RELATIVISTIC VERSUS NEWTONIAN FRAMES: EMISSION COORDINATES arXiv:1001.1337v2 [gr-qc] 1 Feb 2010 J.-F. Pascual-S´anchez, A. San Miguel, and F. Vicente Dept. Matem´atica Aplicada, Facultad de Ciencias, Universidad de Valladolid, Valladolid, 47005, Spain, E.U. Only a causal class among the 199 Lorentzian ones, which do not exists in the Newtonian spacetime, is pri- vileged to construct a generic, gravity free and immediate (non retarded) relativistic positioning system. This is the causal class of the null emission coordinates. Emission coordinates are defined and generated by four emitters broadcasting their proper times. The emission coordi- nates are covariant (frame independent) and hence valid for any user. Any observer can obtain the values of his (her) null emission coordinates from the emitters which provide him his (her) trajectory. Key words: null emission coordinates, location systems, causal class, positioning system, gravimetry. Among the relativistic effects on the rate of the satellite clocks with a time accuracy of nanoseconds and 10 − 12 of frequency accuracy, the most important ones (to first post-newtonian order 1 /c $^{2}$) are: the Einstein effect or gravitational frequency blue shift of the atomic clocks of the satellites (Equivalence principle of General Re- lativity) with respect to Earth bound clocks due to their position in the Earth gravitational field, time dilation or Doppler shift of second order due to the speed of the satellites (Special Relativity) and the kinematical Sagnac effect due to the rotation of the Earth (Special Relati- vity), see Refs. [1] and [2] for reviews. If they were not corrected by imposing an offset, the GNSS would not be operative after few minutes. However, with the present and future more accurate clocks (pico and even femtosecond), it would be nece- ssary in the Newtonian framework to consider other “relativistic corrections” at post-post-Newtonian order as well as metric spatial curvature effects, tidal effects or delay effects of gravity in the light propagation as the Shapiro time delay. In this situation, it can be wondered if it would not be more convenient to change the present Newtonian frame- work to an exact formulation in full General Relativi- ty. This would imply to abandon the classical post- newtonian framework for the description of GNSS. The root of this radical change is the consideration of a new 4D proper relativistic frame (emission coordinates) ins- tead of the usual Newtonian frame, which uses 3D spatial reference systems, such as the ECI (Earth Centered In- ertial system) or the ECEF (Earth Centered Earth Fixed system), and a time reference (GPS time), separately. Emission coordinates were firstly introduced by B. Coll in a pioneering proposal presented at the ERE2000 Spa- nish Relativity Meeting and published in [4]. To dis- cuss and understand the meaning of the null emission coordinate system is necessary to introduce previously some new definitions, as such location systems or causal classification of frames, and mathematical physics tools, mainly geometrical. These new definitions and tools pro- vide a clear way to understand the differences among the special subclass of Newtonian frames and the general class of relativistic frames. At present, the GNSS functioning as global positioning systems, are the GPS and the GLONASS. In general, the satellites (SVs) of the GNSS are affected by Relativity in three different ways: in the equations of motion, in the signal propagation and in the beat rate of the satellite clocks. We will only briefly comment on the clock effects To show this, we will focus on the essential differences between an old newtonian plus “relativistic corrections” coming from the post-Newtonian framework, as in the current operating systems which use only the usual class of Newtonian frames, and the new fully relativistic frame- work which use a new class of relativistic frame: null emission coordinates. Note that there are not “relativis- tic corrections” in relativity, as they are not “Newtonian corrections” in Newtonian theory. Globally, the current situation in the Global Navigation Satellite Systems (GNSS) is almost analogous to the fo- llowing one: imagine that a century after Kepler, the as- tronomers were still using Kepler’s laws as algorithms to correct the Ptolemaic epicycles by means of “Keple- rian effects”. Similarly, a century after Einstein, one still uses the Newtonian theory and corrects it by “relativistic or Einsteinian effects” instead of starting with Einstein’s gravitational theory from the beginning. 1. INTRODUCTION ABSTRACT because they are only the measurable ones in the present GNSS and in the future Galileo.