Maxwell's Equations in Tensor Form

It is a critical prerequisite of QED to understand how the laws of electromagnetism are encoded in tensor form. In this Lesson we build the 4-potential and then create the electromagnetic field tensor. Then we develop the 4-current and then combine everything into an arguably more elegant and versatile form of Maxwell's Equations.
Our next lessons will be a supplemental diversion into yet another formulation of Maxwell's Equations, this time using differential geometry and the language of differential forms.
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[@Nickle314]

It's worth going the whole hog on Geometric Algebra. If nothing else, because the resulting Maxwell equation [singular] is so shockingly beautiful.

[@XylyXylyX]

I know. I am very tempted. The problem is that it is new-ish for me and I may not be the right person for the job. I have noted that no-one has really tackled “spacetime algebra”…it is quite niche. I probably wont be able to resist.

[@Nickle314]

@@XylyXylyX I started by looking at simple GA. The basic idea of a bivector comes across as so obvious, why has it been ignored. Far better than Gibb's formulations. On the books. Hestene's book, Space-Time Algebra is the one to go for the physics side. That Quarternions, Dirac etc all drop in tells me again something is right about it. Vector and Geometric Calculus: 2 (Geometric Algebra & Calculus) Macdonald, Dr. Alan L. Linear and Geometric Algebra (Geometric Algebra & Calculus) Good I suspect if you want to transition from your current knowledge. More vector than tensorial Geometric Algebra for Computer Science (Revised Edition): An Object-Oriented Approach to Geometry (The Morgan Kaufmann Series in Computer Graphics) I really like this one. Partly because my first background was computing. Dorst is the one to google if you want to see what software can do with GA. It's worth getting to see the other usages for GA, such as Robots. kzbin.info/www/bejne/bGHdkJumeqanepo&ab_channel=sudgylacmoe Is good if you want videos.

[@mastershooter64]

@@Nickle314 yea GA is super apt even for basic physics like you can think of torque as a bivector which makes more sense!

[@josephmellor7641]

@@XylyXylyX sudgylacmoe has you covered: kzbin.info/www/bejne/m2jEeomJi8hrmas

[@tomgraupner171]

Thanks. I like the way you present that stuff. One thing at 47:50 is spooky. You said that the number of unknown still matches the number of equations. huh? The are 8 ME - 2 scalars and 2*3 vector equations , while we only search for 6 unknowns (Ex, Ey, Ez, Bx, By, Bz).

[@TrailersReheard]

General comments: 1) Representation theory makes a lot of this obvious after a lot of playing around. The E and B fields rotate like Euclidean vectors but they boost rather confusingly. If one finds how they boost and change in order to satisfy Maxwells equations in a new frame, that transformation is exactly how the components of anti-symmetric tensors transform via SO(1,3). One can justify the move to tensor notation without it feeling arbitrary or like 4-potentials become fundamental 2) The bianchi identity is free. It is just as free as the exterior algebra approach. If the bianchi identity is a second maxwell equation, then so is the fact that d^2=0 where d is the exterior derivative. There really is just 1 physical law expressed, regarding the 4 divergence of an anti-symmetric tensor in Minkowski spacetime

[@XylyXylyX]

I am aware of number 2…. I probably should have mentioned it. I wanted to express the second pair in tensor form. I guess you are arguing that we are already down to one equation? I think that is fair, actually. But if we don’t explicitly list the Bianchi identity as the second pair, it suddenly becomes a bit obscure. That is, we need to account for all four ME, even if two are actually free after we define our potentials and EM tensor. As for 1, … it is pretty well known that the relativistic transformation of the fields preserves Maxwell’s equations…I think trying to figure out the exact form of the EM field tensor would be confusing/hard starting from that more elementary move. I like the potential angle, mostly because the potential formulation dominated QM and QED.

[@TrailersReheard]

@@XylyXylyX oh yeah, I agree the bianchi identity is important to acknowledge, so that it's a known fact. Additionally I agree this method is probably better for where you're aiming re: QED, QFT. Just thought it was important to mention the motivation for the tensor need not appeal to the potential formulation. Though I do disagree about it being very hard to figure out the form if one does have even a little experience identifying 2-forms and bi-vectors with psudeovectors, and then just checking which way of writing the EM fields as a tensor is compatible with the 3-vector expressions for the Lorentz force via F^{μν}u_μ where u is the four velocity.

[@keshavshrestha1688]

1st-> After Choosing suitable lagrangian for electromagnetic field can't we write both inhomogeneous and homogeneous maxwell equation into single one.? OR 2nd-> generalising those equation to curved spacetime helps to unify those two separate equation into single one.?

[@XylyXylyX]

Hmmm….I am not aware of this…however, my “second” Maxwell equation is actually automatic. The first one comes from the Lagrangian. That is, any set of fields defined the way we have (from a potential) will satisfy what I called the “second” Maxwell equation. So..in a way…you can argue that we are already down to one. Is that what you mean?

[@keshavshrestha1688]

@@XylyXylyX Thanks but does your argument solve the unification of maxwell field equations into single one? I have searched a lot but none able to address unification apart from geometric algebra is there any source you can recommend about unification without using geometric algebra.?

[@XylyXylyX]

@@keshavshrestha1688 It is a bit semantic…it depends on whether or not you feel the “automatic” Bianchi identity should count as a separate equation. Ultimately it rests on our definition of the field via a potential, so I personally think it counts as a separate equation. It basically embodies the definition of the electromagnetic field tensor. I am unaware of anything aside from geometric algebra that claims to wrap everything up in on one tidy expression.

[@keshavshrestha1688]

@@XylyXylyX Ok, to describe gravitational field around some celestial body we have to solve complicated EFE for many years which as a result we obtain metric tensor but to describe electromagnetic field we have simply electromagnetic field tensor which i think rather easier to obtain in comparison to metric tensor why is that so.? Why electromagnetic field are linear in nature but gravitational field are non-linear. ?

[@XylyXylyX]

@@keshavshrestha1688 I suppose a good answer is that the mathematics of field theory is set inside spacetime and fields can be very simple or very complex. However the mathematics of spacetime itself can only be done in the setting of differential geometry which doe not have a simple version (in 4 dimensions). Easy question with a hard answer!

[@sahhaf1234]

Actually I think that maxwell's equations in tensor form are analogous to stress-energy tensor in continuum mechanics..