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

In this methodological note, we consider, in a weak-fleld limit, the relativistic linear motion of two particles with masses of opposite signs and a small difference between their absolute values: $m_{1,2}=\pm (\mu\pm \Delta \mu) $, $\mu > 0$, $|\Delta \mu | \ll\mu$. In 1957, H Bondi showed in the framework of both Newtonian analysis and General Relativity that, when the relative motion of particles is absent, such a pair can be accelerated indefinitely. We generalize the results of his paper to account for the small nonzero difference between the velocities of the particles. 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 the 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 by Havas and Goldberg in 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. If we 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 the physical bodies of interest, this scheme could provide a persistent transfer of positive energy to the bodies, which may be classified as ‘perpetual motion of the third kind’. Additionally, we critically evaluate some recent claims regarding the problem.