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.

 

[paranormal]

The correct transformation for a 4-vector in special relativity is given by the Lorentz transformation tensor, which is a 4x4 matrix that relates the coordinates of an event in one frame to the coordinates of the same event in another frame. This transformation takes into account the effects of time dilation and length contraction in special relativity.

In part A, you have correctly calculated the transformation for the coordinates in the x, y, and z directions. However, there seems to be a mistake in your expression for the time coordinate. It should be d/d\overline{t} = \gamma(d/dt - v*d/dx), where the subtraction sign is used instead of addition. This is because time dilation causes time to run slower in the moving frame, so the time coordinate in the moving frame should be smaller than the time coordinate in the stationary frame.

In part B, you have correctly transformed the components of the 4-vector potential. However, there seems to be a mistake in the expression for the x component of the potential. It should be \overline{A}x = \gamma(Ax + v\varphi/c^2), which includes a multiplication by v\varphi/c^2 instead of just v\varphi/c. This is because the Lorentz transformation takes into account the mixing of space and time coordinates in special relativity.

In part C, you have correctly transformed the components of the electric and magnetic fields. However, there seems to be a mistake in the expression for the y component of the electric field. It should be \overline{E}y = \gamma(Ey + vBz), where the vector components are assigned to the correct axis. This is because the y and z components of the electric field will also experience length contraction and time dilation in the moving frame.

Overall, your calculations are mostly correct but there are some minor errors in the signs and vector components. It is important to pay attention to these details when working with special relativity as they can greatly affect the final result. Keep up the good work!