content_image/385772.jpg
( l l l l ) = { d τ $^{1}$, d τ $^{2}$, d τ $^{3}$, d τ $^{4}$} , which is neither or- thogonal nor normalized, the contravariant spacetime metric is symmetric with null diagonal elements and it has the general expression [16, 2]: 

<img src='content_image/385772.jpg'>

where g AB > 0 for A ̸ = B . Four null covectors can be linearly dependent although none of them is propor- tional to another. To ensure that the four null covectors are linearly independent and span a 4-dim spacetime, it is sufficient that det(g $^{AB}$) ̸ = 0 . Finally, this metric has a Lorentzian signature (+ , − , − , − ) iff det(g $^{AB}$) < 0 . 

The expression (2) of the metric is observer independent and has six degrees of freedom. In the terminology of [9], the proper times τ A are partial observables , while the components of the metric g AB are complete obser- vables , i.e, gauge independent or invariant quantities un- der diffeomorphisms in the Lorentzian spacetime. 

A splitting of this metric can be considered, see [16], changing from the six independent components (ten com- ponents minus four gauge degrees of freedom of coordi- nate transformations) of g AB to a more convenient set, which neatly separates two shape parameters depending only on the direction of the covectors dτ A or equiva- lently depending exclusively on the trajectories of the emitters, from other four scaling parameters depending on the length of the covectors or depending on the proper time of each satellite. 

## 4.3. SYPOR project: autolocated positioning system 

SYPOR project is the anagram (in French) of Relativistic Positioning System project. The basic idea of this project, that was conceived by Coll in [6] and also exposed in [2, 3], is the following one: A satellite constellation pro- vided with clocks that interchange its proper time among them (interlinks) and with Earth receivers, is a fully rel- ativistic autonomous or autolocated positioning system. Note that, nowadays, this procedure of proper time auto navigation can be technically fulfilled. 

In the SYPOR, the segment of Control is in the constella- tion of satellites , see Figure 3. The function of this new Control segment is not to determine the ephemerides of the satellites with respect to geocentric coordinates as in the newtonian GNSS, but to determine the null emission coordinates of the receivers with respect to the constella- tion of SVs. Therefore, the procedure used until now in the newtonian GNSS is inverted. 

Let us define properly what means autonomous or autolo- cated. Four satellites emitting, without the necessity of a synchronization convention, not only its proper times τ $^{A}$, but also the proper times τ AB of the three close satellites received by the satellite A in τ A (in total sixteen emitter 

positioning data { τ $^{A}$, τ$^{AB}$} ; A ̸ = B ; A, B = 1 , 2 , 3 , 4 ), constitute an autolocated positioning system. 

In an autolocated positioning system, the receivers can know not only its spacetime path but also the trajectories of the four satellites in the grid$_{R}$ 4 of emission coordi- nates. 

## 5. GRAVIMETRY AND POSITIONING 

In General Relativity, the gravitational field is described by the spacetime metric. If this metric is exactly known a priori, the system just described will constitute an ideal positioning system. In practice, the actual spacetime me- tric (i.e., the gravitational field) is not exactly known (in the GPS it is supposed to be essentially the Schwarzschild one) and the satellite system itself has to be used to infer it. This problem arises when a satellite system is used for both positioning and gravimetry. 

To solve this joint problem, the considered satellites should have more than one clock: they may carry an ac- celerometer providing information on the spacetime con- nection. Of course, in first approximation the satellites are in free-fall and consequently have zero acceleration. However, we are considering here the realistic case where the acceleration is nonzero due, for instance, to a small drag in the high atmosphere and this is measured by the accelerometers. Also, the satellites may have a gradiome- ter, this would give additional information on the metric (in fact, on the Riemann tensor of the spacetime). With these data (and perhaps some additional ones) an opti- mization procedure could be developed (see [14]) to ob- tain the best observational gravitational field acting ac- tually on the constellation. The problem of obtaining the spacetime metric is a kind of inverse problem since one wants to recover the spacetime metric from the observed data in the Coll positioning system 

## 5.1. Two dimensional case 

Coll positioning systems are yet now quite well devel- oped for two-dimensional spacetimes, see [11, 12] were several results have been developed. For instance, the knowledge that the positioning system is stationary and that the space-time is created by a given mass, allows to know the accelerations of the emitters, their mutual radar distances and the spacetime metric in null emission co- ordinates. The important point for gravimetry is that the Schwarzschild mass may be substituted by that of the ac- celeration of one of the emitters. 

## 5.2. Realistic four dimensional case 

For applications of a autolocated positioning system on or near the Earth’s surface, the primary emission coor- 

