Vanishing component theorem for a Lorentz four-vector

If a fixed component of a Lorentz four-vector vanishes in every Lorentz frame, the vector is zero. For its time component, arbitrary infinitesimal boosts force every spatial component to vanish; for a spatial component, first use rotations and then a boost. Applied to the incoming-minus-outgoing total four-momentum, this shows that energy balance in every inertial frame implies full four-momentum conservation. The statement fails for general tensors: a nonzero antisymmetric tensor has zero diagonal components in all frames.