Fermi Coordinates of an Observer Moving in a Circle in Minkowski Space: Apparent Behavior of Clocks

TL;DR

This paper derives Fermi coordinates for an observer moving in a circle in Minkowski space, analyzes clock behaviors, and discusses implications for high-precision time transfer systems like GPS.

Contribution

It provides explicit coordinate transformations and metric calculations for circular motion in Minkowski space, revealing complex clock relations and effects relevant to advanced navigation systems.

Findings

Fermi coordinate time and proper time are intricately related.

An orbital Sagnac-like effect influences portable clocks.

Kinematic effects are small but significant for future high-precision systems.

Abstract

Coordinate transformations are derived from global Minkowski coordinates to the Fermi coordinates of an observer moving in a circle in Minkowski space-time. The metric for the Fermi coordinates is calculated directly from the tensor transformation rule. The behavior of ideal clocks is examined from the observer's reference frame using the Fermi coordinates. A complicated relation exists between Fermi coordinate time and proper time on stationary clocks (in the Fermi frame) and between proper time on satellite clocks that orbit the observer. An orbital Sagnac-like effect exists for portable clocks that orbit the Fermi coordinate origin. The coordinate speed of light is isotropic but varies with Fermi coordinate position and time. The magnitudes of these kinematic effects are computed for parameters that are relevant to the Global Positioning System (GPS) and are found to be small; however, for future high-accuracy time transfer systems, these effects may be of significant magnitude.