Coordinate time and proper time in the GPS

Matolcsi, T. and Matolcsi, Máté (2008) Coordinate time and proper time in the GPS. EUROPEAN JOURNAL OF PHYSICS, 29 (6). pp. 1147-1151. ISSN 0143-0807

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Abstract

The global positioning system (GPS) provides an excellent educational example of 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 [ 1]) uses the Schwarzschild spacetime to deduce the approximate formula, ds/dt approximate to 1 + V - \v\(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 v is its velocity ( with light-speed being normalized as c = 1). In this paper we give a different derivation of this formula, without using approximations, to arrive at ds/dt = root 1 + 2V - \v\(2) - 2V/1+2V(n.v)(2), where n is the normal vector pointing outwards from the centre of Earth to the satellite. In particular, if the satellite moves along a circular orbit then the formula simplifies to ds/dt = root 1 + 2V - \v\(2). We emphasize that this derivation is useful mainly for educational purposes, as the approximation above is already satisfactory in practice.

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