We explore the role of symmetry in the theory of Special Relativity. Using the symmetry of the principle of relativity and eliminating the Galilean transformations, we obtain a universally preserved speed and an invariant metric, without assuming the constancy of the speed of light. We also obtain the spacetime transformations between inertial frames depending on this speed. From experimental evidence, this universally preserved speed is c, the speed of light, and the transformations are the usual Lorentz transformations. The ball of relativistically admissible velocities is a bounded symmetric domain with respect to the group of affine automorphisms. The generators of velocity addition lead to a relativistic dynamics equation. To obtain explicit solutions for the important case of the motion of a charged particle in constant, uniform, and perpendicular electric and magnetic fields, one can take advantage of an additional symmetry—the symmetric velocities. The corresponding bounded domain is symmetric with respect to the conformal maps. This leads to explicit analytic solutions for the motion of the charged particle.

Relativity theory assumes that force fields propagate with the speed of light. We show that such force fields generated by a single source can be described by a pre-potential, which is a complex-valued function on spacetime outside the worldline of the source. The pre-potential is invariant under a spin-half representation of the Lorentz group acting on complexified spacetime. The complex four-potential of such a field is defined and calculated explicitly from the pre-potential without assuming any particular force law for the field. The real part of the obtained four-potential coincides with the known Liénard–Wiechert potential. The symmetry of the four-potential is described herein. The pre-potential satisfies the wave equation. The single source electromagnetic field derived from this four-potential is self-dual or anti-self-dual. The pre-potential and the four-potential are extended to a field with several sources.

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 conservative force. We introduce the relativity of spacetime : an object lives in its own spacetime, whose geometry is determined by all of the forces affecting it. We also introduce the Generalized Principle of Inertia which unifies Newton’s first and second laws and states that: An inanimate object moves freely, that is, with zero acceleration, in its own spacetime. We derive the metric of an object’s spacetime in two ways. The first way uses conservation of energy to derive a Newtonian metric. 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 (RND) for both massive objects and massless particles moving in any static, conservative force field, not necessarily gravitational. In the case of the gravitational field of a static, spherically symmetric mass distribution, this metric turns out to be the Schwarzschild metric. This dynamics reduces in the weak field, low velocity limit to classical Newtonian dynamics and also exactly reproduces the classical tests of General Relativity. In the second way, we obtain the RND metric directly, without first obtaining a Newtonian metric. Instead of conservation of energy, we use conservation of angular momentum, a carefully defined Newtonian limit and Tangherlini’s condition. The non-static case is handled by applying Lorentz covariance to the static case.

We use the Frenet frame to define and completely characterize "uniform acceleration" in flat spacetime. We extend the definition to arbitrary curved spacetime and provide an example in Schwarzschild spacetime.

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 \emph{Generalized Principle of Inertia} stating that: \emph{An inanimate object moves inertially, that is, with constant velocity, in \emph{its own} spacetime whose geometry is determined by the forces affecting it}. Classical Newtonian dynamics is treated within this framework, using a properly defined \emph{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 \emph{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.