On relativistic motion of a pair of particles having opposite signs of masses

(abbreviated) In this note we consider, in a weak-field limit, a relativistic linear motion of two particles with opposite signs of masses having a small difference between their absolute values $m_{1,2}=pm (mupm Delta mu) $, $mu > 0$, $|Delta mu | ll mu$ and a small difference between their velocities. Assuming that the weak-field limit holds and the dynamical system is conservative an elementary treatment of the problem based on the laws of energy and momentum conservation shows that the system can be accelerated indefinitely, or attain very large asymptotic values of the Lorentz factor $gamma$. The system experiences indefinite acceleration when its energy-momentum vector is null and the mass difference $Delta mu le 0$. When modulus of the square of the norm of the energy-momentum vector, $|N^2|$, is sufficiently small the system can be accelerated to very large $gamma propto |N^2|^{-1}$. It is stressed that when only leading terms in the ratio of a characteristic gravitational radius to the distance between the particles are retained our elementary analysis leads to equations of motion equivalent to those derived from relativistic weak-field equations of motion of Havas and Goldberg 1962. Thus, in the weak-field approximation, it is possible to bring the system to the state with extremely high values of $gamma$. The positive energy carried by the particle with positive mass may be conveyed to other physical bodies say, by intercepting this particle with a target. Suppose that there is a process of production of such pairs and the particles with positive mass are intercepted while the negative mass particles are expelled from the region of space occupied by physical bodies of interest. This scheme could provide a persistent transfer of positive energy to the bodies, which may be classified as a Perpetuum Motion of Third Kind.