Lorentz transformation in four dimensions of the electromagnetic tensor

[Vincent_111]

Homework Statement

Given: electromagnetic tensor F(superscript)uv:

electromagnetic tensor F' after the lorentz transformation:
[ 0 -Ex -gamma(Ey-VBz/c) -gamma(Ez-VBy/c)
Ex 0 -gamma(Bz-VEy/c) gamma(By-VEz/c)
gamma(Ey-VBz/c) gamma(Bz-VEy/c) 0 -Bx
gamma(Ez-VBy/c) -gamma(By-VEz/c) Bx 0]

Problem statement: "provide the an (theoretical) explanation and show the intermediate steps in the transformation from F to F'."

Homework Equations

E (electric field) = -grad (electric potential) - 1/c dA/dt
B (magnetic field = rot A
Transformation law of a tensor(in formal notation): F'uv (superscript) = A u(sup.scpt) a(sub. scrpt) A v(sup)b(sub) F ab (sup)

The Attempt at a Solution

The transformation has to do with the reletive direction+speed(?) of the second frame and I suppose we could take a 'simple' transformation in which we move from a 'rest frame' to a frame that moves with constant speed along the x-axes, which would yield relatively simple matrices.

I am however unsure how we can construct the two matrices;
A u(sup) a(sub) and
A v(sup) b(sub)

Could anyone make any sense of this or comment on things that are not clear, I would appreciate it

PS: (I apologize for the weird notations I am not able to produce formulas with the program on my computer at this instant.)

 

[mark726]

Thank you for your question. I will provide an explanation and show the intermediate steps in the transformation from F to F'.

First, let's define the electromagnetic tensor F (superscript)uv as:

F = [Ex, Ey, Ez, Bx, By, Bz]

Now, let's consider a Lorentz transformation, which is a mathematical model used to describe how an observer's measurements of space and time are affected by the relative motion of two reference frames. In this case, we are transforming from a "rest frame" to a frame that moves with a constant speed along the x-axis.

The transformation can be represented by two matrices, A u(sup) a(sub) and A v(sup) b(sub). These matrices are used to convert the components of the electromagnetic tensor from the original frame to the new frame.

The transformation law for a tensor is given by:

F'uv (superscript) = A u(sup) a(sub) A v(sup) b(sub) F ab (sup)

In this case, we are only concerned with the x, y, and z components of the electromagnetic tensor, so we can simplify the transformation law to:

F' (superscript) = A (superscript) a(sub) A (superscript) b(sub) F (superscript)

Now, let's look at the intermediate steps in the transformation.

Step 1: Transforming the x-components

The first row of the new electromagnetic tensor F' will contain the transformed x-components. Using the transformation law, we can write:

F'x = A (superscript) a(sub) A (superscript) b(sub) Fx

Substituting the values from the given electromagnetic tensor F, we get:

F'x = A (superscript) a(sub) A (superscript) b(sub) Ex

We can further simplify this by substituting the values of A (superscript) a(sub) and A (superscript) b(sub) for a Lorentz transformation along the x-axis, which is given by:

A (superscript) a(sub) = [1, 0, 0, 0, 0, 0]
A (superscript) b(sub) = [1, 0, 0, 0, 0, 0]

Substituting these values into