Relativity free of coordinates

TL;DR

This paper develops a coordinate-free, physics-based approach to relativity, deriving transformation rules and effects without relying on traditional coordinate systems, emphasizing physical objects and directions for clarity.

Contribution

It introduces a novel, coordinate-free formulation of relativity transformations based on physical objects and directions, clarifying the nature of boost effects.

Findings

Derived transformation rules differ from traditional vector-like relationships.

Provided a physically meaningful interpretation of coordinate transformations.

Enabled easy evaluation of apparent distortions in boosted coordinate systems.

Abstract

The concept of a physical space, which actualizes Euclidean geometry, is not confined to the statics of solids but extensible to the phenomena where Newtonian mechanics is valid, defining its concept of time. The laws of propagation of electromagnetic disturbances modify Newtonian formalism for sufficiently fast free motions within each spatial domain of its validity for slow motions and introduce the extended concept of time by uniting those of Newtonian which can exist in different spatial domains of their validity. A boost direction for a pair of physical spaces is that spatial direction in one of the spaces along which the other space moves. Free motions of point particles make an instrumentation for identifying the boost direction as well as events on a straight line along that direction. The concept of a boost direction secures the physics-based formulation of the basic relativity effects, which eventually results in the relation between two arbitrary spaces in terms of their position vectors and time of a given event. The obtained transformation rules for the components of the position vector differ from the vector-like relationship known in the literature because the latter actually deals with column vectors made of Cartesian coordinates of true vectors and appears identical to a coordinate transformation referred to as a boost. Within the physics-based approach, the transformation of coordinates implies specifying coordinate systems in each frame in terms of physical objects/directions. This yields a logically consistent and physically meaningful presentation of the coordinate transformations commonly exploited in the special relativity theory, which makes manifestations of those transformations evident. Specifically, for a Cartesian coordinate system subjected to a boost, the coordinate-free technique of reasoning allows one to evaluate its apparent distortion easily.