GPS - Spacetime Observables
Modern Satellite Positioning Systems have to be modelled in the framework of General Relativity which affects the high precision orbit computation, the propagtion of electromagnetic waves and the behaviour of clocks (oscillators), moving in the gravitational field of the celestial bodies. This has to be combined to obtain observation equations in satellite geodesy.
The starting point is the definition of an observable on a pseudo-riemannian manifold which consists of two steps. The first step is the description of electromagnetic wave propagation within the framework of general relativity. As it has been shown in (V.S. Schwarze et ah 1993) the electromagnetic wave follows a geodesic line in the vacuum, refractive and dispersive case with respect to the corresponding space-time metric. For verification see also the reference mentioned above. The second step considers the fact that an observer moving on a pseudo-riemannian manifold refers his measurements to a pseudo-orthonormal (anholonomic) frame, whereas the required quantities refer to a coordinate induced (holonomic) frame which is in general not pseudo-orthogonal. The relations between these two reference frames can be given by applying a proper normalization technique and a sequence of spatial and timelike rotations which will be published elsewhere.
These two steps are combined to derive observation equations for the time-difference and phase-difference technique. The relativistic model can be separated in orbit, propagation and oscillator terms. Numerical investigations have shown that the contribution of the relativistic medium terms (refractive, dispersive) is below the measurement accuracy. Only the kinematical influence of the Earth’s rotation and the monopole part of the Earth’s gravitational field have to be taken into account for the relativistic propagation model. In most cases it is sufficient to describe the relativistic oscillator model by the monopole and quadrupole part of the Earth’s gravitational field. By using a geocentric chart the influence of the other planets can be neglected in the case of satellite positioning whereas they have to be considered for the definition of high accuracy time standards.
- Schwarze,V.S., T. Hartmann, M. Leins and M.H. Soffel (1993): Relativistic Effects in Satellite Positioning, manuscripta geodaetica, 18, 306–316Google Scholar