On the Invariance of the Spacetime Interval

TL;DR

The paper addresses deriving the invariance of the spacetime interval from the constancy of the speed of light $c$ and spacetime homogeneity/isotropy, defining the invariant $( abla S)^2 = - c^2 ( abla t)^2 + ( abla x)^2 + ( abla y)^2 + ( abla z)^2$ as the central quantity. It introduces a geometric light-rectangle construction, where the area $rl$ is proportional to $-( abla S)^2/4$ and remains unchanged across inertial frames, and from this derives the Lorentz transformations in one dimension. The main contributions are a complete geometric proof of $ abla S^2$ invariance for lightlike, timelike, and spacelike intervals and a direct derivation of the Lorentz transformations in one spatial dimension, highlighting the link between light propagation and spacetime structure in Minkowski space. This approach provides a conceptually geometric alternative to algebraic proofs and has pedagogical value for understanding relativistic kinematics.

Abstract

We present a geometric proof of the invariance of the relativistic spacetime interval based solely on the constancy of the speed of light, and the homogeneity and isotropy of spacetime. The derivation is based on a simple construction involving light rectangles, whose areas remain invariant across inertial frames. Based on this construction, we also derive the Lorentz transformations.

Figures (7)

• Figure 3: How to translate an arbitrary spacelike vector ${\cal O P}$ to the vector ${\cal LR}$.
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