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Relativistic versus Newtonian frames: emission coordinates

Only a causal class among the 199 Lorentzian ones, which do not exists in the Newtonian spacetime, is privileged 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 coordinates 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.


INTRODUCTION
Globally, the current situation in the Global Navigation Satellite Systems (GNSS) is almost analogous to the following one: imagine that a century after Kepler, the astronomers were still using Kepler's laws as algorithms to correct the Ptolemaic epicycles by means of "Keplerian 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.
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 framework which use a new class of relativistic frame: null emission coordinates. Note that there are not "relativistic corrections" in relativity, as they are not "Newtonian corrections" in Newtonian theory.
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 because they are only the measurable ones in the present GNSS and in the future Galileo.
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 Relativity) 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 Relativity), 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 necessary 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 framework to an exact formulation in full General Relativity. This would imply to abandon the classical postnewtonian framework for the description of GNSS. The root of this radical change is the consideration of a new 4D proper relativistic frame (emission coordinates) instead of the usual Newtonian frame, which uses 3D spatial reference systems, such as the ECI (Earth Centered Inertial 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 Spanish Relativity Meeting and published in [4]. To discuss 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 provide a clear way to understand the differences among the special subclass of Newtonian frames and the general class of relativistic frames.

LOCATION SYSTEMS
Location systems are physical realizations of 4D coordinate systems. Hence there is a differentiation of a coordinate system as a mathematical object from its realization through physical objects and protocols. A location system is thus a precise protocol on a set of physical fields allowing to materialize a coordinate system. However, different physical protocols, involving different physical fields, may be given for a unique mathematical coordinate system.
A location system must include the protocols for the physical construction of the coordinate lines, coordinate surfaces or coordinate hypersurfaces of specific causal orientations of the coordinate system that it realizes. Thus, for instance, these coordinate elements may be realized by means of clocks for timelike lines, laser pulses for null lines, synchronized inextensible threads for spacelike lines, laser beams or inextensible threads for time like surfaces and light-front surfaces for null hypersurfaces. The different protocols involved in the construction of location systems give rise to coordinate elements (lines, surfaces and hypersurfaces) of different causal orientations, i.e., they realize coordinate systems of different causal nature.

Reference systems
Location systems are of two different types: reference systems and positioning systems. The first ones are 4D reference systems which allow one observer, considered at the origin, to assign four coordinates to the events of its neighborhood by means of electromagnetic signals. In relativity due to the finite speed of the transmission of information, this assignment is retarded with a time delay. A paradigmatic reference system in relativity is the radar system which is based in the Poincaré protocol of synchronization which uses two-way light signals from the observer to the events to be coordinated. Unfortunately, the radar system suffers from the bad property of being constructed from a retarded protocol due to the finite speed of the transmission of information.

Positioning systems
The second kind of location systems are 4D positioning systems, which allow to every event of a given domain to know its proper coordinates in an immediate or instantaneous way without delay. In addition to be immediate, the positioning systems must verify other two conditions, they must be generic and free of gravity. A positioning system is generic, if it can be constructed in any spacetime and, it is free of gravity, if the knowledge of the gravitational field is not necessary to construct it. Reference systems privilege one specific observer among all others, whereas in positioning systems no observers are necessary at all and hence there is no necessity of any synchronization procedure between different observers.
In relativity, a (retarded) reference system can be constructed starting from an (immediate) positioning system, it is sufficient that each event sends its coordinates to the observer at the origin of the reference system, but not the other way around. In contrast, in Newtonian theory, 3D reference and positioning systems are interchangeable and as the velocity of transmission of information is infinite, the Newtonian reference systems are not retarded but immediate. The reference and positioning systems defined here are 4-dimensional objects, including time location.

CAUSAL CLASSIFICATION OF FRAMES
In the Lorentzian spacetime of general relativity, directions and planes or hyperplanes of directions at any event are said to be spacelike, lightlike (or null or isotropic) or timelike oriented if they are respectively exterior, tangent or secant to the light-cone of this event. These causal orientations can be extended in a natural way to vectors, covectors and volume forms on these sets of directions. Thus, every one of the vectors e A of a frame of the tangent space {e A } (A = 1, ..., 4) has a particular causal orientation c A .
However, the causal orientations C AB (A < B) of the six different associated or adjoint planes Π(e A , e B ) of the frame {e A } are not determined by the specific causal orientations c A of the vectors of the frame. For instance, the plane associated to two spacelike vectors may have any causal orientation. So, in general, the causal characters c A and C AB are independent. Moreover, in order to give a complete description of the causal properties of the frames, one needs also to specify the causal orientations c A of the four covectors or 1-forms θ A giving the dual coframe {θ A }, i.e. θ A (e B ) = δ A B . Following [5], the best way to visualize and characterize a spacetime coordinate system is to start from four families of coordinate 3-surfaces, then, their mutual intersections give six families of coordinate 2-surfaces and four congruences of coordinate lines.
Alternatively, one can use the related covectors or 1forms θ A , instead of the 3-surfaces, and the vectors of a coordinate tangent frame {e A } , instead of four congruences of coordinate lines which are their integral curves. The covector θ A is timelike (resp. spacelike) iff the 3-plane Π(e B , e C , e D ) generated by the three vectors {e B } B =A is spacelike (resp. timelike). This applies for both Newtonian and Lorentzian spacetimes. In addition, for the latter, the covector θ A is lightlike (or null) iff the 3plane generated by {e B } B =A is lightlike (or null). Thus, to specify the causal orientations of hyperplanes is not necessary because is redundant with the causal orientation of the covectors.
In this way, for a specific domain of a Lorentzian or New-tonian spacetime, each frame {e A } is fully characterized by its causal class. The causal class of a frame is the set of all the frames that have same causal signature, which is defined by a set of 14 causal characters: (1) As notation for the causal characters, we will use lower case roman types (s, t, l) to represent the causal character of vectors (resp. spacelike, timelike, lightlike), and capital types (S, T, L) and lower case italic types (s, t, l) to denote the causal character of 2-planes and covectors, respectively.

Relativistic frames
This new degree of freedom (lightlike) in the causal character, which is proper of Lorentzian relativistic spacetimes but which does not exist in Newtonian spacetimes, allows to obtain (see [5]), as it has been commented in the abstract, the following theorem: In a relativistic 4dimensional Lorentzian spacetime, there exists 199, and only 199, causal classes of frames. These 199 causal classes have been completely classified.
We shall see that among the 199 Lorentzian causal classes, only one is privileged to construct a generic, gravity free and immediate positioning system.
The notion of causal class extends naturally to the set of coordinate lines of the coordinate system and so, to the coordinate system itself. By definition, the causal class of a coordinate system {x α } 4 α=1 in a domain is the causal class {c α , C αβ , c α } of its associated natural frame at the events of the domain. In relativity, a specific causal class, among the 199 ones, can be assigned to any of the different coordinate systems used in all the solutions of the Einstein equations. However, for the same coordinate system and the same solution, the causal class can change depending on the region of the spacetime considered and the coordinate system in this case is said to be inhomogeneous.
In fact, see [7], in any spacetime every coordinate x α plays two extreme roles: that of a hypersurface for every constant value x α = const, of gradient dx α , and that of a coordinate line of tangent vector ∂ α , when the other coordinates remain constant. This simple fact shows that, in spite of the historical custom of associating to a coordinate a causal orientation, saying that it is timelike, lightlike or spacelike, this appellation is not generically coherent. Causal orientations are generically associated with directions of geometric objects, but not with spacetime coordinates associated to them. In the case of a coordinate x α , this generic incoherence appears because its two natural variations in the coordinate system, dx α and ∂ α , have generically different causal orientations. Only when both causal orientations coincide, it is possible to extend to the coordinate x α itself the character of the common causal orientation of its two mentioned variations.

Newtonian frames
The differences in the geometric description of Lorentzian and Newtonian frames come from the causal structure induced by the different metric descriptions of Lorentzian and Newtonian spacetimes. The main difference comes essentially from the absence of the lightlike character in the Newtonian case. In relativity, the spacetime metric is non-degenerate and defines a one-to-one correspondence between vectors and covectors at the tangent and cotangent space of every event.
In contrast, in a Newtonian space-time no non-degenerate metric structure exists and one have two different metrics, see [8]. This degenerate metric structure is given by a rank one covariant time metric T = dt 2 and an orthogonal rank three contravariant space metric γ. Correspondingly, a covector θ = 0 is timelike if it has no instantaneous part with respect to the contravariant space metric γ, i.e. if γ(θ) = 0 and it is necessarily of the form θ = N dt with N = 0, being future (resp. past) oriented if N > 0 (resp. N < 0). Otherwise, the covector θ is spacelike. Thus, attending to the causal orientation of their covectors, there only exist two causal types of Newtonian coframes bases: {tsss}, {ssss}.
In summary, it can be shown (see [7]) that one has the following implications valid only for Newtonian frames: The simplicity of the Newtonian causal structure with respect to the Lorentzian one lies in the fact that the causal type of a Newtonian frame determines completely its causal class. This is related to the fact that, in Newtonian space-time, any set of spacelike vectors always generates a spacelike subspace. As a consequence, the number of causally different Newtonian classes of frames is equal to the dimension of the spacetime. Hence, see [7], in the 4-dimensional Newtonian spacetime there exist four, and only four, causal classes of frames. For instance, the standard spatial coordinates ECI and ECEF used in the GPS more the GPS time, i.e. those that are locally realized with three rods and one clock, belong to the same causal class {tsss, TTTSSS, tsss}, the first one above. The history of the clock is a timelike coordinate line. The other coordinate lines are spacelike straight lines tangent to the rods at every time. Also the reference systems adopted by the I.A.U. for the Earth and the barycenter of the Solar system as, respectively, the 3- 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 belong to this usual causal Newtonian class.

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 system. 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 coordinates 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 homogeneous 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 spacelike 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 emitter 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 intersect 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 satellites ({τ A }; A = 1, 2, 3, 4) read at an event by a receiver or user constitute the null (or light) proper emission coordinates or user positioning data of this event, with respect to four SVs, see Figure 2. These four numbers or parameters 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 parameters {τ A }, any user receiving continuously his null emission coordinates from four satellites may know his trajectory 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 interplanetary 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 millisecond pulsars and a conventional origin of the emission 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 complicated because post-post-Newtonian corrections must be implemented.

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 asymmetry 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 (l l l l ) = {d τ 1 , d τ 2 , d τ 3 , d τ 4 }, which is neither orthogonal nor normalized, the contravariant spacetime metric is symmetric with null diagonal elements and it has the general expression [16,2]: 0 g 12 g 13 g 14 g 12 0 g 23 g 24 g 13 g 23 0 g 34 g 14 g 24 g 34 0    , (2) where g AB > 0 for A = B. Four null covectors can be linearly dependent although none of them is proportional 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 observables, i.e, gauge independent or invariant quantities under diffeomorphisms in the Lorentzian spacetime.
A splitting of this metric can be considered, see [16], changing from the six independent components (ten components minus four gauge degrees of freedom of coordinate 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 equivalently 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.

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 provided with clocks that interchange its proper time among them (interlinks) and with Earth receivers, is a fully relativistic 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 constellation 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 constellation of SVs. Therefore, the procedure used until now in the newtonian GNSS is inverted.
Let us define properly what means autonomous or autolocated. 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 coordinates.

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 metric (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 accelerometer providing information on the spacetime connection. 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 gradiometer, 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 optimization procedure could be developed (see [14]) to obtain the best observational gravitational field acting actually 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

Two dimensional case
Coll positioning systems are yet now quite well developed 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 coordinates. The important point for gravimetry is that the Schwarzschild mass may be substituted by that of the acceleration of one of the emitters.

Realistic four dimensional case
For applications of a autolocated positioning system on or near the Earth's surface, the primary emission coor- dinates should be related to some terrestrial secondary 4dimensional Newtonian coordinate system. This problem has been solved for a general configuration of the emitters in flat Minkowski spacetime [17] and also for the case of a special configuration of the emitters in a Schwarzschild spacetime [18].
However, in general the known results for the two dimensional case are not trivially generalizable for the realistic four dimensional one [16] and much work remains to be done in the future.

ACKNOWLEDGMENTS
One of the authors (JFPS) is grateful to Bartolomé Coll by communications on this subject along several years and to Joan Josep Ferrando and Juan Antonio Morales by some friendly discussions. The authors are currently partially supported by the Spanish Ministerio de Educación y Ciencia MEC-FEDER ESP2006-01263 grant.