RELATIVISTIC STABILITY. PART 1: RELATION BETWEEN SPECIAL RELATIVITY AND STABILITY THEORY IN THE TWO-BODY PROBLEM

With reference to the restricted two-body problem we show that Stability Theory (ST) and Special Relativity (SR) can be joined together in a new theory that explains a large class of physical phenomena (e.g. black-holes, cosmological dynamics) and overcomes the dualism between SR and General relativity (GR). After recalling the main features of ST (from the Method of Lyapunov to more recent developments up to analysis of fractals) we determine the canonic relativistic equations of the restricted two-body problem. A substantial novelty with respect to noted formulations is pointed out: three state variables (and not two only) are needed for "defining" said equations. They include variable v (magnitude of the rotation speed) in addition to radius and to radial speed. By means of eigenvalue analysis and by application of the Lyapunov theorem on stability in the first approximation we show that linearized system analysis gives a necessary condition only for stability: the radius must be greater than half the Schwarzschild radius. The derivation of a sufficient condition passes through the definition of a convenient Lyapunov function that represents the "local energy" around a given Equilibrium Point. Such derivation is deferred to Part II and results in the proof that the Schwarzschild radius actually represents the reference stable radius of the two-body problem.