Lorentz-invariant three-vectors and alternative formulation of relativistic dynamics

Besides the well-known scalar invariants, there also exist vectorial invariants in special relativity. It is shown that the three-vector $(dp⃗/dt)∥+γv(dp⃗/dt)⊥$ is invariant under the Lorentz transformation. The subscripts $∥$ and $⊥$ denote the respective components with respect to the direction of the velocity of the body $v⃗$⁠, and $p⃗$ is the relativistic momentum. We show that this vector is equal to a force $F⃗R$⁠, which satisfies the classical Newtonian law $F⃗R=ma⃗R$ in the instantaneous inertial rest frame of an accelerating body. Therefore, the relation $F⃗R=(dp⃗/dt)∥+γv(dp⃗/dt)⊥$⁠, based on the Lorentz-invariant vectors, may be used as an invariant (not merely a covariant) relativistic equation of motion in any inertial system of reference. An alternative approach to classical electrodynamics based on the invariant three-vectors is proposed.