors). This is actually helping to give me a lot of insight into the reasons Lie Algebras are used so heavily in quantum physics and why their commutation relations are so important- it's nice to see some useful applications at the classical level for a change.

Anyhow if I manage to show that rotations indeed do what I expect they must to EM field vectors, and then move on to infinitesimal boosts and consider those results, if I can solve the remaining issues with Lie Algebra methods, then I will make note of such and then put forth the simpler derivation I worked out based on physically-motivated mathematical postulates and some physical reasoning added in, so we can then show that it works out whether you go on classical physical reasoning (i.e. the idea of bringing in charges and currents from infinity and having them superimpose), or whether you insist on working exclusively from Maxwell's equations.

I think some form of physically-based mathematical postulates along the lines of what I did in Section II might still be of utter necessity, though. We could systematically deduce all the local symmetries of Maxwell's equations as Guest suggested, but that still leaves us with an infinite selection of symmetry parameters and whatnot, leaving us to determine the physical meaning of various changes of variables, when such meaning exists. I.e. no doubt Lorentz boosts are a symmetry of Maxwell's equations, but how do we argue purely on Lie symmetry analysis alone that this actually corresponds to the relationship between certain coordinate systems in two different inertial frames? On the other hand, if we stick with what I did in Section II, then we have a physical interpretation of rotations and boosts in the \(\left(t,\vec{x}\right)\) coordinates, and we can use Lie-type analysis to perhaps uniquely determine the parameter \(\kappa\) in those boosts as well as the corresponding EM field transformations.

Question: To apply Lie-type analysis with the coordinate transformation rules of Section II as a starting point, would we not need to use the full Maxwell's equations and not just in vacuum, get an even heavier set of equations to deal with and thus determine how charges and currents transform as well, all in one step? Or would it be sufficient to make an argument like the one with which I've attempted to start off Section III, where I argue that the fields in two inertial frames share a strictly local relationship which depends strictly on the fields themselves?

If I can legitimately use the same logic to show that the field correspondence depends only on the content of the local fields themselves and not on the possible presence of local charges and currents, then could I not just apply Lie's method with Maxwell's equations in vacuum to show \(\kappa=-1/c^2\) and determine the general E-M transformation rules, then apply these same transformation rules to Maxwell's equations with charges and current afterwards to find how charge and current transforms? Maybe they both get equivalent results but the physical reasoning introduces heuristics while reducing rigour?

P.S. przyk, I will reply to your post very soon. For the time being let me just say, I certainly don't plan on playing Russell until some Godels comes along to kick my ass. I figured most classical theories of physics are taught with a lot of hand-waving and common acceptance of this hand-waving as physically reasonable, so I wasn't terribly opposed to hand-waving, as long as it's restricted to areas of classical physics where it's already generally accepted, and then trying to show that no further hand-waving or guessing is then needed to derive Special Relativity as a direct and inescapable consequence.

 

I have to say, it's been rather time-consuming, but I'm really loving the information and discussions coming up in this conversation, I'm certainly learning lots of new things and I hope everyone else is getting some entertainment out of it too. Someone should have mentioned this kind of Lie symmetry analysis stuff when I was saying in Rpenner's recent Relativity thread how there are more fundamental derivations of Relativity directly from Maxwell, rather than the typical ones presented where you have to accept the speed of light postulate as a starting point (any comments on my previously-stated objections/counter-examples to the lightspeed postulate approach?). So cheers everyone, I hope to get back on track soon and start wrapping up the essential results.

 

Mod note : Chinglu has had a post removed because he's rehashing the 'Prove it using the recursion theorem!' stuff and ruining an otherwise excellent thread. If he wishes to discuss that then, as I've offered many times, I'll reopen the thread on it provided he agrees to first address my retort of his "It maps to two places!" claim. Unless that condition is met any further attempts to discuss the recursion theorem in regards to SR will be met with warnings and holidays. If he agrees to said condition he is to PM me and confirm it.

 

Much appreciated as always. I was considering reporting the post, but figured you or one of the other mods would deal with it pretty rapidly. Ah, the eternal problem of dealing with Wiki-plagiarizing, and people who misrepresent advanced material in the hopes that no one here is knowledgeable enough to catch them... do they have friends they brag to about their "research" and this is their way of making it look like they're actually up to something? They have some kind of "reputation" here they need to defend? Weird why anyone would need to bother doing that, considering most of us are posting under pseudonyms in the first place.

Anyhow, today maybe I can sit down and finish up my playing around with Lie Algebras, and try to summarize/integrate some of my results into this topic. That was pretty awesome advice on Guest's part, I have to admit.

 

Yes, I should try to acquire some older materials on the subject (i.e. Heaviside's approach), and see how they originally worked it out, and how far they were able to take their conclusions. All the same, I intend to show here that we have a specific symmetry which we should already select as the mathematical interpretation of a physical velocity boost, and I will see what constraints we can put on this interpretation via Lie symmetry analysis and compare the results to those produced from the "physical reasoning" I was originally planning to use. But the general idea is to show that if you accept Newton's laws for objects at or infinitesimally close to rest, and you accept the complete laws of electrodynamics, then whether by advanced mathematical argument or via simplified mathematical argument supplemented with physical reasoning, Special Relativity becomes not only convenient, but fundamentally inescapable.

Yeah I was thinking more along the lines of "mathematically airtight, if you already accept the standard postulates of classical mechanics including electrodynamics."

You're correct- when I speak of an inertial frame, I think of a physical observer not being acted on or interacting in any substantial manner with anything else, and thus moving at a constant velocity with respect to other "inertial frames". When I speak of systems, I'm generally referring to systems of coordinates, which could apply to a single inertial frame, multiple frames, or even non-inertial frames.

 

All this to say what? That 1905 or 6 could have occurred in say 1861?

Not sure if I get the point of that. Um, do you get a feeling you're having an epiphany cuz there's a helluva lotta energy, the frenetic kind, going on here. Maxwell himself might even say you're amped up.

So fer ijjits such as me, could you maybe restate your thesis, lest my eyes glaze over with a sense of skeptical resistance to the tether your lengthy derivation drags me through. On the upside, all I can say is wow..

 

It's to show that all of the core results follow directly from physical facts which were known and well-established as of 1861 or earlier, so any critics of Relativity must argue with these basic facts in addition to the results of more sophisticated experiments and alternative, simpler ways of deriving the theory. Not a "thesis", just a compendium of different efforts in this regard, with my own calculations thrown in when I can't find the relevant work elsewhere.

 

How are you getting on? It's been nearly two weeks now.

 

I've decided to shelve this just momentarily, until I've had a chance to look into the recommended Lie Algebra stuff more. I worked out some of the relevant details, but I want to make sure I understand everything I need. I actually acquired Olver's book as Guest recommended, so I'm trying to read a bit of it to get a firm grasp on what I can and can't do with this technique.

If there's enough demand, I can just go ahead and post the arguments I already worked out, with a small bit of physical reasoning where necessary, and then go through the Lie Algebra stuff and see if any of that physical reasoning is actually not necessary to prove the argument, when advanced mathematics is applied instead.

 

I'm interested, but I don't want to distract you. On the other hand there's something you said that makes me wonder if you're going to get bogged down. Apologies in advance if it turns out I've wasted your time, but I won't feel comfortable if I don't mention it.

You said:

"I'm actually using your suggested Lie-type symmetry method to see if the EM fields can be shown to uniquely transform as spatial vectors under spatial rotations".

That doesn't seem to square with what Minkowski said:

"Then in the description of the field produced by the electron we see that the separation of the field into electric and magnetic force is a relative one with regard to the underlying time axis; the most perspicious way of describing the two forces together is on a certain analogy with the wrench in mechanics, though the analogy is not complete."

Picking something at random from the internet, see the depiction of an electric field at Andrew Duffy's Physics 106. See where he says an electric field can be visualized on paper by drawing lines of force. This isn't "the field produced by the electron". It's just a depiction of the linear aspect of the force one electron can exert upon another. The rotational aspect is depicted on this page. To visualize "the field produced by the electron", the electromagnetic field, you have to combine them. Maxwell didn't talk about "molecular vortices" for nothing. You end up with something like this:

Please Register or Log in to view the hidden image!

Public-domain vector-field image by "Fibonacci"

Place two of these things down next to one other and they move apart in a linear fashion. Throw one past the other and it swirls around too.

 

Minkowski wasn't talking about rotating the observer's reference frame there. It's generally taken for granted that we can treat electric and magnetic fields as 3-dimensional spatial vectors whose components are functions of position and time. This implies that a rotated observer will see the fields rotate in a corresponding manner. With Lie's symmetry analysis method, I found that this rotational property is automatically required by Maxwell's equations, but there are some small caveats involved, and I want to go through Olver's text a bit to make sure I'm properly dealing with those caveats.

One thing I will say from this methodology: Lie symmetry analysis does indeed show that we must set \(\kappa=-1/c^2\) in order to preserve Maxwell's equations, i.e. it requires that the spacetime transforms in Section II reduce to the Relativistic case, but I haven't sorted out the full implications for how EM fields transform for their own part using this approach, so I'm not completely sure if that portion of the argument yields a unique result, and hence I'm delaying further additions 'til I have a chance to cover the necessary background.

 

Well done captian, just sat here and read all of this. Very good post, nice one!

Please Register or Log in to view the hidden image!

 

All points noted Cpt. Good luck with this.