How to Derive Relative Velocity from 4-Momentum Invariants?

[scottJH]

Homework Statement

In a particular frame of reference a particle with 4-momentum $P_p$ is observed by an observer moving with 4-momentum $P_o$. Derive an expression for the speed of the particle relative to the observer in terms of the invariant $P_p.P_o$

I am completely stuck on this question. Any help would be appreciated.

Homework Equations

The Attempt at a Solution

I used the equations

$P_p = m_p\gamma(u_p, ic)$

and

$P_o = m_o\gamma(u_o, ic)$

I tried to obtain $P_p.P_o$ in the rest frame of the observer and obtained

$P_p.P_o = m_pm_o\gamma(-c^2)$ where $\gamma$ is in terms of $u_p$

This is because $\gamma$ in terms of $u_o$ is equal to 1 in the rest frame of the observer

I'm pretty sure that my answer is incorrect. However even if it is correct I am unsure how I would be able to determine the relative velocity from it.

 

[Orodruin]

I tried to obtain $P_p.P_o$ in the rest frame of the observer and obtained

$P_p.P_o = m_pm_o\gamma(-c^2)$ where $\gamma$ is in terms of $u_p$

This is because $\gamma$ in terms of $u_o$ is equal to 1 in the rest frame of the observer

The $\gamma$ here is based on the relative velocity between the observer and the particle (which in the rest frame of the observer happens to be the same as the velocity of the particle in that frame). Since you have expressed the invariant in terms of invariants ($m_p$, $m_o$ and $c$) and the relative velocity $v$ (through $\gamma$) you can solve this equation to find the relative velocity.