Electrodynamics/Lorentz Transformation

Fields and Forces

The forces caused by electric and magnetic fields are mostly what we can actually measure in electromagnetism. These vector quantities are related to the scalar and vector potentials as follows:

$\mathbf {E} =-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}$

$\mathbf {B} =\nabla \times \mathbf {A}$

The E is the electric field vector, and the B is the magnetic field vector. Because of these equations, electric fields are frequently called "E Fields", and magnetic fields are frequently called "B Fields". This book may use either of these notations.

Lorentz Equation

By comparison of these equations with the general expression for force in gauge theory, we find that the electromagnetic force on a particle with charge q is

$\mathbf {F} =q\mathbf {E} +q\mathbf {v} \times \mathbf {B}$

where v is the velocity of the particle. For historical reasons this is called the Lorentz force.

Relativity

Relativity is, in brief, the study of reference frames. A reference frame is a fixed coordinate system against which local measurements are taken. Consider the common example of two observers: Observer A is located on a moving train, and Observer B is standing in a field watching the train. Here is what the two observers see:

Observer A

Observer A sees the train as being stationary, and the field as moving.

Observer B

Observer B sees the field as stationary, and the train as moving.

Both observers are essentially the origin of their own coordinate system. Clearly, the two coordinate systems can be related together through some sort of transformation, that is that things that Observer A can see can be translated into coordinates according to Observer B, and vice-versa.

In linear algebra, a coordinate system has a basis, a small set of unit vectors that can be used to describe all the points in that system. If we can relate the basis vectors of Observer A and Observer B, then we can relate any point in either system to the other system.

Because basis vectors are vectors (rank-1 tensors), the transformations between them are typically matrices (rank-2 tensors).

Special Relativity is based on the idea that the laws of physics are the same in all inertial reference frames, and that the speed of light, c, is a constant regardless of the frame. An equation is said to be "Invariant under Lorentze Transformation" if it satisfies these requirements when a lorentz transformation is applied to it. We will see that the requirement of lorentz invariance is an important one in electrodynamics.

Another subject, General Relativity, expands the mathematical ideas of relativity to non-inertial frames. We will not consider general relativity topics in this book.

Lorentz Transformations

It is tempting, but naive, for us to consider only 3 basis vectors. These vectors, the spacial vectors, can be called without any lack of generality "X", "Y", and "Z". However, one of the important results from Einsteins work on relativity is that time is also dependent on the reference frame, and that therefore we need to consider all points in both space (X, Y, and Z vectors) but also in time (a T vector). All our vectors then have a length of 4, and our transformation matrices must be 4×4 matrices. A 4×4 transformation matrix that uses three spatial coordinates and 1 time coordinate is known as a lorentz transformation matrix, or simply a "lorentz transformation".

If we have two coordinate systems, (X, Y, Z, T), and (X', Y', Z', T') and they are non-inertial systems, we can relate the two systems using the L transformation functions:

$X'=L_{X}(X)$

$Y'=L_{Y}(Y)$

$Z'=L_{Z}(Z)$

$T'=L_{T}(T)$

Now, if each element X', Y', Z', and T' are linearly related to X, Y, Z, and T, we can convert L into a matrix:

${\begin{bmatrix}X'\\Y'\\Z'\\T'\end{bmatrix}}=\mathbf {L} {\begin{bmatrix}X\\Y\\Z\\T\end{bmatrix}}$

As we can see from this equation, if we are going to apply a Lorentz transformation to a coordinate system, the coordinates must be specified in vectors of length 4. We will call all vectors with a length of 4 a "four vector". We will discuss four vectors in a later chapter.

Inertial vs. Non-Inertial Frames

An inertial frame is a frame with no net acceleration. Consider the case above with the two observers, Observer A, and Observer B. Observer A is on a train, and Observer B is looking at the train from a distance. If the train is moving at a constant speed and in a straight line, then the two frames are inertial. However, if the train is accelerating or decelerating, or if the train is not moving in a straight line, then the two frames are non-inertial.

The study of inertial frames is a field known as Special Relativity. The study of non-inertial frames is known as General Relativity. In this book, we will consider special relativity only.

Ohm's Law

The continuum form of Ohm's Law is only valid in the reference frame of the conducting material. If the material is moving at velocity v relative to a magnetic field B, a term must be added as follows:

$\mathbf {J} =\sigma \cdot \left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)$

The analogy to the Lorentz force is obvious, and in fact Ohm's law can be derived from the Lorentz force and the assumption that there is a drag on the charge carriers proportional to their velocity.