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A geometric relativistic dynamics under any conservative force. (English) Zbl 1408.83008

Summary: Riemann’s principle “force equals geometry” provided the basis for Einstein’s General Relativity – the geometric theory of gravitation. In this paper, we follow this principle to derive the dynamics for any static, conservative force. The geometry of spacetime of a moving object is described by a metric obtained from the potential of the force field acting on it. We introduce a generalization of Newton’s First Law – the Generalized Principle of Inertia stating that: An inanimate object moves inertially, that is, with constant velocity, in its own spacetime whose geometry is determined by the forces affecting it. Classical Newtonian dynamics is treated within this framework, using a properly defined Newtonian metric with respect to an inertial lab frame. We reveal a physical deficiency of this metric (responsible for the inability of Newtonian dynamics to account for relativistic behavior), and remove it. The dynamics defined by the corrected Newtonian metric leads to a new Relativistic Newtonian Dynamics for both massive objects and massless particles moving in any static, conservative force field, not necessarily gravitational. This dynamics reduces in the weak field, low velocity limit to classical Newtonian dynamics and also exactly reproduces the classical tests of General Relativity, as well as the post-Keplerian precession of binaries.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C30 Asymptotic procedures (radiation, news functions, \(\mathcal{H} \)-spaces, etc.) in general relativity and gravitational theory
83C10 Equations of motion in general relativity and gravitational theory
70A05 Axiomatics, foundations
70B05 Kinematics of a particle
70H40 Relativistic dynamics for problems in Hamiltonian and Lagrangian mechanics
53Z05 Applications of differential geometry to physics
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