In classical mechanics, if
$$\frac{\mathrm d}{\mathrm d t}\sum_i m_i\vec{v_i}=0$$is true for one frame of reference, then it is easy to prove that this is true for all frames (since different frames are related by Galilei transformations, the transformation of velocities is trivial).
What about Special Relativity? I was told that the equation
$$\frac{\mathrm d}{\mathrm d t}\sum_i \vec{p_i}=0$$with$$\vec{p_i}=m_i\vec{\frac{\mathrm d x_i}{\mathrm d\tau}}=\gamma_im_i\vec {v_i}$$is invariant under Poincaré transformations "if interactions are localized in events". Can someone elaborate on that?
Best Answer
It is easy if viewed a posteriori. Consider a finite number of material points which may interact only in single events where the four momentum is conserved. In each event the total in going four momentum is the same as the outgoing total four momentum. Outside these events describing the interactions the worldlines of the points are future directed timelike straight lines (Minkowskian geodesics). On each such line the four momentum and the mass of the particle are constant. The number of lines entering an interaction event may be different of the lines exiting that event, but the total four momentum immediately before and immediately after the event is the same.
Under these hypotheses you can define a total four momentum at time $t$, referred to a time slice of a Minkowskian reference frame: it is the sum of the four momenta of the lines which intersect that time slice. It is easy to see that this total four momentum does not depend on $t$ (and on the reference frame). The conservation law of the spatial components of the four momentum takes, in the considered reference frame, the form you wrote.
Also the temporal component of the four momentum is conserved. The physical interpretation of that law leads to the famous principle $E=mc^2$ as well as to the relativistic failure of the law of the conservation of the mass.
I stress that this is just a mathematical model not very realistic. It can serve as a first step in understanding conservation laws in relativity. Physically speaking interactions are carried by fields which, in turn, transport four momentum. This amount of four momentum has to be included in the conservation law through a suitable integration procedure. In that case, the worldlines of the involved particles cease to be geodesics in general. In general relativity, the elementary model described above is not feasible from scratch, because the spacetime is not an affine space and we cannot add together vectors placed at different events.
On the other hand, the naive picture above strongly resembles the picture of Feynman diagrams, with the crucial difference that there the internal lines (corresponding to virtual particles) may be also spacelike.