What is the correct transformation for a 4-vector in special relativity?

[physicsforumsfan]

Hi all,

I got a 3 part Qs: γ=1/√1-v^2-c^2

Part A

Homework Statement

Consider the Lorentz transformation tensor

Matrix
Row 1: [ γ 0 0 -vγ/c]
Row 2: [ 0 1 0 0 ]
Row 3: [ 0 0 1 0 ]
Row 4:-[vγ/c 0 0 γ ]

for transforming 4-vectors from frame S to $\overline{S}$ according to$\overline{A}$$^{\mu}$ = L$^{\mu}$ $_{v}$ A$^{v}$ . The coordinate system is x$^{0}$ =ct, x$^{1}$ = x, x$^{2}$ = y, x$^{3}$ = z .

The Attempt at a Solution

Doing the transformation and then solving for it gives the answer:

d/d$\overline{t}$=γ(d/dt-vd/dx), d/d$\overline{x}$=γ(v/c^2 d/dt - d/dx), d/d$\overline{y}$ = d/dy, d/d$\overline{z}$=d/dz

That's the answer I get but I am not sure about if I have the addition and substraction signs correct.

Part B

Homework Statement

In above question, if the 4-vector potential is given by $\underline{A}$=($\phi$/c, Ax, Ay, Az) in frame S what are its components in frame $\overline{S}$?

The Attempt at a Solution

Again solving for and getting the answer, I am confused on the addition and subtraction signs:

$\overline{A}$=(γ$\varphi$/c + γv/c Ax, γAx+ γv$\varphi$/c^2, Ay, Az)

Part C

Homework Statement

In Part B, the electric and magnetic fields are defined in frames S and $\overline{S}$ by

E$^{(3)}$=-∇$\varphi$-dA$^{(3)}$/dt, $\overline{E}$$^{(3)}$=-∇$\overline{\varphi}$-d$\overline{A}^{(3)}$/d$\overline{t}$, B$^{(3)}$=∇xA$^{3}$, $\overline{B}$$^{(3)}$=$\overline{∇}$x$\overline{A}^{(3)}$,
$\overline{A}$=($\overline{\varphi}$/c, $\overline{A}$x,

If

$\overline{A}$y, $\overline{A}$z)=($\overline{\varphi}$/c, $\overline{A}^{(3)}$)

what is value of $\overline{E}$x?

The Attempt at a Solution

Again solving for it I get my answer in which I am unsure of the addition and subtraction signs.

$\overline{E}$x=Ex, $\overline{E}$y=γ(Ey+vBz), $\overline{E}$z=γ(Ez-vBy)

I am also not sure if the have the vector components assigned to the correct axis.

Help would be appreciated.

 

[physicsforumsfan]

Hi,

no reply?

Help?

 

[MisterX]

I'm not even sure what the questions are.

 

[physicsforumsfan]

I'm not even sure what the questions are.

In Part A - I am supposed to find the transformation of the L matrix using that tensor equation. Is my transformation correct? It was my attempt at the question.

In Part B - Again, are the components of $\overline{S}$ correct (ie. is $\overline{A}$ correct)? It was my attempt at the question.

In Part C - It is a bit crowded (the formulae) but essentially they are the electric and magnetic field equations E, E (dashed), B and B (dashed) of the S and S (dashed) frames.

A (dashed, the 'if' was supposed to start before the A dashed equation and not in the middle)

I am supposed to find the E (dashed, the 'x' is a typo, sorry) components of this system (from the A dashed equation of part B). If the above is wrong then so is my following working. Are the + and - signs in the answer? It was my attempt.

Thanks for brings that up.