The Combinatorial Structure of Special Relativity

In this paper, we investigate the rise of elementary symmetric polynomials in compositions under the hyperbolic tangent function. We first split the composition into two cases, where we examine the structure when an odd number and an even number of elements are composed. Through the use of floor functions, we unify the two cases, resulting in a closed form generalization. We prove this structure using mathematical induction.

Our contribution lies in applying this structure to special relativity, where we convert rapidities into velocities. The structure is valid for only collinear velocity compositions in 1 dimension. We then further prove that the structure obeys the postulates of special relativity by showing that it does not exceed the speed of light.

In addition, we develop a generalized formula for the total effect of the Lorentz factor for any number of collinear or parallel boosts. A boost is a transformation of coordinates between two inertial coordinate systems moving at constant relative velocity to one another without rotation.

In a purely theoretical scenario, we implement the structure to analyze the sufficient conditions by which the Lorentz factor converges for an infinite number of velocity additions. Finally, we derive generalizations of the Lorentz transformations.