Relativity from Relative Entropy

Spurred by Einstein's appendix to \emph{The Meaning of Relativity}, we embark on an algebraic voyage from von Neumann algebras to the emergence of special relativity from algebraic structure. We identify time with modular flow (in line with Connes and Rovelli's Thermal Time Hypothesis), space with a suitable commutative subalgebra, and observers themselves with cyclic, separating states. Comparisons between observers are governed by the Connes cocycle, with the real-time cocycle near $t=0$ yielding a relative entropy $S$ that we interpret as comparing the frequencies of their respective modular ``clocks'', and the Radon-Nikodym point $t = -i/2$ giving a fidelity $\mathcal{F}$ that compares the measure density of modular ``rods''. In a Type I Gibbs model, these quantities exactly obey $\mathcal{F}^2 = e^{-S}$ in the well-localized (single eigenvalue) limit. We then analyze the Rindler wedge, where the vacuum modular Hamiltonian generates Lorentz boosts, and discuss localized coherent excitations for which the relative entropy is computable in terms of boosts and the cocycle admits a Gaussian closed form at the Radon-Nikodym point. In this regime, special-relativistic kinematics emerges from modular data. We end with a survey of promising directions to continue voyaging.