On the formal statement of the special principle of relativity

The aim of the paper is to develop a proper mathematical formalism which can help to clarify the necessary conceptual plugins to the special principle of relativity and leads to a deeper understanding of the principle in its widest generality.

💡 Research Summary

The paper by Gömöri and Szabó sets out to give a precise mathematical formulation of the special relativity principle (RP), distinguishing it clearly from the often‑conflated notion of covariance. The authors begin by introducing two collections of physical quantities: ξ₁,…,ξₙ defined by operations performed with measuring devices at rest in an inertial frame K, and ξ′₁,…,ξ′ₙ defined by the same devices when they move with constant velocity V relative to K (i.e., co‑moving with a second inertial frame K′). Although the numerical values of corresponding ξ’s and ξ′’s may coincide, the physical situations differ, so the authors model them on two distinct n‑dimensional manifolds Ω and Ω′, each equipped with a global coordinate map (φ for Ω, φ′ for Ω′) into a common value space Σ = ×₁ⁿσᵢ ⊂ ℝⁿ.

Two key bijections are then defined. The “prime” map PV = φ′⁻¹∘φ simply carries a point of Ω to the point of Ω′ that has the same coordinate tuple; it reflects a change of labels without any physical transformation. The “physical‑quantity” map TV : Ω ⊇ R → R′ ⊆ Ω′ (where R and R′ are the physically admissible subsets of Ω and Ω′) is a genuine transformation of physical states: it tells which state in K′ corresponds to a given state in K. TV’s existence is not guaranteed in general; the authors illustrate this with three examples: (a) a static ideal gas where pressure and temperature transform trivially, (b) an electric field configuration where no one‑to‑one TV exists, and (c) the full electromagnetic field where the Lorentz transformation provides a suitable TV.

The behavior of a physical system is represented by a set of equations F ⊂ R, i.e., a subset of the admissible state space that encodes a particular law or solution. The collection of all such subsets is denoted E. For each F ∈ E the authors define PV(F) (the “primed” version of the same formal expression) and TV(F) (the same physical situation expressed in the primed variables). They also postulate a map MV : E → E that takes a description of a system at rest in K to a description of the same system when it moves as a whole with velocity V relative to K (i.e., co‑moving with K′).

The core statement of the RP in this formalism is the equality
 TV(MV(F)) = PV(F) for all F ∈ E.
In words, the description of the moving system, after being transformed by the genuine physical‑state map TV, coincides with the formally “primed” description obtained by merely relabeling the variables. This condition guarantees not only that the equations retain the same form in different inertial frames, but also that the specific solutions correspond appropriately—a stronger requirement than mere covariance.

Covariance is then defined as the condition TV(E) = PV(E) (or equivalently TV(E) ⊇ PV(E)). This ensures that the set of equations is invariant under the transformation of variables, but it does not specify which solution in K maps to which solution in K′. Consequently, covariance is necessary but not sufficient for the RP. The authors emphasize Bell’s observation that Lorentz invariance alone tells us that for any rest‑state there exists a “primed” state, but it does not guarantee that a system set in motion will evolve into the primed version of its original rest‑state. To bridge the gap, an additional condition MV(E) = E (the equations themselves must be invariant under the motion of the whole system) is required for covariance to imply the RP.

The paper further discusses the role of initial and boundary conditions, treating them as extra subsets ψ ⊂ 2Ω that restrict the solution space. The transformation properties of these conditions must be compatible with PV, TV, and MV for the RP to hold in a fully specified problem.

Through detailed examples—static versus uniformly moving point charges, the transformation of electric and magnetic fields, and the failure of a simple “form‑preserving” approach in certain contexts—the authors demonstrate how their formalism captures the subtle logical structure underlying the special relativity principle. They conclude that the RP is a precise law‑like regularity linking the behavior of measuring equipment and that of the physical system under motion, and that a rigorous distinction between RP and covariance clarifies many conceptual confusions in the foundations of relativistic physics.