I understand that the inner product of two 4-vectors is conserved under the Lorentz transformations, so that the absolute value of the four momentum is the same in any reference frame. This is what I (most likely mistakenly) thought was meant by the conservation of momentum. I don't understand why equations such as

$P_1=P_2+P_3$

($P_i$ are 4-momentum vectors for different particles in a collision for example)

should hold, within a reference frame. I've been told that you can't just add four velocities together on collision of particles, so why should you be able to do this with the momentum vectors?

Thanks for any replies!

I understand that the inner product of two 4-vectors is conserved under the Lorentz transformations, so that the absolute value of the four momentum is the same in any reference frame. This is what I (most likely mistakenly) thought was meant by the conservation of momentum. I don't understand why equations such as

$P_1=P_2+P_3$

($P_i$ are 4-momentum vectors for different particles in a collision for example)

should hold, within a reference frame. I've been told that you can't just add four velocities together on collision of particles, so why should you be able to do this with the momentum vectors?