2006/11/30 by T. Matolcsi, Tamas Matolcsi, Máté Matolcsi +3
Earth and Planetary Sciences · Mathematics · Physics and Astronomy · #53C80 #83A05 #FOS: Physical sciences #Geophysics and Gravity Measurements #Mathematical Physics (math-ph) #Pulsars and Gravitational Waves Research #Relativity and Gravitational Theory #math-ph #math.MP #msc:53C80 #msc:83A05
paper · pdf · doi:10.48550/arxiv.math-ph/0611086
5 pages, revised, over-over-simplified... Does anyone care that the GPS uses an approximate formula, while a precise one is available in just a few lines??? Physicists don't
openalex publication_date 2006/11/30 · arxiv created 2008/10/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Global Positioning System (GPS) provides an excellent educational example as to how the theory of general relativity is put into practice and becomes part of our everyday life. This paper gives a short and instructive derivation of an important formula used in the GPS, and is aimed at graduate students and general physicists. The theoretical background of the GPS (see \citeashby) uses the Schwarzschild spacetime to deduce the \it approximate formula, ds/dt≈ 1+V-(|\vv|2)/(2), for the relation between the proper time rate s of a satellite clock and the coordinate time rate t. Here V is the gravitational potential at the position of the satellite and \vv is its velocity (with light-speed being normalized as c=1). In this note we give a different derivation of this formula, \it without using approximations, to arrive at ds/dt=√(1+2V-|\vv|2 -(2V)/(1+2V)(\n⋅\vv)2), where \n is the normal vector pointing outward from the center of Earth to the satellite. In particular, if the satellite moves along a circular orbit then the formula simplifies to ds/dt=√(1+2V-|\vv|2). We emphasize that this derivation is useful mainly for educational purposes, as the approximation above is already satisfactory in practice.