In the Schrödinger picture relativistic and non-relativistic scattering are closely analogous. On scattering states the invariant mass operator M′ of the interacting time evolution is unitarily equivalent to the free invariant mass M acting on many-particle states. M′ must not commute with the total momentum P =∑iPi of the many-particle states but only with the four-velocity \(U= P / \sqrt {P^2}\) of their center. In center coordinates M′ generates the interacting relative motion and U0 the motion of the center. The relativistic Hamiltonian P0′ = U0M′ of many-particle states does not separate into a sum for the motion of the center and the relative motion but factorizes as their product. The probability of scattering of massive particles is approximately proportional to the spacetime overlap of their position wave functions. This is basic to macroscopic locality and the decoupling of the environment.