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Help with the basics of Maxwell and Lorentz equations 
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#1
Feb2310, 07:46 PM

P: 63

Hi there.
I have just started to read about the Relativity part of Maxwell's equations and would like to get some clarification from any one here were I can discuss and/or get answers in a more " every day life " way of writing. 1) What is the most simple way of transforming Maxwells equations into a Covariant form? I know that it is done by using fourvectors/tensors but how is it actually done? 2) I know that from Lorents Transformation, that magnetic and electric fields are simply different aspects of the same force. Also that when looking from a frame either in rest or moving, in reference to charged particles we get 2 different results, an electric field and an magnetic field. What I would like to know is if there is a quick way of showing that this is true ? Like a set of equations so that when put in a moving frame you get the Dimensions of a Magnetic field. 3) How do I prove in the most simple way how the Lorentz invariance ( invariant ) is true for Maxwell's Equation. What I know is this: Quantities which remain the same under Lorentz transformations are said to be Lorentz invariant. According to the representation theory of the Lorentz group, Lorentz covariant quantities are built out of scalars, fourvectors, fourtensors, and spinors. How do I use this knowledge ( if I even should ) to confirm that Maxwell's equations is L. invariance Thanks! 


#2
Feb2310, 09:04 PM

Mentor
P: 11,770

See the following page and related pages on the same site:
http://farside.ph.utexas.edu/teachin...s/node121.html 


#3
Feb2410, 01:07 AM

Emeritus
Sci Advisor
PF Gold
P: 5,597

This may be helpful: http://www.lightandmatter.com/html_b...ch11/ch11.html



#4
Feb2410, 03:05 AM

P: 894

Help with the basics of Maxwell and Lorentz equations
Consider a moving charge in a frame where there is only a magnetic field. The charge feels a force. Now imagine the same situation from the charge's (instantaneous) rest frame. The only way that it could feel a force (since it is at rest) is if there was an electric field in this frame. So even without getting into the nitty gritty details, it is clear that relativity demands electric and magnetic fields must be able to transform into each other when changing frames. Does that help? If you want all the details, the links already provided look pretty good. 


#5
Feb2410, 03:10 AM

PF Gold
P: 4,087

Quick answer 
The EM field has a relativistically invariant scalar [itex]F^{ab}F_{ab}=2(E^2B^2)[/itex]. The field tensor is [itex]F^{ab}=\partial^aA^b\partial^bA^a[/itex]. The invariance is manifest, because if we boost the fieldtensor the scalar becomes, [tex]\lambda_p^a\lambda_q^bF_{ab}\lambda_a^p\lambda_b^qF^{ab}=(\lambda_p^a\l ambda_a^p)(\lambda_q^b\lambda_b^q)F^{ab}F_{ab}=F^{ab}F_{ab}[/tex] The [itex]\lambda[/itex] are not tensors but Lorentz transformation matrices, and the pairs in parentheses are inverses. 


#6
Feb2410, 03:50 AM

PF Gold
P: 4,087

How to make a magnetic field appear 
If we start with an electric field in the y direction, [tex] F=\left[ \begin{array}{cccc} 0 & 0 & E_y & 0 \\\ 0 & 0 & 0 & 0 \\\ E_y & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 \end{array} \right] [/tex] and boost in the x direction using this matrix ( twice ) [tex] \left[ \begin{array}{cccc} \gamma & \beta\gamma & 0 & 0 \\\ \beta\gamma & \gamma & 0 & 0 \\\ 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & 1 \end{array} \right] [/tex] the field tensor becomes [tex] F'=\left[ \begin{array}{cccc} 0 & 0 & \gamma E_y & 0 \\\ 0 & 0 & \beta\gamma E_y & 0 \\\ \gamma E_y & \beta\gamma E_y & 0 & 0 \\\ 0 & 0 & 0 & 0 \end{array} \right] [/tex] which has a magnetic field in the z direction. Note that [itex]E^2+B^2=E_y^{2}\gamma^{2}+\beta^{2}E_y^{2}\gamma^{2}=E_y^2[/itex]. I transcribed my last two posts from some Latex notes I already had. I hope it's correct in every particular but not guaranteed. Right in principle, though. 


#7
Feb2410, 05:41 AM

P: 4,512

You might begin by reading a short diversion into covariant fields within Sean Carroll's Text, Notes on General Relativity chapter 2. He derives one covariant form and but not the hardest. Each of two pairs of Maxwell's equations combine to one covariant form. It can take a few pages to get from Maxwell's three dimensional equations to the fully covariant 4 dimentional form for the two charge and current equations. The first step is to recognize that the elements B^{i} are equal to B_{i} and E^{i} are equal to E_{i} in orthonormal coordinates. The second point to understand is that [rho, J] is a 4vector in Minkowski space, and that [rho, J] is it's covariant form. The entire derivation is not too obtuse, just tedious. (The really cool thing that drives all this, is that what is true in one coordinate system for tensors, is true in any other well behave coordinate systemwithout singularities 'n stuff. We can derive things in orthonormal coordinates and they are equally true in other coordinate systems. This means that the covariant forms of Maxwell's equations are just as good in most of general relativity as they are in nice flat Minkowski space. This is an awfully nice thing to know. It means that the simple impression of a vector bundle upon the spacetime manifold has simple global rules. And this simple impression can be coerced into a metric (but I degress). Even better for mathematical simplicity, the connection coefficients of Riemannian geometry vanish for pforms (antisymmetric covariant forms, such as the electromagnetic field tensor).) Ultimately, you obtain two compact equations, dF=0 and d*F=J, or just one physical equation d*F=J, where dF=0 is a mathematical identity that rules out magnetic charge (All exact forms are closed on well behaved manifolds. This is applicable when F=dA, and A is the socalled vector potential. For the dizzying extremes where exact forms are not closed, see de Rham cohomologies). I find this all very exciting, or I wouldn't be blathering like this. Anyway. I think the undoctored derivation is in my notes somewhere that should be alot less obtusive than the above. But you have to be willing to take the risky step of asking for an MS word file that I would send to you. 


#8
Feb2410, 06:46 AM

Sci Advisor
P: 1,261

abotiz:
I think you would do better looking at a textbook than trying to wade through lots of posts. 


#9
Feb2410, 12:20 PM

#10
Feb2610, 04:34 AM

P: 63

Anyone?



#11
Feb2610, 08:15 PM

Emeritus
Sci Advisor
PF Gold
P: 9,382

It sounds like you want an explanation both without math and without words that you may have to look up at Wikipedia or dictionary.com. I think that's expecting too much. You should probably try to follow a derivation from a book or a web site, and ask about the specific details that you find difficult.
I was going to copy and paste some stuff from my personal notes, but it looks like I messed up the signs somewhere and I'm not sure where yet. Can someone tell me if there's a standard convention for how to use the LeviCivita symbol with a + metric? Is [itex]\varepsilon^{123}=1[/itex] or [itex]\varepsilon_{123}=1[/itex]? 


#12
Feb2710, 03:10 AM

P: 4,512

I can't tell you if there's a dominant standard in +, but I can make a guess. The LeviCivita tensor is usually introduced as a tensor with all lower indices, where the elements of even permutations are equal to one. 


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