Classical Electrodynamics · Problem 11.2

The key insight is that the group structure of inertial transformations—composing two successive boosts must equal a single direct boost—imposes a powerful algebraic constraint. By comparing the composition $(K'' \to K' \to K)$ with the direct transformation $(K'' \to K)$, you'll find that the transformation's functional form must satisfy a consistency condition that forces the ratio $|h(v^2)/f(v^2)|$ to be independent of velocity. This ratio must have dimensions of inverse speed squared to make the algebra work out dimensionally, revealing that relativity's structure emerges purely from the requirement that boosts form a closed group—no light-speed postulate needed yet.

The key insight is that the group structure of inertial transformations—composing two successive boosts must equal a single direct boost—imposes a powerful algebraic constraint. By comparing the composition $(K'' \to K' \to K)$ with the direct transformation $(K'' \to K)$, you'll find that the transformation's functional form must satisfy a consistency condition that forces the ratio $|h(v^2)/f(v^2)|$ to be independent of velocity. This ratio must have dimensions of inverse speed squared to make the algebra work out dimensionally, revealing that relativity's structure emerges purely from the requirement that boosts form a closed group—no light-speed postulate needed yet.