4-momenta
As was previously shown for the space-time 4-vector, a similar 4-vector can be composed of momentum. Using index notation, the energy and momentum components can be combined into a single "4-vector" [math]\mathbf{p^{\mu}},\ \mu=0,\ 1,\ 2,\ 3[/math], that has units of momentum(i.e. E/c is a distance).
[math]\mathbf{P} \equiv
\begin{bmatrix}
p^0 \\
p^1 \\
p^2 \\
p^3
\end{bmatrix}=
\begin{bmatrix}
E/c \\
p_x \\
p_y \\
p_z
\end{bmatrix}[/math]
As shown earlier,
[math]\mathbf R \cdot \mathbf R = x_0^2-(x_1^2+x_2^2+x_3^2)[/math]
Following the 4-vector of space-time for momentum-energy,
[math]\mathbf P \cdot \mathbf P = p_0^2-(p_1^2+p_2^2+p_3^2)[/math]
[math]\mathbf P \cdot \mathbf P = \frac{E^2}{c^2}-\vec p\ ^2[/math]
Using the relativistic equation for energy
[math]E^2=\vec p\ ^2+m^2[/math]
[math]\mathbf P \cdot \mathbf P = \frac{E^2}{c^2}-E^2+m^2[/math]
[math]\mathbf P \cdot \mathbf P = m^2[/math]
A 4-momenta vector can be composed of different 4-momenta vectors,
[math]\mathbf \equiv \mathbf P_1 +\mathbf P_2[/math]
This allows us to write
[math]\mathbf P^2 \equiv (\mathbf P_1+\mathbf P_2)^2[/math]
[math]\mathbf (\mathbf P_1 +\mathbf P_2)^2 \equiv \mathbf P_1^2+2 \mathbf P_1 \mathbf P_2+\mathbf P_2^2[/math]
Using
[math]\mathbf (\mathbf P_1 +\mathbf P_2)^2=m^2[/math]
[math]\mathbf (\mathbf P_1 +\mathbf P_2)^2 \equiv m_1^2+2 \mathbf P_1 \mathbf P_2+m_2^2[/math]