Getting the Lorentz transformations without requiring an invariant speed (1504.02423v1)

Abstract: The structure of the Lorentz transformations follows purely from the absence of privileged inertial reference frames and the group structure (closure under composition) of the transformations---two assumptions that are simple and physically necessary. The existence of an invariant speed is \textit{not} a necessary assumption, and in fact is a consequence of the principle of relativity (though the finite value of this speed must, of course, be obtained from experiment). Von Ignatowsky derived this result in 1911, but it is still not widely known and is absent from most textbooks. Here we present a completely elementary proof of the result, suitable for use in an introductory course in special relativity.

• The paper derives the Lorentz transformations using only the relativity principle and the group structure of transformations, without assuming the invariance of light speed.
• This derivation shows that the existence of an invariant speed naturally emerges from the relativity principle itself.
• The work offers an elementary, rigorous derivation suitable for education and highlights the fundamental role of symmetry in physics independent of specific phenomena.

Derivation of Lorentz Transformations Without an Invariant Speed

In the paper "Getting the Lorentz transformations without requiring an invariant speed," Andrea Pelissetto and Massimo Testa explore an alternative derivation of the Lorentz transformations distinct from the conventional Einsteinian approach that emphasizes the invariance of light speed. The authors revisit the analysis initiated by W. von Ignatowsky in the early 20th century, providing a more elementary proof suited for educational purposes, particularly in introductions to special relativity.

Alternative Approach to Derivation

Traditionally, Lorentz transformations are deduced within the framework of special relativity via the invariance of the speed of light. However, von Ignatowsky's approach circumvents this requirement by focusing on two fundamental assumptions: the absence of privileged inertial reference frames and the group structure of transformations. The primary insight gained from this analysis is that the existence of an invariant speed naturally emerges from the relativity principle, irrespective of prior knowledge about electromagnetic phenomena.

Key takeaways of the presented derivation include:

• The requirement that inertial frames are equivalent, which leads to constraints on transformation parameters.
• The necessity of transformations forming a group, ensuring closure under composition, which further restricts the form these transformations can take.

Detailed Analysis

Pelissetto and Testa carefully outline the formal proof by examining the transformations between any two inertial reference frames. Each transformation is encapsulated in a matrix form, and through systematic exposition, the authors explicate how constraints arise when ensuring that physical laws appear identical in each inertial frame. They emphasize the importance of linearity in these transformations, requiring that spatial and temporal intervals transform linearly between frames.

The derivation proceeds by analyzing transformations that maintain the symmetry and isotropy of space, confirming that the permissible transformations align with either Lorentz or Galilei transformations. This exploration reveals that Lorentz transformations manifest independently of electromagnetic properties, focusing instead on the more universal aspects of spacetime symmetry.

Implications and Future Directions

This work's implications are predominantly educational, offering a comprehensible yet rigorous derivation for undergraduates that does not rely on concepts such as the invariance of light speed until the transformation is fully formulated. By presenting the derivation in this manner, educators can introduce relativity's mathematical structure while preserving its conceptual elegance.

The derivation also suggests further inquiries into fundamental physics' axiomatic structures. It invites exploration into other phenomena where symmetry and transformation principles might yield new insights or alternative formulations. Understanding how fundamental transformations can emerge from core symmetry principles without specific experiment-driven assumptions enriches both theoretical and practical endeavors.

In the broader scientific context, this form distinguishes the boundary between theoretical constructs and empirical validation, reiterating the importance of physical interpretation in symmetry and group theory. Future research could extend these ideas to general relativity or alternate cosmologies where conventional transformation assumptions have broader implications.

Ultimately, the authors provide valuable educational material and a fresh perspective on classical derivations, promoting re-evaluation of foundational concepts within physics. Their work fosters a deeper appreciation for the symmetry-centric perspectives in theoretical developments, potentially driving new interpretations and applications in various areas of physics and beyond.