Published Paper Details:

This article provides a comprehensive examination of Cauchy's principle in the context of special and general relativity, tracing its origins from classical partial differential equations to its modern role in the formulation of deterministic spacetime evolution. We review the mathematical structure of the Cauchy problem, including PDE classification, constraint equations, and well-posedness criteria, before applying these concepts to relativistic field theories. In special relativity, the principle manifests in Lorentz-invariant hyperbolic systems, ensuring consistent propagation of initial data across inertial frames. In general relativity, the interplay between the Einstein field equations, foliation techniques, and gauge choices defines a more intricate Cauchy problem, where spacetime geometry itself is a dynamical variable. We analyze the ADM decomposition, constraint propagation, and the implications of local versus global hyperbolicity, emphasizing the physical significance of domains of dependence and Cauchy horizons. The discussion connects these mathematical results to the principles of relativity, determinism, and cosmic censorship, situating Cauchy's principle at the core of modern theoretical physics.