The Dynamic Doppler Spectrum Induced by Nonlinear Sensor Motion: Relativistic Kinematics and 4D Frenet-Serret Spacetime Geometry

TL;DR

This paper develops a comprehensive framework to analyze dynamic Doppler effects caused by nonlinear relativistic motion, using higher-order kinematic vectors and 4D Frenet-Serret geometry, with applications in radar and communication systems.

Contribution

It introduces new analytical expressions for Doppler spectrum transformations due to relativistic acceleration and jolt, and links signal fluctuations to geometric path parameters.

Findings

Exponential spectral broadening from relativistic acceleration.

Nonlinear skewed chirps induced by relativistic jolt.

Amplitude and phase fluctuations described by curvature and torsion.

Abstract

Fundamental to the analysis of nonlinear relativistic motion is the precise characterization of the induced dynamic Doppler effects. In this work, we analyze the electromagnetic signals observed by non-inertial receivers using two frameworks to describe the relativistic motion. We first consider observer paths described by higher-order kinematic 4 vectors: relativistic acceleration and jolt. The dynamic Doppler effects of relativistic acceleration and jolt are exponential spectral broadening and exponential amplitude growth or decay. We derive compact expressions for the spectrum transformation resulting from relativistic acceleration and jolt. The jolt induces nonlinear skewed chirps in observed signals. Next we consider observer paths described by the 4D Frenet-Serret frame and the curvature and torsion of the observer path. We obtain descriptions of the amplitude and phase fluctuations of the signal in terms of the geometric parameters of curvature and torsion. Concise, interpretable descriptions of non-inertial dynamic Doppler effects provide a useful diagnostic and predictive tool for engineering applications including radar, sensing, and communications systems.