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.