Computational Relativity: A Geometric Theory of Algorithmic Spacetime

Computation has historically been framed in Newtonian terms: time and space as separate, absolute measures of algorithmic cost. Yet modern algorithms—from randomized heuristics to quantum circuits—demand a relativistic view where time, space, energy, entropy, and coherence form a unified manifold. This work introduces computational relativity: a geometric theory of algorithms built on spacetime geodesics, entropy trade-offs, and quantum coherence dynamics. We show how classical complexity results like the Hopcroft-Paul-Valiant theorem and recent improvements by Williams emerge naturally as specific geodesic types in this framework. The theory extends through thermodynamic principles to encompass stochastic algorithms and energy consumption, then to quantum coherence for quantum computing applications. This progression motivates living algorithms—self-monitoring systems that dynamically optimize their computational trajectories in real-time. We demonstrate applications across machine learning, quantum computing, robotics, and optimization, concluding with a comprehensive algorithm compendium that classifies computational methods by their geometric properties and resource trade-offs.