Coordinate time and proper time in the GPS

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.Comment: 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'

Absolute Time Derivatives

A four dimensional treatment of nonrelativistic space-time gives a natural frame to deal with objective time derivatives. In this framework some well known objective time derivatives of continuum mechanics appear as Lie-derivatives. Their coordinatized forms depends on the tensorial properties of the relevant physical quantities. We calculate the particular forms of objective time derivatives for scalars, vectors, covectors and different second order tensors from the point of view of a rotating observer. The relation of substantial, material and objective time derivatives is treated.Comment: 26 pages, 4 figures (minor revision

On the Radiation Reaction Force

The usual radiation self-force of a point charge is obtained in a mathematically exact way and it is pointed out to that this does not call forth that the spacetime motion of a point charge obeys the Lorentz--Abraham--Dirac equation.Comment: 22 pages, 1 figur

DYNAMICS OF PHASE TRANSITIONS

Stability of equilibria in first order phase transitions is investigated by Lyapunov's method. If both phases are present then the set of equilibria is strictly asymptotically stable. The 'metastable' states (only one of the phases is present) are unstable states having a peculiar feature

Thomas rotation and Thomas precession

Exact and simple calculation of Thomas rotation and Thomas precessions along a circular world line is presented in an absolute (coordinate-free) formulation of special relativity. Besides the simplicity of calculations the absolute treatment of spacetime allows us to gain a deeper insight into the phenomena of Thomas rotation and Thomas precession.Comment: 20 pages, to appear in Int. J. Theo. Phy

Can material time derivative be objective?

The concept of objectivity in classical field theories is traditionally based on time dependent Euclidean transformations. In this Letter we treat objectivity in a four-dimensional setting, calculate Christoffel symbols of the spacetime transformations, and give covariant and material time derivatives. The usual objective time derivatives are investigated. © 2006 Elsevier B.V. All rights reserved