Relativistic Timekeeping, Motion, and Gravity in Distributed Systems

Abstract

Timekeeping and related functions such as navigation are important in distributed systems with manufactured nodes such as satellites, spacecraft, distributed computing, and communications. With increases in frequencies, velocities, clock rates, and bit rates, as well as the tightening timekeeping and positioning accuracy demanded of applications, relativistic effects increasingly enter as a design consideration. In this paper, representative applications are surveyed to illustrate relativistic effects in timekeeping. The general theory of relativity is avoided by neglecting gravity in the presence of motion and neglecting motion in the presence of gravity. The special theory of relativity combined with the equivalence principle then provide closed-form models that aid system designers in understanding the nature and relevance of relativistic effects. This is applied to modeling relativistic clock rate and time synchronization in the presence of motion or gravity. The kinematics of special relativity is developed using a new simplified approach in which velocity and acceleration are referenced to the frame-invariant proper time rather than the conventional frame-dependent coordinate time. This approach considerably narrows the scope of relativistic effects, and is particularly appropriate for manufactured systems. Further, by decomposing acceleration into components longitudinal and transverse relative to the direction of motion, simple scalar frame-invariant relations between proper and coordinate perspectives follow. The dynamics and conservation laws of special relativity are illustrated by the relativistic rocket equation. Finally, the Lorentz transformation and its implications are derived from first principles, completing a self-contained exposition. Our perspective and approaches are tailored to the needs of engineering practitioners, educators, and students, and our new methodology may be appropriate in an introductory physics education as well. Mathematically the treatment is based on first-order vector differential equations and the chain rule of differentiation.