Aspects of Relativity in Flat Spacetime

A monograph on the mathematical aspects of Special Relativity, focusing on the Lorentz group and the properties of relativistic transformations in mechanics and electrodynamics. Manuscript of published book, with an added appendix.

💡 Research Summary

The manuscript “Aspects of Relativity in Flat Spacetime” is a compact yet thorough monograph that treats Special Relativity (SR) from a purely mathematical standpoint, emphasizing the role of symmetry, group theory, and covariant formulations. The author begins with a philosophical preface, arguing that symmetry is not merely aesthetic but a dynamical principle that underlies conservation laws via Noether’s theorem, and that the Lorentz symmetry is the cornerstone of SR.

Chapter 1 lays out the basic postulates of SR: the existence of a four‑dimensional flat spacetime equipped with a constant Minkowski metric (g_{\mu\nu}= \mathrm{diag}(1,-1,-1,-1)), the equivalence of all inertial frames, and the invariance of the speed of light. The failure of Galilean transformations to preserve Maxwell’s equations is highlighted, motivating the introduction of the Lorentz transformation (LT) as the unique linear map that leaves the spacetime interval (ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}) invariant. The chapter also clarifies the geometric meaning of world‑lines, light cones, and the distinction between proper and coordinate time.

Chapter 2 is devoted to the Lorentz group itself, treated independently of its physical applications. The group is defined as the set of real (4\times4) matrices (\Lambda) satisfying (\Lambda^{T}g\Lambda=g). The author distinguishes proper ((\det\Lambda=+1)) from improper ((\det\Lambda=-1)) transformations, and then isolates the connected component of the identity— the restricted Lorentz group, denoted (L=SO(3,1)^{\uparrow}). Detailed proofs of closure, existence of inverses, and continuity are provided. A subtle condition (\Lambda^{4}{}_{4}\ge 1) is derived to guarantee connectivity with the identity, and the author notes that this selects the physically admissible boosts.

The Lie algebra (\mathfrak{so}(3,1)) is introduced via the exponential map (\Lambda=e^{\omega}). The matrix (\omega) is shown to be traceless and to satisfy (g\omega) antisymmetric, leading to the familiar six‑parameter parametrization in terms of three rotation generators (A_i) and three boost generators (B_i). The commutation relations (