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 &amp; -\gamma \beta &amp; 0 &amp; 0 \\<br /> - \gamma \beta &amp; \gamma &amp; 0 &amp; 0 \\<br /> 0 &amp; 0 &amp; 1 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 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?