Lorentz Transformation of Vectors from S to S' Frame

[physicsforumsfan]

Homework Statement

The question is quite basic; what is the Lorentz transformation of the follows 4-vectors from S to S' frame:

A photon (P) in S frame with 4-momentum

P = (E/c,p,0,0) and

frequency f where

hf = pc = E. h is the planks constant, p is the magnitude of 3-momentum and E is energy.

S' frame travels in positive x direction with position v speed (ie. not an ANTIPARTICLE).

...so how's P' related to P? P' is momentum in S' frame

My attempt:

Lorentz boost, simple gamma factor (sqrt (1-(v/c)^2)) relationship with P to give a P' answer as

P' = √(1-(v/c)^) * P

Correct?

How above the relationship of f and f'? f' is the frequency of photon in S' frame.

My attempt:

Same as above, gamma relationship with f to give

f' = √(1-(v/c)^) * f

Correct?

Thanks everyone

 

[hilbert2]

You have to write the 4-momentum as a column vector
$\begin{bmatrix}E/c\\p\\0\\0\end{bmatrix}$
and operate on it with the Lorentz boost matrix, which is
$\Lambda=\begin{bmatrix}\gamma&-\beta\gamma&0&0\\-\beta\gamma&\gamma&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}$
for boosts in the $x$-direction. Here $\beta=v/c$, where $v$ is the relative velocity of the frames. Then you get the components of the four-momentum in the new frame. The frequency changes in the transformation by the same factor as the energy component of the 4-momentum vector.

 

[vanhees71]

In addition note that for a photon the four-momentum is
$$(p^{\mu})=\begin{pmatrix} |\vec{p}| \\ \vec{p}, \end{pmatrix}$$
because a photon's four-momentum is light-like.

 

[physicsforumsfan]

You have to write the 4-momentum as a column vector
$\begin{bmatrix}E/c\\p\\0\\0\end{bmatrix}$
and operate on it with the Lorentz boost matrix, which is
$\Lambda=\begin{bmatrix}\gamma&-\beta\gamma&0&0\\-\beta\gamma&\gamma&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}$
for boosts in the $x$-direction. Here , where $v$ is the relative velocity of the frames. Then you get the components of the four-momentum in the new frame. The frequency changes in the transformation by the same factor as the energy component of the 4-momentum vector.

Hi,

Thank you for replying (hilbert and vanhees!). I cross multiply the matrix, factor in the fact that E=pc to get:

γP * matrix

[ 1, -v/c]
[-v/c, 1 ]

How do I solve this?

 

[paranormal]

for your responses. Yes, your attempts are correct. The Lorentz transformation of a 4-vector from one frame to another is given by the following equation:

P' = γ(P - (v/c^2)E)

where γ is the Lorentz factor, v is the relative velocity between the two frames, and c is the speed of light. This equation can be applied to any 4-vector, including the 4-momentum of a photon.

In the case of a photon, the energy and momentum are related by the equation E = hf = pc, where h is the Planck constant. This means that the transformation of the 4-momentum can also be written as:

P' = γ(E/c, p, 0, 0)

This shows that the energy and momentum components are transformed separately, with the energy component being boosted by a factor of γ and the momentum component remaining unchanged.

Similarly, the frequency of a photon can be transformed using the same equation, since frequency is directly proportional to energy. This gives us the equation:

f' = γf

which is the same as your attempt. So, you are correct in your understanding of the Lorentz transformation of 4-vectors and its application to photons.