A Categorical View on the Principle of Relativity

TL;DR

This paper explores how category theory offers a new perspective on the principle of relativity, highlighting the importance of frame transformations beyond traditional group theory approaches.

Contribution

It introduces a categorical framework for understanding frame changes in physics, addressing limitations of group theory in describing reference frame transformations.

Findings

Category theory provides a novel foundation for relativity.

Frame changes are characterized by categorical constructs beyond group actions.

The approach clarifies the role of boosts in classical and relativistic physics.

Abstract

Category theory plays a special character in mathematics - it unifies distinct branches under the same formalism. Despite this integrative power in math, it also seems to provide the proper foundations to the experimental physicist. In this work, we present another application of category in physics, related to the principle of relativity. The operational construction of (inertial) frames of reference indicates that only the movement between one and another frame is enough to differentiate both of them. This fact is hidden when one applies only group theory to connect frames. In fact, rotations and translations only change coordinates, keeping the frame inert. The change of frames is only attainable by boosts in the classical and relativistic regimes for both Galileo and Lorentz (Poincar\'e) groups. Besides providing a non-trivial example of application of category theory in physics, we also fulfill the presented gap when one directly applies group theory for connecting frames.