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Coordinate time and proper time in the GPS
Mathematical Physics · Submitted 30 Nov 2006
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this note we give a different derivation of this formula, {\it without using approximations}, to arrive at $ds/dt=\sqrt{1+2V-|\vv|^2 -\frac{2V}{1+2V}(\n\cdot\vv)^2}$, where $\n$ is the normal vector pointing outward from the center of Earth to the satellite.
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The abstract, sentence by sentence
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 \cite{ashby}) uses the Schwarzschild spacetime to deduce the {\it approximate} formula, $ds/dt\approx 1+V-\frac{|\vv|^2}{2}$, 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 $ds/dt=\sqrt{1+2V-|\vv|^2 -\frac{2V}{1+2V}(\n\cdot\vv)^2}$, 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 $ds/dt=\sqrt{1+2V-|\vv|^2}$. We emphasize that this derivation is useful mainly for educational purposes, as the approximation above is already satisfactory in practice.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 0.98 | 0.08 | 0.04 | asserted 1.00 |
| s1 | 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. | contribution 0.98 | 0.79 | 0.06 | asserted 0.98 |
| s2 | The theoretical background of the GPS (see \cite{ashby}) uses the Schwarzschild spacetime to deduce the {\it approximate} formula, $ds/dt\approx 1+V-\frac{|\vv|^2}{2}$, for the relation between the proper time rate $s$ of a satellite clock and the coordinate time rate $t$. | background 1.00 | 0.06 | 0.07 | asserted 1.00 |
| s3 | Here $V$ is the gravitational potential at the position of the satellite and $\vv$ is its velocity (with light-speed being normalized as $c=1$). | background 0.97 | 0.11 | 0.03 | asserted 0.97 |
| s4 | In this note we give a different derivation of this formula, {\it without using approximations}, to arrive at $ds/dt=\sqrt{1+2V-|\vv|^2 -\frac{2V}{1+2V}(\n\cdot\vv)^2}$, where $\n$ is the normal vector pointing outward from the center of Earth to the satellite. | contribution 0.86 | 0.96 | 0.79 | asserted 1.00 |
| s5 | In particular, if the satellite moves along a circular orbit then the formula simplifies to $ds/dt=\sqrt{1+2V-|\vv|^2}$. | result 0.99 | 0.85 | 0.13 | asserted 1.00 |
| s6 | We emphasize that this derivation is useful mainly for educational purposes, as the approximation above is already satisfactory in practice. | limitation 0.70 | 0.44 | 0.10 | asserted 0.94 |
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Original abstract
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 \cite{ashby}) uses the Schwarzschild spacetime to deduce the {\it approximate} formula, $ds/dt\approx 1+V-\frac{|\vv|^2}{2}$, 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 $ds/dt=\sqrt{1+2V-|\vv|^2 -\frac{2V}{1+2V}(\n\cdot\vv)^2}$, 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 $ds/dt=\sqrt{1+2V-|\vv|^2}$. We emphasize that this derivation is useful mainly for educational purposes, as the approximation above is already satisfactory in practice.
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In such a case our method and result ([ref]) might be implemented — the difference between the approach presented here and those discussed in [ref] is obvious: we apply the key element of the Schwarzschild geometry, an absolute rest system.
The Doppler shift in a Schwarzschild spacetime · Introduction · 0.97
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