On the Invariance of the Spacetime Interval

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.

💡 Research Summary

The paper revisits the foundations of special relativity by deriving the invariance of the spacetime interval solely from the constancy of the speed of light together with the homogeneity and isotropy of spacetime. The author introduces a geometric construction called a “light rectangle”: a set of four events (origin, two mirror reflections, and their meeting point) whose sides are light‑like world‑lines at ±45° in any inertial frame. By expressing the coordinates of the rectangle’s vertices in terms of two parameters r and l (the distances to the right and left mirrors), the author shows that the product r l equals (c²t² – x²)/4, which is proportional to the spacetime interval ΔS². One diagonal of the rectangle yields the timelike case (ΔS² < 0), the other the spacelike case (ΔS² > 0), while the lightlike case (ΔS² = 0) follows trivially because the rectangle collapses to zero area.

The linearity of the transformation follows from spacetime homogeneity, leading to relations r′ = K⁺ r and l′ = K⁻ l between frames. The product condition K⁺K⁻ = 1 is imposed by the requirement that the rectangle’s area be invariant. Writing K⁺ = e^{ϕ} and K⁻ = e^{-ϕ} introduces a rapidity parameter ϕ, which is linked to the relative velocity β = v/c through tanh ϕ = β. Substituting these expressions yields the standard one‑dimensional Lorentz transformations: ct′ = γ(ct – βx) and x′ = γ(x – βct), with γ = 1/√(1 – β²). Perpendicular lengths are argued to remain unchanged, completing the full set of transformations.

The author contrasts this geometric derivation with traditional algebraic approaches, emphasizing its pedagogical appeal: the invariance of a simple area replaces more abstract algebraic invariants. Nevertheless, the paper acknowledges limitations: the proof is confined to a single spatial dimension, relies on the assumption that linearity already encodes Lorentz‑type behavior, and does not address the full four‑dimensional Minkowski symmetry group. The work thus offers an intuitive, visual route to the core results of special relativity while highlighting areas where a more rigorous, higher‑dimensional treatment would be required.