# Special Relativity; 3 Momentum Conservation

In SR, I understand you can use 4 momentum conservation, but what are the special cases where you can use 3 momentum/energy conservation?

An example I have seen is with $$P_1=(M_1, 0) \\ P_2=(M_2,0) \\ P_3=\left(\sqrt{(m_3^2+p^2)},p\right) \\ P_4 =(|p|,-p)$$

where $$P_1$$ is the initial state and $$P_2$$, $$P_3$$, $$P_4$$ are the final. From here the example said

$$M_1 = M_2 + \sqrt{(m_3^2+p^2)} + |p|$$

Could somebody explain why we are allowed to say that, I thought that energy and momentum became "intertwined" and energy and momentum conversation were coupled into four momentum conservation.

The fact that the four-momentum vector $$p^\mu=(E,p_x,p_y,p_z)$$ is conserved,
$$p^\mu_\text{after}=p^\mu_\text{before},$$
simply means that $$E$$ is conserved, $$p_x$$ is conserved, $$p_y$$ is conserved, and $$p_z$$ is conserved. A vector equation is simply a set of equations for each component. The equation that puzzles you is simply the conservation of energy.