Lorentz transformation & relativistic momentum

[EsmeeDijk]

Homework Statement

We now specify the velocity v to be along the positive x¹-direction in S and of magnitude v. We also consider a frame [itex] \overline{S} [/itex] which moves at speed u with respect to S in the positive x¹-direction.

question 1 : Write down the transformation law for [itex] p^\mu [/itex].
question 2: Write [itex] \overline{p} ^\mu [/itex] also in terms of the speed [itex] \overline{v} [/itex] and its corresponding gamma factor.

For the first question I got the following answer which I believe is right:
[itex] \begin{pmatrix}
\overline{p}^0 \\ \overline{p}^1 \\ \overline{p}^2 \\ \overline{p}^3 \end{pmatrix}
\begin{pmatrix} \gamma & -\gamma \beta & 0 & 0 \\
- \gamma \beta & \gamma & 0 & 0 \\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} p^0 \\ p^1 \\ p^2 \\ p^3 \end{pmatrix}[/itex]
[itex] \overline{p}^0 = \gamma (p^0 - \beta p^1) [/itex] (1)
[itex] \overline{p}^1 = \gamma (p^1 - \beta p^0) [/itex] (2)
[itex] \overline{p}^2 = p^2 [/itex] (3)
[itex] \overline{p}^3 = p^3 [/itex] (4)
[itex] \overline{p}^\mu = \Lambda ^\mu _\nu p^\nu [/itex] (5)

Homework Equations

[itex] \textbf{p} = m\eta = \frac{ m \eta}{\sqrt{1 - u^2/c^2}} [/itex] (6)
[itex] p^0 = m \eta ^0 = \frac{ mc}{\sqrt{ 1 - u^2/c^2}} [/itex] (7)
[itex] p^\mu p_\mu = -(p^0)^2 + ( \textbf{p} \bullet \textbf{p}) = -m^2c^2 [/itex] (8)

The Attempt at a Solution

Now I believe I have to substitute equation 7 for p⁰ in equation 1/2
and equation 6 for p^1/2/3 in equation 1-4
The only problem now is that equations 6 and 7 assume a certain u² but I need to get something expressed in [itex] \overline{v}[/itex]. According to my textbook u in equation 6/7 is :the velocity of a traveling object of mass m .
To me it sounds like this means that I can replace all the u's by
[itex] \overline{v}[/itex]'s.
But I already know that in the next question I need to get something expressed in [itex] \overline{v}[/itex], [itex] v[/itex] and [itex] u [/itex]. So maybe that probably means that my theory of how to substitute the u's is wrong.

Of course we can also use Einstein's velocity addition law: [itex] \overline{u} = \frac{u - v}{ 1 - uv/c^2} [/itex] where [itex] \overline{u} [/itex] is the velocity between the two reference frames.

Thanks in advance for any help!

 

[ChrisVer]

In the equations (1) and (2), can you write the expression of [itex]\gamma[/itex] in terms of velocity?