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dimensional International Terrestrial Reference System (ITRF), which is also an ECEF, plus the International Atomic Time (TAI) and the 3-dimensional International Celestial Reference Frame (ICRF) plus the TCB time be- long to this usual causal Newtonian class. 

## 4. RELATIVISTIC POSITIONING 

## 4.1. Coll positioning system 

As it has been commented above, among the 199 Lorentzian causal classes, in which the four Newtonian ones are included, only one is privileged to construct a generic (valid for a wide class of spacetimes), gravity free (the previous knowledge of the gravitational field is not necessary) and immediate relativistic positioning sys- tem. This is the causal class { s s s s , S S S S S S , l l l l } of the Coll homogeneous coordinate system [4, 2, 5]. In this causal class the null emission coordinates of the Coll positioning system are included. These emission coordi- nates have been also studied in [9, 10, 11] in the special case of a flat Minkowski spacetime without gravity. 

The coordinate system of this causal class is always ho- mogeneous and it has associated four families of null 3-surfaces or equivalently a real non-orthogonal null coframe, whose mutual intersections give six families of spacelike 2-surfaces and four congruences of space- like lines. Such a coordinate system does not exist in a Newtonian space-time where the light travels at infinite speed. One satellite clock broadcasting its proper time is des-cribed in the spacetime by a timelike line γ $^{A}$( τ $^{A}$) in which each event of proper time τ A is the vertex of a future light cone. The set of these light cones of a emit- ter constitutes a one-parameter (proper time) family of null hypersurfaces. So, four satellite clocks broadcasting their proper times determine four one-parameter families of lightlike 3-surfaces (future light cones), see Figure 1. Thus, the Coll positioning system makes use of the ma- 

thematical fact that four future light cones generically in- tersect in an unique event, which is just the spacetime position of the receiver or user. 

In this relativistic positioning system, any receiver or user at any event in a given spacetime region can know its proper coordinates. The four proper times of four sate- llites ( { τ $^{A}$} ; A = 1 , 2 , 3 , 4 ) read at an event by a receiver or user constitute the null (or light) proper emission coor- dinates or user positioning data of this event, with respect to four SVs, see Figure 2. These four numbers or pa- rameters can be understood as the “distances” between the reception event and the four satellites. 

In a certain domain Ω ⊂ R 4 of the grid of parame- ters { τ $^{A}$} , any user receiving continuously his null emis- sion coordinates from four satellites may know his tra- jectory in the grid of parameters. If the observer has his own clock, with proper time denoted by σ , then he can know his trajectory with proper time parametrization, τ A = τ $^{A}$( σ ) , and his four-velocity, u $^{A}$( σ ) = dτ $^{A}$/dσ . 

For positioning out a GNSS constellation, i.e. for inter- planetary missions in the Solar system, a “pulsar” Coll relativistic positioning system can be conceived, based on the X-ray signals of four properly selected stable mi- llisecond pulsars and a conventional origin of the emi- ssion coordinates. On the other hand, a navigation project called XNAV (based in pulsars) is being developed during the last years by DARPA and NASA but unfortunately, see [13], is based in the same Newtonian concepts that the GPS or Galileo. However, in this case, it is more compli- cated because post-post-Newtonian corrections must be implemented. 

## 4.2. Contravariant metric in emission coordinates 

As the emission coordinates belong to the causal class { s s s s , S S S S S S , l l l l } , there is not a spacetime asym- metry like in the standard Newtonian coframe ( t s s s ) (one timelike “ t ” and three spacelike “ s ”). In emission coordinates obtained from a general real null coframe 

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