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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.

We apply Relativistic Newtonian Dynamics (RND), a Lagrangian-based, metric theory to a static, spherically symmetric gravitational field. Using a variational principle and conserved momenta, we construct several metrics, analytic everywhere except at r = 0, which have g₀₁ ≠ 0 yet still leads to the same trajectories as in the Schwarzschild model. These metrics passes all classical test of GR. However, this model and GR predict different velocities on the trajectories, both for massive objects and massless particles. The total time for a radial round trip of light in RND is the same as in the Schwarzschild model, but RND allows for light rays to have different speeds propagating toward and away from the massive object. One of theses metrics keeps the speed of light toward the object to be c. We present possible experiments to test whether g₀₁ = 0. RND extends to multiple non-static forces, each of which obeys an inverse square law and whose field propagates at the speed of light.