Covariance of a selected particle momentum sum ($P_A^{\mu }={\sum }_{i\in A}{p}_i^{\mu }$)

A fixed set of four-momenta at fixed events sums to a four-vector, whether or not the set is the whole physical system. Its Minkowski norm is invariant. Reapplying a frame-dependent particle-selection rule or sampling different events need not give the transformed same sum. Physical conservation across different hypersurfaces is a separate issue involving interactions and boundary flux.