For a particle, we always have $Eu=p$ where $u$ is the speed of the particle and correspondingly the momentum is always timelike. For a system of particles, if the potential energy is not considered then (since $E>p$ holds for each particle and $E>0$) the system will also have a timelike momentum. But can there exist a system with potential energies of such a nature (which essentially decreases the total energy of the system more than it increases the spatial interval of momentum) that the momentum of the entire system is spacelike? It would suggest strange implications like its energy (in some frames) being negative or zero with spatial parts of momentum being normal so it seems quite unlikely. But can it be proven rigorously that the momentum of any arbitrary system (that, of course, follows relativity, and local conservation laws as normally assumed) must not be spacelike?
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