Born reciprocal (non-inertial) relativity, phase space trajectories and strings with variable tension

Pursuing further our work on Born Reciprocal (Non-inertial) Relativity theory, we find that the family of hyperbolic trajectories of particles living in a four-dim phase space of signature [Formula: see text] automatically satisfy the maximal speed of light and the maximal proper force conditions. A direct link between the maximal proper force b and the minimum value of the initial positions [Formula: see text] of these hyperbolic paths is found. Furthermore, when the maximal proper force is set to be equal to the Planck-mass squared [Formula: see text] then [Formula: see text] happens to coincide with the numerical value of the horizon radius of the Schwarzschild black hole in four spacetime dimensions. We proceed with a rigorous analysis of how open strings sweeping a two-dim world-sheet region inside a Rindler wedge in a [Formula: see text] spacetime can be seen as a continuum of point masses that are being collectively accelerated along the Rindler hyperbolic trajectories. The upshot of this analysis reveals that these strings can naturally acquire a [Formula: see text] tension which is consistent with the dynamical tension model of strings and branes proposed by Guendelman in the past years [Guendelman, arXiv:2508.17333; Guendelman, Class. Quantum Grav. 17, 3673 (2000); Guendelman, Kaganovich, Nissimov and Pacheva, Phys. Rev. D 66, 046003 (2002); Guendelman, Phys. Rev. D 63, 046006 (2001)]. We finalize by analyzing strings moving in phase space backgrounds within the context of non-inertial relativity theory.