How does special relativity follow from classical electrodynamics

[Wox]

I'm trying to understand how one derives the relativistic treatment of the electromagnetic interaction from the classical one and which are the extra postulates made. We can start from Maxwell's equations and the Lorentz force. From the Galilean invariance of Newton's second law of motion $F=\frac{dp}{dt}=q(E+v\times B)$ one can derive (ref) how electric and magnetic field change when transforming to another inertial frame of reference. From this we realize (same ref) that Maxwell's equations don't have the same form in all inertial frames. Therefore the Lorentz force is now Galilean invariant, meaning that it must be described in another space (where inertial frames of reference transform in a different way). But how do we proceed and end up with the relativistic treatment? Let's say we take the Minkowskian geometry of space-time as postulated. Do we need any more postulates to end up at $F=\frac{dp_{rel}}{dt}=q(E+v\times B)$ where $p_{rel}=m\frac{dx}{d\tau}=m\gamma v$ the relativistic momentum instead of the classical $p=m\frac{dx}{dt}=mv$?

Additionally, can we derive Lorentz invariance of the Maxwell equations from $F=\frac{dp_{rel}}{dt}=q(E+v\times B)$ in the same way as we (same ref) tried showing their Galilean invariance? I'm not sure how to handle the effect of 4D Lorentz transformations on 3D vector fields.

 

[Simon Bridge]

Yep - you need a 4-vector form.
http://en.wikipedia.org/wiki/Electromagnetic_tensor

Hint: have a look at what happens for the field about a uniform line of charge in difference reference frames.

Overview:
http://staff.kfupm.edu.sa/PHYS/tahmed/Relativistic Electrodynamics.pdf

 

[Mentz114]

An interesting thing is that electric and magnetic fields transform like time and space under Lorentz transformations.

The EM invariant F^abF_ab is

invariant = -2c²E² + 2B²

from which looks rather like the metric ds² = -c²dt² + dx²

 

[Wox]

Ok, that concerns my second question and I will try the 4-vector approach to see where Maxwell's equations are Lorentz invariant. But concerning my first question, where does this 4-vector formulation come from? If we break down the 4-vector from of the Lorentz force (using the electromagnetic tensor) then we get for the spatial part $$\frac{dp_{rel}}{dt}=q(E+v\times B)$$ and for the time part $$qE\cdot v=\frac{dE_{p}}{dt}$$ where $E_{p}=m\gamma(t)c^{2}$ the energy (rest + kinetic) of the particle. The time part follows from the spatial part and Minkowski geometry since $\frac{dp_{rel}}{dt}\cdot v=qE\cdot v$ and $\frac{dp_{rel}}{dt}\cdot v=mc^{2}\frac{d\gamma}{dt}$ (which follows from four-velocity and four-acceleration being perpendicular in Minkowski space). So in other words, the four-vector formulation of the Lorentz force comes from $\frac{dp_{rel}}{dt}=q(E+v\times B)$ and the geometry of Minkowski space. Now is $\frac{dp_{rel}}{dt}=q(E+v\times B)$ also postulated (maybe founded by experiments) or does it follow somehow from the geometry of Minkowski space and classical electrodynamics?

 

[Muphrid]

I expect that people studied four-vector dynamics before they realized how best to translate classical EM theory into that language, and while it is accepted to translate EM theory in this way, there are more consequences to how we should view EM fields than a lot of people realize, I think.

Look at the Faraday tensor $F^{\mu \nu}$. What I feel a lot of people don't realize is that it lends itself to a clean geometric interpretation: that of a linear combination of planes in Minkowski space. The Faraday tensor has 6 independent components. There are 6 linearly independent planes in any 4-dimensional space, and so just like a vector is a linear combination of basis vectors whose magnitudes are given by components, the Faraday tensor is a linear combination of basis planes.

To me, this has profound implications for the nature of electric and magnetic fields themselves. It says to me that the electric and magnetic fields are naturally planes, not vectors themselves, and we've viewed them as vectors incorrectly all this time because it was good enough to model behavior in 3D space, but viewed as a projection of electrodynamics in Minkowski spacetime to our 3D hypersurface, that idea no longer makes sense.

 

[Darwin123]

I'm trying to understand how one derives the relativistic treatment of the electromagnetic interaction from the classical one and which are the extra postulates made. We can start from Maxwell's equations and the Lorentz force. From the Galilean invariance of Newton's second law of motion $F=\frac{dp}{dt}=q(E+v\times B)$ one can derive (ref) how electric and magnetic field change when transforming to another inertial frame of reference. From this we realize (same ref) that Maxwell's equations don't have the same form in all inertial frames. Therefore the Lorentz force is now Galilean invariant, meaning that it must be described in another space (where inertial frames of reference transform in a different way). But how do we proceed and end up with the relativistic treatment? Let's say we take the Minkowskian geometry of space-time as postulated. Do we need any more postulates to end up at $F=\frac{dp_{rel}}{dt}=q(E+v\times B)$ where $p_{rel}=m\frac{dx}{d\tau}=m\gamma v$ the relativistic momentum instead of the classical $p=m\frac{dx}{dt}=mv$?

Additionally, can we derive Lorentz invariance of the Maxwell equations from $F=\frac{dp_{rel}}{dt}=q(E+v\times B)$ in the same way as we (same ref) tried showing their Galilean invariance? I'm not sure how to handle the effect of 4D Lorentz transformations on 3D vector fields.

There is a book by H. A. Lorentz entitles "The Theory of Electrons." It does what you are asking about. Lorentz derives the "Lorentz transform" using "classical physics". However, he had to make ad hoc assumptions as to the nonelectromagnetic forces that hold and electron together. He made some assumptions about a so called ether wind. However, this book is closer to anything else I read as to "explaining" special relativity.
Lorentz takes into account the stresses on an extended charged body caused by the interaction between different parts of the charged body and the electromagnetic field. Lorentz also takes into account the delay in the electromagnetic interaction caused by the speed of light. The self interaction of the body that is moving at constant velocity relative to the ether is what causes the foreshortening of length and the dilation of time. The dilation in time causes the increase in mass.
There is one important point about the Lorentz theory that hasn't been discussed very well. This is that the Lorentz theory is not strictly Newtonian.
Newton's third law postulates that if a first body applies a force to second body, the second body immediately applies a force to the first body that is equal in magnitude but opposite in direction to the force on the second body. Thus, every action has an immediate reaction.
Lorentz postulates that there is a time delay between the action and the reaction of an electromagnetic force that is proportional to the distance between the two bodies. Thus, internal forces of a charged body don't completely cancel out. If the charged body is very big, the electromagnetic force generated inside the charged body will greatly effect its movement and its shape.
Lorentz derived an "effective time" and an "effect position" that was derived from the "actual time" and "actual position". However, he didn't realize that these effective quantities were every bit as real as the actual quantities. He himself said that he didn't realize that the effective quantities can be measured!
Hence the Lorentz derivations are not 100% Newtonian. Newton's Third Law of Motion is written in present tense. Newton described his law of gravity as action at a distance. There is no such thing as a force field with a delay in Principia. According to Lorentz, there is always a delay at least with respect to the electromagnetic forces. Although not Newtonian, relativity is still referred to as part of "classical" physics.
Lorentz did not realize that he had discovered a symmetry principle. Einstein extracted a symmetry principle which is more general than the dynamics described by Lorentz. Einstein's special relativity applies to all forces, both electromagnetic and nonelectromagnetic.
Einstein's symmetry principle even applies to quantum mechanical systems. Lorentz's theory is strictly classical. It requires the electromagnetic field to be continuous and the particles to be discrete although extended. So one can't analyze quantum systems using the Lorentz theory. Thus, Einstein's theory is more general than the Lorentz theory of the electron. Symmetry principles in general tend to be more robust then the detailed dynamics they are derived from. However, that is a topic that deserves a separate discussion.
"The Theory of Electrons" is a great book. The mathematics of Lorentz are far harder then the mathematics of Einstein, but are satisfying in their own way. I couldn't find any edition of this book still being published. I had to buy an expensive used copy of "Theory of Electrons." Still, the book was worth it.

 

[DrGreg]

"The Theory of Electrons" is a great book. The mathematics of Lorentz are far harder then the mathematics of Einstein, but are satisfying in their own way. I couldn't find any edition of this book still being published. I had to buy an expensive used copy of "Theory of Electrons." Still, the book was worth it.

You might not want to hear this, but a bit of googling has revealed you can download the book (which is out of copyright) free of charge from http://archive.org/details/electronstheory00lorerich.

 

[Darwin123]

You might not want to hear this, but a bit of googling has revealed you can download the book (which is out of copyright) free of charge from http://archive.org/details/electronstheory00lorerich.

That is okay by me. They scanned the book in 2006. I bought the book in 2002.
I couldn’t wait for four years. I thought that I needed it right away for something that I was doing. I had immediate gratification when I started to read it.
At least I have it in bound form this way. Printing off of my own printer is expensive. I didn’t know how to get my pdf files printed off-line at a store. I know now, but it was all knew to me then. Getting it bound would have cost extra.
Your link is very is good. Now, there is no excuse for not reading from this great book. I can quote from the book with the confidence that the book is available to all.
Here is the link and a quote showing that the book was scanned in 2006.
http://archive.org/details/electronstheory00lorerich
“Copyright-evidence: Evidence reported by scanner-ian-white for item electronstheory00lorerich on November 29, 2006: visible notice of copyright; stated date is 1916.”

By the way, I should have been more specific in my previous post. Quantum mechanics is fully consistent with special relativity, not general relativity.
De Broglie was supposedly inspired partly by special relativity. That is one reason that he contacted Einstein. The mass-energy equivalence of special relativity was also used by some early pioneers in quantum mechanics. Dirac's theory of the electron combines special relativity and quantum mechanics. There is no serious inconsistency between special relativity and quantum mechanics. The consistency between special relativity and quantum mechanics is actually a strong point in favor of special relativity.
Quantum mechanics is not consistent with the Lorentz theory of the electron. The Lorentz theory of the electron requires classical dynamics with a delay between action and reaction forces. Special relativity is a symmetry principle that can be applied to any system of dynamics, including quantum mechanics.
Quantum mechanics is also not consistent with general relativity. The problem is the inconsistency between general relativity and quantum mechanics. I don't fully understand causes the inconsistency. It has something to do with nonlinearities in Einstein's field equation. Quantum mechanics doesn't work in strong gravitational fields because of these nonlinearities.

 

[harrylin]

There is a book by H. A. Lorentz entitles "The Theory of Electrons." It does what you are asking about. Lorentz derives the "Lorentz transform" using "classical physics". However, he had to make ad hoc assumptions as to the nonelectromagnetic forces that hold and electron together. He made some assumptions about a so called ether wind. However, this book is closer to anything else I read as to "explaining" special relativity. [..]

That book gives, if I'm not mistaken, an overview his pre-1905 theory of electrons, but yes it gives an intuitive explanation of the phenomena. Lorentz improved on that with his new theory of 1904 (also referenced in the OP's paper), and which avoided such ad hoc assumptions. Note that the transformations were assumed to be valid for all forces. Although Lorentz may have been one of the last people to understand the achievements of his own paper, not by coincidence it's included in the bundle "The principle of relativity" about SR, and one can read it here:
http://en.wikisource.org/wiki/Electromagnetic_phenomena

To get back at the OP: you could assume the relativity principle for all laws of physics incl. the Maxwell equations, and that is how it was done at Poincare's request, first by Lorentz in the above-mentioned paper (followed by small corrections in a 1905 paper by Poincare) and next more elegantly by Einstein. Maxwell's laws include independence of the speed of light c of the motion of the source. So, if in addition you postulate a Minkowskian space-time geometry, that's certainly sufficient - perhaps even overkill, for with those you already have the space-time invariance which is equivalent to the Lorentz transformations (see section 4 of: https://en.wikisource.org/wiki/On_the_Dynamics_of_the_Electron_%28July%29 )

Before that, attempts were made with little fixes such as length contraction, and that was not satisfactory. See the ref. 5-11 of that article which give an overview.

PS I didn't yet find the time to carefully look into the questions in post #4 but I guess that the answer is contained in the literature that has been linked.