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 $\overline{S}$ which moves at speed u with respect to S in the positive x¹-direction.

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

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

Homework Equations

$\textbf{p} = m\eta = \frac{ m \eta}{\sqrt{1 - u^2/c^2}}$ (6)
$p^0 = m \eta ^0 = \frac{ mc}{\sqrt{ 1 - u^2/c^2}}$ (7)
$p^\mu p_\mu = -(p^0)^2 + ( \textbf{p} \bullet \textbf{p}) = -m^2c^2$ (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 $\overline{v}$. 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
$\overline{v}$'s.
But I already know that in the next question I need to get something expressed in $\overline{v}$, $v$ and $u$. 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: $\overline{u} = \frac{u - v}{ 1 - uv/c^2}$ where $\overline{u}$ 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 $\gamma$ in terms of velocity?