Positioning systems in Minkowski space-time: Bifurcation problem and observational data

Positioning systems in Minkowski space-time: Bifurcation problem
and observational data

Bartolomé Coll bartolome.coll@uv.es    Joan Josep Ferrando joan.ferrando@uv.es    Juan Antonio Morales-Lladosa antonio.morales@uv.es Departament d’Astronomia i Astrofísica, Universitat de València, 46100 Burjassot, València, Spain.
Abstract

In the framework of relativistic positioning systems in Minkowski space-time, the determination of the inertial coordinates of a user involves the bifurcation problem (which is the indeterminate location of a pair of different events receiving the same emission coordinates). To solve it, in addition to the user emission coordinates and the emitter positions in inertial coordinates, it may happen that the user needs to know independently the orientation of its emission coordinates. Assuming that the user may observe the relative positions of the four emitters on its celestial sphere, an observational rule to determine this orientation is presented. The bifurcation problem is thus solved by applying this observational rule, and consequently, all of the parameters in the general expression of the coordinate transformation from emission coordinates to inertial ones may be computed from the data received by the user of the relativistic positioning system.

pacs:
04.20.-q, 95.10.Jk
preprint: APS/123-QED

I Introduction

To locate the users111The word “user” here denotes any person or device able to receive the pertinent emitted data from the relativistic positioning system and to extract from it the corresponding information. For short, we shall refer to the user as “it”. of a Global Navigation Satellite System (GNSS), several geometric methods and algebraic algorithms have been developed in the past Schmidt-72 (); Bancroft-85 (); Kreuse-87 (); Abel-Chaffee-91 (); Chaffee-Abel-94 () that are still in use Strang-Borre-97 (); Juang-Tsai-09 (). Basically, the algebraic statement of the location problem is rather simple: to find the events where the emission light cones of four broadcast signals intersect. Of course, this idea is implicit in the Bancroft algorithm Bancroft-85 () and other similar ones Kreuse-87 (). In fact, Abel and Chaffee Abel-Chaffee-91 (); Chaffee-Abel-94 () used Minkowskian algebra to state the problem properly, making apparent that the more Lorentzian a description is, the more clear algorithm is performed.

However, in a full relativistic framework (cf. Bahder-2001 (); Coll-ERE-2000 (); Coll-Bucarest-2002 (); Rovelli-2002 (); Hehl-2002 (); Bahder-2003 (); D4a ()), and even in the case of the flat space-time, an explicit form of the solution of the location problem for arbitrary emitters has not been obtained until recently emission-1 ().

In emission-1 (), an exact relativistic formula giving the inertial coordinates of an event in terms of the received emission coordinates is obtained. This formula applies in all the emission coordinate region and involves the orientation of the emission coordinates of the user. Nevertheless, there exists an inherent limitation on the applicability of this formula: only the users in a certain region (named the central region, see Sec. IV.2) of a positioning system can obtain the orientation from the sole standard emission data, that is to say, from the sole set of the positions of the four emitters in inertial coordinates and of the emission coordinates of the user. Consequently, only these restricted users are able to locate themselves in inertial coordinates.

Here, assuming that the users out of the central region may observe the relative positions of the four emitters on their celestial sphere, we will give a simple rule allowing any user of the positioning system to locate itself in inertial coordinates. To show that, we will see that the orientation of the emission coordinates of a user is related to the relative positions of the emitters of the positioning system on the celestial sphere of the user.

In building current GNSS models, the usual assumption consists in picking out an approximate numerical solution. But, because gravitational effects are not taken into consideration at the considered leading order, one should start from the best accurate solution that nowadays we know. Such a solution is precisely the simple, exact, and covariant formula, found in Ref. emission-1 () and improved here, giving the location of a user of a relativistic positioning system in Minkowski space-time.

Let us remark that our result not only concerns GNSS around the Earth, but also general (relativistic) positioning systems anywhere in the Solar System or elsewhere. It is true that for most, but not all, of the present applications of the GNSS the users are near the Earth’s surface. Therefore, they are usually in the central region of the satellites they detect, so that additional data (and in particular our observational rule) are not necessary. But for other applications of the GNSS as well as for general positioning systems, our observational rule may be a simple way to solve the bifurcation problem and hence, the location one.

i.1 Outline of the paper

The paper is organized as follows. In Sec. II, the inertial coordinates of a user are expressed in terms of the emitter configuration and the orientation of the positioning system. This provides a covariant formula for the transformation from emission to inertial coordinates. An analysis of the solution in terms of the configuration of the emitters is also presented. In Sec. III, some properties of the border between the two emission coordinate domains are obtained, and an observational rule to detect it is remembered. Section IV is devoted to define the genuine regions and coordinate domains involved in the problem. We stress the geometrical meaning of the coordinate transformation formula in connection with these regions. In particular, we show that in the central region of the positioning system the orientation is computable from the sole standard emission data. In Sec. V we discuss the bifurcation problem (nonuniqueness of solutions in the determination of the location) which is related to the existence of regions whose events can not be located from the sole standard emission data. We give an observational rule to solve the above indetermination problem. This rule allows us to determine, at any event in the emission region, the orientation of the emission coordinates of the user from the observational data of the relative positions of the emitters on the celestial sphere of any user at this event. The concluding Sec. VI is devoted to summarize and discuss the results. The used notation is explained in an appendix.

Some preliminary results of this work were presented at the Spanish Relativity meeting ERE-2010 NosERE10b ().

i.2 Relativistic positioning terminology: Brief compendium

As pointed out in Ref. Coll-ERE-2000 (), relativistic positioning systems Bahder-2001 (); Coll-ERE-2000 (); Coll-Bucarest-2002 (); Rovelli-2002 (); Hehl-2002 (); Bahder-2003 () and the emission coordinates D4a (); NwReEC () they realize are essential elements to develop the relativistic theory of the GNSS. Starting from scratch, we present here a compendium of basic definitions about this specific subject. Anyway, we consider these definitions necessary not only to make this paper self contained, but also as an incipient piece of concepts to deal with GNSS in a full relativistic perspective.

Relativistic positioning system: set of four emitters (), of worldlines , broadcasting their respective proper times by means of electromagnetic signals.222For simplicity the proper time is taken here, but any other time is valid. For example, the Global Positioning System (GPS) broadcasts the GPS time, a time which, roughly speaking, coincides up to a fixed shift with the International Atomic Time (TAI), a sort of mean proper time on the Earth surface.

Emission coordinates of an event: the four times which are received at each event reached by the emitted signals.333Emission coordinates have received different appellations in the past (see NosERE10a () for a brief and critical account).

Configuration of the emitters for an event x: set of four events of the emitters at the emission times received at .

Emission region: set of events reached by the four signals broadcast by the positioning system. Every is labelled with the corresponding emission coordinates .

Characteristic emission function: map that to every associates its emission coordinates, that is .

The characteristic emission function describes the action of a positioning system and, hence, represents it.

Emission coordinate region: subset of the emission region where the gradients are well defined and linearly independent.

The emitter worldlines are excluded from because every is not defined at the emission event (this event being the vertex of the emission light cone ).

Orientation of a relativistic positioning system at the event : orientation of its emission coordinates at . It is given by the sign of the Jacobian determinant of at , .

In terms of the gradients of the emission coordinates, one has

(1)

where stands for the Hodge dual operator, and is the exterior product (see Appendix A for transcription into index notation).

Ii The location problem in Minkowski space-time

Suppose a given specific coordinate system covering the emission region , let be the worldlines of the emitters referred to this particular coordinate system, and let be the values of the emission coordinates received by a user. The data set is called the standard emission data set.

The location problem with respect to , also called the standard location problem for short, is the problem of finding the coordinates of the user from the sole data .

In Ref. emission-1 (), the above standard location problem was analyzed for arbitrary relativistic positioning systems in Minkowski space-time, assuming that the specific coordinate system is an inertial one. There, the explicit expression was found, giving the coordinate transformation from emission coordinates to inertial ones (Eq. (3) below).

Particular simple cases have already been studied: considering a 2-dimensional D2a (); D2b (); D2c () or a 3-dimensional Pozo-Escola3D () space-time, or for special motions of the emitters in the Schwarzschild geometry from analytical BiniMas () and numerical DelvaMas () approaches. For a recent approach to emission coordinates using the integration of the eikonal equation, and some numerical simulations, see also Ref. Bu-Ca-Matzner-2011 ().

In this and the following sections, we are mainly dealing with relativistic positioning systems in Minkowski space-time.

ii.1 Covariant expression of the solution

Figure 1: The emission of an electromagnetic signal from a satellite at proper time , and its reception by a user at . These events define the future pointing null vector .

From now on, we shall suppose that any user in the emission coordinate region receives the standard emission data set . Let us denote by the position vector (with respect the origin of this inertial system) of an event in the emission region , . If a user at receives the broadcast times , denote the position vectors of the emitters at the emission times, . Then

(2)

are future-oriented light-like vectors that represent the trajectories followed by the electromagnetic signals from the emitters to the reception event (see Fig. 1).

In the standard emission data set , the emission data received at are the emission coordinates of the event and were broadcast when the emitters were at the events , the configuration of the emitters for the event . Generically, these four events determine the configuration hyperplane for .444Here, we always consider that the emitter configuration is regular, i. e., the four events are noncoplanar. Nonregular or degenerate configurations are considered elsewhere (for some remarks concerning these situations, see Refs. Abel-Chaffee-91 (); Chaffee-Abel-94 ()).

For the events in the emission coordinate region , the transformation from emission to inertial coordinates is locally well defined. In emission-1 (), we have obtained a covariant expression of this transformation, given by the following formula:

(3)

where has been chosen as the reference emitter.555The transformation (3) from emission to inertial coordinates may be written in a totally symmetric form without the choice of any emitter worldline as reference origin line. For this purpose, one has to consider the barycenter of the emitters as the convenient reference event rather than one of the emitters. This issue will be addressed elsewhere Minko-baricentric (), in connection with the symmetric formulation of the location problem in flat space-time.

Quantity is given by

(4)

where is the configuration vector

(5)

and is the configuration bivector

(6)

with (see Fig. 2)

(7)

and where is any vector transversal to the configuration, , and stands for the tensor contraction of and the first slot of (see Appendix A).

Figure 2: If we choose the emitter four as origin (reference emitter), the relative positions of the others (referred emitters) are , , and the position vector of the event is .

Quantity is the orientation of the positioning system at , that is now equivalently expressed as

(8)

It is worthy to remark that and are determined by the relative positions associated with a given configuration of the emitters. Therefore, is directly computable from the sole standard emission data.

Nevertheless, if we want to obtain from (3) we also need to determine the orientation , which involves, by substituting (2) in (8), the unknown . In fact, from Eqs. (2) and (7) it is clear that (see Fig. (2)) and one obtains:

(9)

that taking into account (5) allows us to express Eq. (8) as

(10)

which by (2) depends on . Therefore, in order to show that Eq. (3) does not chase its own tail, we must be able to determine the orientation at by using a procedure not involving the previous knowledge of .

ii.2 Analysis of the solution

Figure 3: For the event , the configuration hyperplane is space-like, . In this case, the emitters remain on a 2-sphere laying in , and a sole emission solution exists. In this 3-dimensional representation for three satellites, the 2-sphere reduces to a circle.

Figure 4: For the event , the configuration hyperplane is light-like, . In this case, the emitters remain on a 2-paraboloid laying in , and a sole emission solution exists. In this 3-dimensional representation for three satellites, the 2-paraboloid reduces to a parabola.

In Ref. emission-1 (), Eq. (3) was obtained by separately analyzing three different cases, and gluing together their different solutions in a sole covariant an analytic expression. In gluing them, the role played by the external element is essential.666For a detailed discussion about this point, see Ref. emission-1 ().

The three cases correspond to the different causal characters of the configuration vector . In space–time metric signature , one has for each case:

(i) time-like, : there is a sole emission solution (the other one is a reception solution). The orientation corresponding to the emission solution remains to be calculated;

(ii) light-like, : there is a sole emission solution (the other one being degenerate). The orientation corresponding to the emission solution remains to be calculated;

(iii) space-like, : there are two emission solutions, and . They only differ by their orientation . The problem is how to determine the one corresponding to the real user.

Figure 5: For the events, and , the configuration hyperplane is time-like, . In this case, the emitters remain on a 2-hyperboloid laying in , and both and are emission solutions corresponding to the same emission coordinates (). In this 3-dimensional representation for three satellites, the 2-hyperboloid reduces to a hyperbola.

The above cases are illustrated in Figs. 3, 4 and 5.

For cases (i) and (ii), the matter to determine was solved in emission-1 () (see Sec. IV.2 below). Figure 3 shows the emission solution for the case (i). The configuration hyperplane, being space-like, cuts the past light cone of the solution in a 2-sphere containing the configuration of the emitters. Figure 4 shows case (ii), where the emitter configuration stays on a 2-paraboloid contained in the null configuration hyperplane.

For case (iii), can not be determined from the sole emission data. Figure 5 shows a pair of emission solutions receiving the same emission coordinates. This ‘indetermination’ is known as the bifurcation problem. To solve it is the main subject of this paper (see Sec. V).

Iii The border between the emission coordinate domains

The emission coordinate region contains two emission coordinate domains (see Sec. IV below). The border between these domains is the hypersurface , where the Jacobian determinant of the characteristic emission function vanishes,

(11)

We are going to obtain some related properties showing its interest in relativistic positioning.

First, let us note that, in an adequate and condensed form, Eq. (3) reads as

(12)

where

(13)

As is a null vector, and vectors and generate the same 2-plane, the following relation holds:

(14)

and then , assuring consistence for the above definition of . Consequently, one has the following result, made already evident by Eqs. (14) and (10).

Proposition 1: if, and only if, .

The fact that is non-negative says that the 2-plane generated by and is everywhere time-like, except in the border , where this plane is light-like.

Coming back to Eq. (6), let us note that is a simple bivector, that is,

(15)

for some vector , because of , which is a direct consequence of Eqs. (5) and (6). Therefore, the invariant vanishes, , and the invariant takes the expression (see Appendix A):

(16)

On the other hand, substituting (15) into (4), is expressed as

(17)

and then, Eq. (13) for becomes

(18)

Consequently, really does not depend on the choice of the transversal vector and, by comparing (16) and (18), the following result has been proved.

Proposition 2: Up to sign, the quantity defined in (13) is the scalar invariant of the configuration bivector :

(19)

Moreover, from Eq. (19), the user can determine from the sole standard emission data . Thus, taking into account Proposition 1, the user is able to know, from the sole standard set it receives, when it is crossing the border of the two emission coordinate domains.

Furthermore, it is worth remarking that on the border the location of a user may be unambiguously solved. There, its location is obtained from (12) by taking in Eq. (13).

On the way, taking into account that , will be a null bivector only when the invariant vanishes. Then, we have also proven the following result.

Proposition 3: For an event , the configuration bivector is a null bivector if, and only if .

On the other hand, an observational method allowing the user to detect when it is on the border has been previously studied by Coll and Pozo, who stated the following result Pozo-Escola4D (); Pozo-Varsovia-2005 ().

Proposition 4: The border consists in those events for which any user at them can see the four emitters on a circle on its celestial sphere.

This result is rather counterintuitive. When the GPS satellites are all near the horizon, or are all too close together on our zenith, the error in positioning is great. It would seem then that the optimal conditions for a precise location would be obtained when all the satellites are situated on an intermediate circle of the celestial sphere (say, among 30 or 60 degrees with respect to the zenith). Nevertheless, Proposition 4 shows that the circle corresponds to the most degenerate distribution that a set of satellites may have.

Proposition 4 also makes clear that the border may be plotted from the sole observational data, a result that was not, a priori, evident.

Iv Regions and coordinate domains in relativistic positioning

This section provides a geometrical background to analyze the space-time regions which are relevant in relativistic positioning. In particular, we study the subset of the emission coordinate region , where the orientation is computable from the standard emission data (the central region of the positioning system).

iv.1 Emission configuration regions , and

The emission coordinate region is constituted by three disjoint regions, and one can write . They are the space-like , the null and the time-like emission configuration regions defined by the conditions , , and , respectively.

This means that at every event ( or , respectively) a user receives the signals from four emission events that generate a space-like (null or time-like, respectively) hyperplane.

From Eqs. (5) and (7), which only involve the emitter configuration of the standard emission data , the user is able to determine the sign of . Consequently, from the data set and the above definitions, the user knows in what configuration region, , , or , of the positioning system it is traveling.

iv.2 The central region

We name the central region of the positioning system.

At every event , one has for any future pointing time-like vector , because is not space-like in this region. Taking into account that is a future pointing null vector, the sign of the scalar products and is the same for any future pointing time-like vector , and from Eq. (10) this sign is precisely the orientation of the positioning system on the central region. More precisely, we can prove the following result:

Proposition 5: In the central region , the orientation of a relativistic positioning system is constant, and may be evaluated from the sole standard emission data :

(20)

where is any future pointing time-like vector.

Thus, from (20) any user in the central region is able to determine the orientation of the positioning system, and then, from Eqs. (3)-(7), it can obtain its own position in the inertial system from the sole standard emission data by substituting in (3). The resulting sign of will be positive or negative depending on the time orientation of the computed vector .

iv.3 Front () and back () coordinate domains

As a consequence of Proposition 5, the Jacobian determinant does not vanish, , in the immediate vicinity of . Therefore, the border divides the time-like configuration region of . In other words, the whole region cannot be recovered by a sole coordinate domain.777Remember that, for historical reasons, the coordinate domain of a coordinate system is an open set, but not necessarily a topological domain (i. e. an open and connected set). In emission-1 () we proved the following result.

Proposition 6: The emission coordinate region is not a coordinate domain but the union of two disjoint coordinate domains, called the front , and the back emission coordinate domains, .

The front coordinate domain contains the central region and a proper subset of the time-like configuration region, . This proper subset is the part of adjacent to the central region , so that the whole front domain , , has, by continuity, constant orientation (the same as the central region). However, the orientation at can not be determined from Eq. (20), because Proposition 5 only applies on .

The back coordinate domain is not a simply-connected domain. In fact, the region is not simple-connected, and its leaves are constituted by pairs of events having the same emission coordinates but different inertial ones, defining two well differentiated regions: if , then .

To illustrate these coordinate domains, let us consider the simple case of a symmetric stationary positioning system in flat space-time. In this case, the four emitters define a regular tetrahedron, and is the union of four connected components. The common boundary of the domains and is a four-leaf hypersurface that contains the shadows that each satellite produces on the signals coming from the other ones in the region . The orientation of the positioning system only changes across taking different constant value on each coordinate domain. The analogous, but simpler to draw, stationary and symmetric 3-dimensional case is illustrated in Fig. 6, that shows the involved configuration regions and coordinate domains.

Figure 6: Quotient space of the stationary observers of a stationary and symmetric relativistic positioning system in flat 3-dimensional space-time. Dots represent stationary user worldlines, and solid dots stand for stationary satellites (emitters). The emission configuration regions , and are differently colored. The border between and is : the surface of vanishing Jacobian. Conjugate events, , having the same emission coordinates but different inertial ones, necessarily occur on separate parts of : if , then . In this 3-dimensional situation, is just the union of the shadows that each satellite produces on the signals coming from the other ones in the region .

V The bifurcation problem. Its observational solution

Figure 7: Observational rule to determine the orientation at the location of a user from the relative positions of the emitters on its celestial sphere. In the left diagram, the visual axis is oriented towards the spherical cap that does not contain the fourth emitter, and then . In the middle diagram, the four emitters are on a circle of the celestial sphere of the user, and thus (Proposition 4). In the right diagram, the visual axis is oriented towards the spherical cap that contains the fourth emitter, and thus .

The above results show that the standard emission data are generically insufficient to locate a user of a positioning system in an inertial system.

In the past, and in connection with GNSS, this problem was pointed out by Schmidt Schmidt-72 () and studied by Abel and Chaffee Abel-Chaffee-91 (); Chaffee-Abel-94 () by introducing a “bifurcation parameter” (equivalent to the square of the configuration vector of Eq. (5)). Afterwards, it was referred as the bifurcation problem Grafarend-Shan-1996 ().

In current practical situations in present day GNSS,888A present day GNSS allows locating only part of the interior of a sphere surrounding the satellite constellation.  the bifurcation problem may be solved by hand: simply checking which of the two solutions satisfies an observable pertinent constraint. Thus, for example, if a user stays near the Earth surface the right solution is the nearest to the Earth radius. However, in extended GNSS or more general positioning systems in the Solar System, the bifurcation problem cannot be so easily avoided; it will always be present for users traveling in the time-like configuration region .

One could think that the bifurcation problem could be avoided by continuity for users traveling from the central region (where they are able to calculate the positioning system orientation from the standard emission data) to the time-like configuration region . But the discrete character of true successive location operations, and the fact that it suffices of a sole instant to cross the border , also make this possibility illusory. Not only for theoretical reasons, but also for future practical applications where the role played by Earth based coordinate systems could become secondary (cf. Coll-ERE-2000 (); D4a (); D2a ()), it is essential to learn to solve this important part of the location problem, the bifurcation problem.

We have seen that, from the sole standard emission data , the users can know the configuration region that they are traveling. The bifurcation problem appears when this configuration region is the time-like one, , because this region is constituted by pairs of conjugate events, and , separated by the border , receiving the same standard emission data (see Figs. 5 and 6). Conjugate events belong to different (back and front) coordinate domains, of different orientation. As Eq. (3) shows, the knowledge of the orientation in addition to the data set solves completely the bifurcation problem. Thus, how to extend the standard emission data so as to be able to determine the orientation of the positioning system for the user?

We shall suppose here that, in addition to the standard emission data , the users are able to observe the relative positions of the emitters on their celestial sphere. We shall denote this extended data set by .

Consider an arbitrary user of unit velocity , at the event of . With respect to this user, the null vectors may be decomposed as

(21)

where and denote vectors of the proper 3-dimensional space of the user, (cf. Eq. (30)).

Let us consider the unit vectors giving the relative directions of propagation of the signals. Because the vectors point to the positions of the emitters , i.e., are the unit vectors along the apparent line of sight of the emitters , we say that is a set of observational data. It is this set of data (or any equivalent one) which, added to the standard emission data, is included in .

By direct substitution of (21) in the expression (8) of , one has

(22)

where we have taken into account that for emission vectors . And because any 3-form in satisfies , the above expression gives (see Appendix A),

(23)

where the triple product is defined according to Eqs. (32) and (33). Then, the following result holds.

Proposition 7: The orientation of a relativistic positioning system is given by

(24)

with .

Thus, from the relative positions of the emitters on the celestial sphere of the user, we can obtain the orientation . For instance, if the referred emitters 1, 2, 3 are counterclockwise aligned on a circle of the celestial sphere of the user and the fourth emitter is inside this circle, then . Then, analyzing separately all the possible situations we arrive to the following rule to obtain the orientation.

Observational rule to determine . For any user in the coordinate region receiving the extended data set , the orientation of the positioning system may be obtained as follows:

i) consider the circle of the celestial sphere of the user containing the three referred emitters, ,

ii) turn this circle around its center in the increasing sense to orient the visual axis of the user by the rule of the right-hand screw,

iii) if the fourth emitter is in the spherical cap pointing out by this oriented axis, then the orientation is , otherwise .

By applying this observational rule, the users receiving the extended emission data set can determine the orientation and, from Eq. (3), their position in inertial coordinates.

For a better geometric comprehension of the above observational rule, we can consider an alternative approach to its proof. Indeed, let us focus on the generic situation in which is a basis of . Then the solution of the linear system

(25)

is given by , with the vectors expressed in terms of the know data as

(26)

Now, by substituting (26) into (23) we arrive to the following expression for the orientation,

(27)

where .

That when and only when the Jacobian vanishes has been known since Pozo-Escola4D (); Pozo-Varsovia-2005 (). From Eq. (27), and according to the result stated in Proposition 4, the events of the emission coordinate region are all those for which the four emitters are not aligned on a circle of the celestial sphere of the users at these events.

Then, it is possible to state that the factor in (27) is positive or negative if is interior or exterior, respectively, to the oriented half cone containing the three emitters (). The unit vector axis of this cone is given by

(28)

Moreover, in terms of the basis given by Eq. (26), the unit axis has the expression

(29)

as can be directly verified.

Therefore, a unit vector is in the interior of the half cone or at its exterior if the quantity is greater or less than , respectively, or by (29) if , or . Thus, by taking , from (27) one has the following result.

Proposition 8: Consider the oriented half-cone containing , and . If is in its interior, then . Otherwise, .

From this proposition we can recover the observational rule by considering all the possible relative positions of the unit vectors . Fig. 7 illustrates the application of the rule when is a negative-oriented basis of , that is for .

Let us remark that the relative positions of the emitters in the celestial sphere of a user are Lorentz invariant: by Lorentz transformations between users at an event, the diameter of the circle as well as the positions of the emitters on it may change, but their increasing sense as well as the interior or exterior position of the fourth emitter will remain unchanged.

Vi Discussion and ending comments

The main result of this paper is the observational rule giving the orientation of the emission coordinates for the user. Together with the standard emission data, it gives a full operational character to formula (3), allowing any user to obtain the coordinate transformation from emission coordinates to inertial ones and, in particular, to locate itself in inertial coordinates.

In the central region , where the orientation may be deduced from the sole standard emission data (Proposition 6), both the observed and the computed orientations may be contrasted.

It is worth to remark here that the sole standard emission data allows the users to detect when they are on the border separating the two coordinate domains (Proposition 1), a situation that may be also contrasted with the limit of the observational rule (when the four emitters are on a circle of the celestial sphere of each user). In spite of the fact that the border does not belong to any coordinate domain, the user can also locate itself in it (taking in Eqs. (12) and (13).

Relativistic positioning concepts have been recently implemented in an algorithm giving the Schwarzschild coordinates of the users in terms of their emission coordinates (see DelvaMas ()). If the conditions of applicability of our rule are given (observation of the emitters), the rule extends the region of validity of this algorithm.

It is important to note that, in dealing with approximate methods, or iterative algorithms, to solve the location problem in weak gravitational fields, Eq. (3) is the best zero order solution to start with.

A numerical analysis of the quantities appearing in (3) has been recently implemented Neus-Diego-ERE10 (); Neus-Diego (). This analysis provides a numerical test of the results obtained in emission-1 (), and a promising via to deal with numerical simulations in modeling GNSS by starting from a fully relativistic conception.

Acknowledgements.
This work has been supported by the Spanish Ministerio de Ciencia e Innovación, MICINN-FEDER Project No. FIS2009-07705.

Appendix A Notation

The notation of this paper has not been chosen for academic reasons but for practical ones. This notation allows, generically, more compact and shorter expressions (in occupied space and in expended time) than index notation, improving a best understanding of the formula, but overall, it suggests more compact and shorter calculations. In this subject, where expressions and calculations are determined to become more and more complicated, the choice of appropriate symbols from the beginning is more than a matter of habit or of preference. Almost all our expressions have been calculated many times in different notations, including the index one, and the symbols in the manuscript have been chosen as the best ones from the above criteria. For readers for which this notation is not usual, we indicate here the relation between tensor index notation and ours.

(i) Interior or contracted product: denotes the contraction of a vector and the first slot of a tensor . For instance, if is a covariant 2-tensor, .

(ii) Exterior or wedge product: . If and are both covariant (or contravariant) antisymmetric tensors, the wedge product is the anti-symmetrized tensorial product . For instance, for vectors and , one has

and, for a covector and a 2-form ,

(iii) Space-time metric: , defined as a four dimensional Lorentzian metric, has components (). The signature of is taken here as .

The scalar or inner product of two vectors and is denoted as (in particular, ), and it is naturally extended to bivectors, and , according to . Indices are raised or lowered by using the metric .

(iv) Metric volume element: , given by

where stands for the Levi-Civita permutation symbol, .

(v) Hodge dual operator: . Let be a vector, a 2-form, and an 3-form. Their associated Hodge duals, the 3-form , the 2-form , and the 1-form , are respectively given as , , and .

If are space-time vectors, one has

and

(vi) Invariants associated with a 2-form . A space-time 2-form has associated two independent invariants, and , which are given as:

(vii) Relative splitting. For an arbitrary user of a relativistic positioning system (space-time observer of unit 4-velocity , ), any space-time vector may be written as:

(30)

where and are the time-like and space-like components, respectively, of relative to , .

(viii) Induced volume on . The 3-dimensional Euclidean space orthogonal to , , has induced volume element, , given by , that is, . The Hodge dual operator with respect is denoted as .

(ix) Cross and triple products in . Vectors in are denoted with an arrow above them. Thus, for vectors , the vector or cross product is expressed as

(31)

The scalar triple product is then given by

(32)

or, equivalently,

(33)

References

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