#### Abstract

Surprisingly, the issue of events localization in spacetime is poorly understood and* a fortiori* realized even in the context of Einstein’s relativity. Accordingly, a comparison between observational data and theoretical expectations might then be strongly compromised. In the present paper, we give the principles of relativistic localizing systems so as to bypass this issue. Such systems will allow locating users endowed with receivers and, in addition, localizing any spacetime event. These localizing systems are made up of relativistic autolocating positioning subsystems supplemented by an extra satellite. They indicate that spacetime must be supplied everywhere with an unexpected local four-dimensional projective structure besides the well-known three-dimensional relativistic projective one. As a result, the spacetime manifold can be seen as a generalized Cartan space modeled on a four-dimensional real projective space, that is, a spacetime with both a local four-dimensional projective structure and a compatible (pseudo-)Riemannian structure. Localization protocols are presented in detail, while possible applications to astrophysics are also considered.

#### 1. Introduction

The general principles of the* relativistic localizing systems* have been defined in a previous paper [1] with just a few details on the projective underlying structure provided by these localizing systems. The latter are based on the so-called* relativistic positioning systems* [2–9]. The protocols of relativistic positioning are a priori rather simple. For instance, in a four-dimensional spacetime, we can consider four emitting satellites and users with their respective (timelike) worldlines. The four emitters broadcast “emission coordinates” which are no more and no less than time stamps generated by onboard clocks and encoded within EM signals propagating in spacetime. Then, a so-called four-dimensional* emission grid* can be constructed from this relativistic positioning system and its system of emission coordinates. This grid can be eventually superposed to a* grid of reference* supplied by a “system of reference” (e.g., the well-known WGS84 system). And then, from this superposition, the positions of the users can be deduced in the given system of reference. More precisely, in relativistic positioning systems, the emitters broadcast not only their own time stamps, but also the time stamps they receive from the others. This process of echoes undergone by the time stamps enables users to construct the four-dimensional emission grid because they can then deduce the spacetime positions of the four emitters. And then, because the positions of the four emitters can be known also in a given system of reference the users can deduce their own positions in this system of reference from their positions in the emission grid.

Here, we focus on the relativistic localizing systems which are systems incorporating relativistic positioning subsystems. We show how* causal axiomatics* [10–13] and particular projective structures (actually, compasses) homeomorphic to and attached all along the worldlines of the emitters of the localizing systems are sufficient to justify a four-dimensional projective structure of the spacetime; in addition to the well-known three-dimensional projective structure.

Beforehand, to proceed in the difficult and delicate description of the relativistic localizing systems, we first need to define as clearly as possible the terminology and the different conventions and notations.

#### 2. Notations and Conventions

We consider a constellation of satellites called “*emitters*” which typically broadcast numerical values (called “*time stamps*”) generated, for instance, by embarked onboard clocks.The “*main*” emitters are denoted by , , , and with their respective worldlines , , , and . The “*ancillary*” emitter and the “*user*” have their worldlines denoted, respectively, by and .The* main* emitters constitute the* relativistic positioning system*.The* ancillary* emitter and the main emitters constitute the* relativistic localizing systems*.The event to be localized is always denoted by the small capital letter .The user collects along its worldline all the data—in particular, the time stamps—from which the localization of the event is deduced. Among these data, there are those for identifying physically the event such as, for instance, its shape, its spectrum, and so forth, and which are obtained from apparatus making physical analyses embarked onboard each mean emitter.Any explicit event will be marked by symbols like “,” “,” “~,” “,” and so forth, or also by small capital letters like “,” “,” and so forth. Non-marked or numbered events will refer to general or generic, unspecified events. For instance, will be a specified event while or will be generic, unspecified events.The generic events , , , , , and belong, respectively, to the worldlines , , , , , and .The* time stamps* will be denoted by the Greek letters “,” “,” “,” “,” and “.” The first four previous time stamps are “generated” and broadcast, respectively, by the main emitters , , , and , and the last one is “generated” and broadcast by the ancillary emitter . The four main emitters not only generate their own time stamps but transmit also the time stamps they receive. These main emitters constitute the various* autonomous autolocating* relativistic positioning systems from which the relativistic localizing systems presented further are constructed.Two classes of* time stamps* are considered:(i)The time stamps which are generated and then broadcast by the emitters at given events on their worldlines. Then, we agree to mark the corresponding time stamps like the given events. For instance, if an emitter generates and broadcasts a time stamp at the specified event or at the generic event , then the respective time stamps will be denoted by or .(ii)The time stamps which are the* emission *(or* time*) coordinates of an event —specified or not—will be denoted by “,” “,” “,” “,” and “.”The ancillary emitter generates and broadcasts its own time stamp and it broadcasts also its time (emission) coordinates provided by the relativistic positioning system. In other words, it is also a particular user of the relativistic positioning system like the user . Contrarily to the ancillary emitter, the user does not necessarily broadcast its emission coordinates.Projective frames at events will be denoted by . There are sets of “canonical projective points ” which are the following:(i) for projective frames of the real projective line , and(ii) for projective frames of the 2-dimensional real projective space . The subscripts will be canceled out if there are no ambiguities on the referring event.The celestial circles/spheres are denoted by , and then is the celestial circle/hemisphere at the event . The celestial circles are invoked in the definition of the “echoing systems” of relativistic localizing systems in ()-dimensional spacetime presented in Section 4. Considering relativistic localizing systems and their corresponding echoing systems (Section 5) in ()-dimensional spacetime, then 2-dimensional projective spaces are also considered. But, contrarily to the relativistic localizing systems in ()-dimensional spacetime, the 2-dimensional real projective spaces cannot be immersed in spheres (or ). Then, as well-known from the cell decomposition of , the Euclidean space is identified in a standard way with a hemisphere of while is identified with half of the equatorial boundary (see, e.g., [14, p. 10–14] for details).We denote by (see [15, Def. 3.1, p. R16] and [16])^{1}(i)“” the* causal* order,(ii)“” the* chronological* order, and(iii)“” the* horismos* (or* horismotic* relation/order).We call “*emission* (or* positioning*)* grid *” the Euclidean space of positioning, and “*localization* (or* quadrometric*/*pentametric*)* grid *” and “*anisotropic localization* (or* quadrometric*/*pentametric*)* grid *” two different Euclidean spaces ascribed to two different, particular sets of time coordinates used for the localization.The acronyms RPS and RLS mean, respectively, “*Relativistic Positioning System*” and “*Relativistic Localizing System*.”

#### 3. RLSs in -Dimensional Spacetime

In this -dimensional case, there are two main emitters and constituting the RPS, and with the ancillary emitter they constitute the RLS. We first give the causal structures of the RPS and the associated RLS. In Figures 1 and 2, and also, in all other subsequent figures representing a causal structure, the arrows represent always the horismotic relation between two events.

##### 3.1. The Causal Structure of the RPS

We have the causal structure (see Figure 1 and Table 1) for the autolocating RPS from which the positioning of the user is realized.

Then, the position of the user at the event in the* emission grid * is . Also, the user can know from the autolocating process the positions of the two emitters: and . Moreover, ephemerides are regularly uploaded onboard the main emitters which broadcast with their time stamps these ephemerides to the users. From these data, that is, ephemerides and positions of the main emitters, the users can deduce their own positions with respect to a given system of reference (e.g., the terrestrial frame of WGS84). This is the core and the important interest of the autolocating positioning systems to immediately furnish the positions of the users with respect to a given system of reference.

##### 3.2. The Causal Structure of the RLS

In this very specific -dimensional case, the localized event is necessarily the intersection point of two null geodesics. The causal structure is shown in Figure 2 and Table 2.

Then, the protocol of localization gives the following time coordinates for in the* localization grid *: and .

*Remark 1. *It matters to notice that the two events of reception and are matched by the user on the basis of a crucial identification of the physical data transmitted by the two main emitters (see convention 5) which allow explicitly identifying the physical occurrence of an event . And then, the whole different time stamps collected at these two events can be therefore considered by the user as those needed to make the localization of .

##### 3.3. Consistency between the Positioning and Localizing Protocols: Identification

*Definition 2 (consistency). *We say that the localizing and the positioning protocols or systems are “*consistent*” if and only if the time coordinates ascribed to each event belonging to an emitter’s worldline and provided by the localization (resp. positioning) system are the same as those provided by the positioning (resp., localization) system.

*Remark 3. *In this -dimensional case, when we* identify* the time stamps and with, respectively, and , then the localization is equivalent to the positioning. This leads to the general Definition 5 below.

*Remark 4. *The consistency between the localizing and the positioning protocols is not an absolute necessity. We can obtain different time coordinates for the same event belonging to an emitter’s worldline from the positioning system or the localizing one if we change the time stamps ascriptions in the protocols of localization presented further. Then, we can choose arbitrarily the emission grid or the grid of localization to position the event , and, then, we can refer to the preferred grid for the time coordinates ascribed to any other event, positioned or localized. In other words, because the systems of localization include implicitly by construction derived positioning systems, the latter can differ from the initial ones. In this case, the consistency is not satisfied but we can still refer the time coordinates of any event with respect to the localization grids rather than to the emission grids. The only advantage of the consistency is that once the events are localized then the time coordinates can be ascribed indifferently to any of the two grids.

*Definition 5. *Let a localized event and an event on the worldlines of a* main* emitter or of the ancillary emitter be such that or or . Then, we call “*identification*” in the emission (position) grid the ascription of an emission coordinate of to an emission coordinate of .

#### 4. RLSs in -Dimensional Spacetime

In this case, there are three main emitters , , and constituting the autolocating RPS and, again, an ancillary emitter with which they constitute the RLS.

##### 4.1. The Causal Structure of the RPS

This causal structure is described in Figure 3 and Table 3.

Then, the position in the emission grid of the user at is , and those of , , and are, respectively, , , and .

*Remark 6. *It matters to notice that in autolocating RPSs the time stamp broadcast by each main emitter is also one of its emission coordinates, for example, for at in Table 1 and for at in Table 3. This property is common to any RPS whatever is the spacetime dimension.

##### 4.2. The Description of the RLS and Its Causal Structures

The determination of the first emission coordinate for the event to be localized is obtained from a first system of light “echoes” associated with the privileged emitter . And then, this system is linked to one event of reception where all the time stamps are collected by the user. We denote by this system of light “echoes” on the worldline of the given, privileged emitter .

Also, one of the key ingredient in the echoing process presented below is the way any event in the past null cone of is associated with a “bright” point on the celestial circle (see Figure 8). Because , we can only consider null “directions” at the origin and tangent at to the null geodesic joining to . The abstract space whose element are these past null directions we call . This space can be represented by the intersection of the past null cone with a space-like surface passing through an event in the past vicinity of , that is, . Then, the exterior of this celestial circle represents space-like directions.

In physical terms, the significance of is the following. Light rays reaching the event and detected by the “eye” of the satellite correspond to null lines through whose past directions constitutes the field of vision of the “observing” satellite. This is and it is represented by the celestial circle which is an accurate geometrical representation of what the satellite actually “sees.” For, the satellite can be considered as permanently situated at the center of a unit circle (his circle of vision) onto which the satellite maps all it detects at any instant. Then, the mapping of the past null directions at to the points of we can call the* sky mapping*. Additionally, because the circle is homeomorphic to the real projective line and we need angle measurements to frame the points of associated with any event in the past null cone of to be furthermore localized, then a particular production process of projective frame for must be devised and incorporated in the echoing system definition now given below.

*Definition 7 (the echoing system ). *The echoing system associated with the* privileged* emitter is based on the following features (see Figure 4 and Table 4):(i)One* primary* event with its celestial circle .(ii)Three* secondary* events , , and , associated, respectively, with the canonical projective points , , and of the projective frame defined on .(iii)Two* ternary* events: and .(iv)A compass on with a moving origin* anchored* on the projective point of associated with .(v)An event of reception at which all the data are collected and sent by the emitter .

The determination of the second (resp., third) emission coordinate (resp., ) for the event to be localized is obtained from a second (resp., third) system of “echoes” associated with the privileged emitter (resp., ). It is also linked to one event of reception (resp. ) where all the time stamps are collected. We denote by (resp., ) this second (resp., third) system of “echoes” on the worldline of the privileged emitter (resp., ).

Then, we have the following.

*Definition 8 (the echoing systems and ). *The definitions of the echoing systems and are obtained when making the following substitutions of events and marks in the definition of :(i)For : and ,(ii)For : and .

Then, we have the causal structure of the echoing system (Figure 4 and Table 4); the other two causal structures for and (Figures 5 and 6) are deduced from the causal structure of by making the substitutions indicated in Definition 8. We indicate also the three structures with the event (Figure 7).

*Remark 9. *Again (Remark 1), it matters to notice that the three events of reception , , and (Figure 7) are matched by the user on the basis of an identification of the physical data for transmitted by the main emitters (see convention 5).

##### 4.3. The Projective Frames and the Time Stamps Correspondences

The realization of the RLS is based on a sort of spacetime parallax, that is, a passage from angles “” measured on celestial circles to spatiotemporal distances. And thus, because spatiotemporal distances are evaluated from time stamps “” in the present context, we need to make the translation of angles into time stamps. This involves onboard compasses embarked on each main emitter to find somehow the bearings. Then, this translation is neither more nor less than a change of projective frames.

To make this change of projective frames effective, we need to define the projective frames on the celestial circles attached to each main emitter. This can be done ascribing to specific “bright points” detected on the celestial circles both angles and time stamps. This ascription is then naturally achieved if these bright points are the main emitters themselves since they broadcast the time stamps. But, if we have three emitters for the RPS, then only two bright points can be detected on each celestial circle attached to each main emitter. And, we need three bright points to have a projective frame on the celestial circle homeomorphic to , hence the need for the ancillary emitter . The change of projective frames is described in Table 5 and Figures 8 and 9. For instance, the main emitter broadcasts the time stamp at the secondary event , and the former is then received by the emitter at the primary event . Also, if is always associated by convention with the canonical projective point on the celestial circle of , then we deduce that corresponds by a projective transformation to . And then, we proceed in the same way with the other two canonical projective points.

As a result, the relations between the angles and the time stamps are the following:where is the cross-ratio of the four projective points , , , and :Conversely, the time coordinates for the event are then obtained from the angles measurements and the following formulas:And thus, the event is localized in the localization grid . Then, we deduce the following lemma.

Lemma 10. *The mapwhere is an automorphism.*

*Proof. *This is obvious from the relations (1a)–(1c), because , , and are bijective Möbius transformations.

##### 4.4. The Consistency between the Positioning and Localization Protocols

Theorem 11. *The localization and the positioning protocols or systems in a -dimensional spacetime are consistent.*

*Proof. *The consistency must be satisfied if is an element of the emitters’ worldlines. Indeed, the localization protocol is consistent with the positioning protocol if the set of events on the emitters’ worldlines from which the localization of any event is possible are themselves localizable.*Case 1* (). We consider two cases: and . The other cases with or instead of give the same results. Now, we start with the assumption from which we deduce the following causal structure:In particular, from , we find^{2} that , and then, with , we obtain^{3} . But then^{2}, we have . With the assumption , we deduce also , and therefore . Hence, we consider that and with the other two sets of events with or and , we deduce finally that . Therefore, we conclude that the time coordinates of provided by the positioning system are , , and .

Besides, from the projective frames, we have alsoAnd therefore, we obtainwhich are the coordinates of .

In conclusion, the localization protocol is consistent with the positioning one.*Case 2* ( is a* primary *event: ). In this case, we obtain the causal structure (Figure 10).

Then, from the three echoing causal structures , , and , we have where is associated with the projective point and is associated with the projective point . Consequently, we have and . Also, we have and from which we deduce from the positioning system that their time coordinates are equal; that is, we have (one of the emission coordinates is equal to the broadcast one in the positioning protocol; see Remark 6)Besides, from the localization protocol, we haveHence, we deduceAnd from the positioning protocol, because is a positioned point with emission coordinates , we have alsoand therefore, with (8), we deduce the consistency for two time stamps. Actually, is not obtained by localization but by* identification* (Definition 5). Indeed, we know that is an element of and that is broadcast by the* identified* main emitter . This determination of is then similar to the emission coordinate ascription presented in the -dimensional case for which localization is equivalent to positioning (Remark 3), hence the consistency.*Case 3* ( is a* secondary *event: or ). Then, the causal structure is the following whenever (Figure 11):(i). Then, the localization protocol at gives the formula because is associated with the projective point . Therefore, we have . But, from the positioning protocol, the emission coordinate of relative to the main emitter is broadcast at the ternary event . Hence, and we deduce the consistency of the localization protocol with the positioning protocol for one emission coordinate.(ii). The reasoning is similar to the previous one. Then, we deduce the consistency for because is the primary event for and is the ternary event for , and which involves .Now, we consider two distinct causal structures of localization and such that from which we deduce the consistency for and . Furthermore, as in Case 2, we deduce by* identification* (Definition 5) and we obtain , hence the consistency.*Case 4* ( is a* ternary* event). For instance, we can set . But then, we have also on which is impossible since we have only the chronological order on the emitters’ worldlines.

*Remark 12. *From this theorem, we can then notice that RLSs are based on* autolocalization* protocols similarly to RPSs which are based on* autolocation* protocols. As a result, RLSs and RPSs are independent of any* system of reference*.

##### 4.5. The Local Projective Structure

*Definition 13. *We call(i)*Emission grid* the Euclidean space of the positioned events ;(ii)*Localization* (or* quadrometric*)* grid* the Euclidean space of the localized events , where is provided by the ancillary emitter by* identification* from the horismotic relation or the “*message function*” [11] ; that is, the time stamp broadcast by at is such that ;(iii)*Anisotropic localization* (or* quadrometric*)* grid* the Euclidean space of events .

*Definition 14. *We denote by the bijective map such that . And we denote by the submersion such that .

*Remark 15. *In these definitions, the time coordinate must be nonvanishing. If this condition is not satisfied we can, nevertheless, always consider that the ancillary emitter * generates* a time number and* broadcasts *. This can be realized from a real-time computer with as the generated input and as the broadcast output. Obviously, we can assume the same for the main emitters.

Let be an element of such that . And thus, acts linearly on . Then, the action of on and is nonlinear and locally transitive and it defines homographies (i.e., conformal transformations):where , , and . Let us notice that does not intervene in (12a). Moreover, we deduce that acts locally transitively on . Therefore, we obtain the following.

Theorem 16. *The -dimensional spacetime manifold has a local 3-dimensional projective structure inherited from its causal structure.*

*Proof. *Let , , and () in be such thatThen, the relations (1a)–(1c) can be put in the forms (; no summation on )where the coefficients of the tensors take only the values 0 or and the only nonvanishing coefficient of is . Then, it is easy to show that for all and in there exists such thatIn particular, if and if the are fixed, then the set of localized events is an orbit of and the set of corresponding events is an orbit of the projective group .

And then, because the relations (14) are homogeneous polynomials, we deduce that has a projective structure as expected.

*Remark 17. *The map defined* locally* by on is the so-called “*soldering map*^{4}” of Ehresmann defined on to the spacetime manifold :And the set of homogeneous equations when the are fixed defines leaves in the trivial bundle . After reduction of the bundle to this projective bundle, the* projective Cartan connection* in the sense of Ehresmann [17] is defined as the differential with respect to the vertical variables and the horizontal variables ; and thus, the tangent spaces of these horizontal leaves are the* annihilators/contact elements* of .

*Remark 18. *Also, as a result, the spacetime manifold can be considered as a “*generalized Cartan space*” which is endowed with both a “*projective Cartan connection*” (of dimension 4) providing a* local* projective structure, and a compatible (pseudo-)Riemannian structure viewed for instance as a horizontal section in the four-dimensional anisotropic grid.

Also, we can eventually define a Ehresmann connection providing a horizontal/vertical splitting in the tangent space of the principal bundle of projective frames of the spacetime manifold. And then, once this Ehresmann connection is given, we can define from this splitting a projective Cartan connection^{5} which can be viewed as the infinitesimal changes of the projective frames with respect to themselves.

#### 5. RLSs in -Dimensional Spacetime

We need similarly four main emitters , , , and providing a RPS and, again, one ancillary emitter emitting its time coordinates and its own time stamp from a clock to get a RLS.

##### 5.1. The Causal Structure of the RPS

The protocol becomes more and more complex to implement. Indeed, sixteen time stamps are needed to provide to the users their positions in a given system of reference. These positions are obtained from the knowledge the users acquire of their own positions and those of the main emitters both in the emission grid and in the system of reference; and this is due to the ephemerides that the emitters upload to the users and the autolocating process. The causal structure of the RPS is shown in Figure 12 and Table 6.

The position in the emission grid of the user at is .

##### 5.2. The Description of the RLS

As in the -dimensional case, we need a system of light echoes associated with each privileged emitter, each linked to an event of reception on the user’s worldline. Again, we denote by the system of light echoes for the privileged emitter with as* primary* event. But now, contrarily to the -dimensional case, we must consider celestial spheres rather than celestial circles. And then, we have again sky mappings from the past null cones directions at the primary events to the “bright” points on the associated celestial spheres. Nevertheless, we have only homeomorphisms between hemispheres with half of their boundaries and . Thus, a problem occurs a priori in this -dimensional case because we have two disjoint hemispheres for each celestial sphere. And then, consecutive problems appear for the choice and the realization of these hemispheres in the localizing protocol. However, we show in the sequel this problem vanishes completely when considering the full set of echoing systems and the particular hemispheres implementation we present for the emitters. We need, first, the following definition for the determination of the first emission coordinate .

*Definition 19 (the echoing system ). *The echoing system associated with the* privileged* emitter is based on the following features (see Figure 15):(i)One* primary* event with its celestial sphere .(ii)Four* secondary* events , , and with the* ancillary* event , associated, respectively, with the canonical projective points , , , and of the projective frame defining one specific hemisphere of the celestial sphere (Figure 15).(iii)One* ternary* event for , two* ternary* events and for , two* ternary* events and for , and none for ,(iv)Two compasses on the specific hemisphere of defined above with a moving origin* anchored* on the projective point associated with .(v)One event of reception at which all the data are collected and sent by the emitter .

Then, we have the following hierarchy of events in the four different echoing systems , , , and :(i)Four* primary* events , , , and , each with three* secondary* events and one* ancillary* event (Table 7).(ii)Four horismotic relations , , , and , where the chronologically ordered events of reception , , , and belong to the user worldline .(iii)One or two (*normal* and* shifted*)* ternary* events by* secondary *event except for the ancillary event:(iv)Two events associated with the projective points and define the equatorial circle dividing the celestial sphere in two celestial hemispheres which are identified to a unique projective space . In other words, the directions of propagation of the light rays detected as bright points on the hemispheres are not considered. This could be a problem a priori. Actually, this difficulty is completely canceled out from the operating principles of RLSs as we will see in the sequel.(v)Two compasses on each celestial hemisphere , , , and with a common moving origin for angle measurements* anchored* on the projective point .(vi)We recall that broadcasts as a particular user its own emission coordinates obtained from the positioning system for all . It broadcasts also all along its own time coordinate denoted again by .

##### 5.3. The Causal Structure of the RLS

We represent (Figures 13 and 14 and Tables 8 and 9) only the causal structure for the echoing system ; the other echoing systems , , and can be easily obtained from the symbolic substitutions deduced from Table 7 and (17a)–(17d).

##### 5.4. The Projective Frame, the Time Stamps Correspondence, and the Consistency

We consider the projective frame at the primary event and the time stamps correspondence associated with the change of projective frame on (Table 10 and Figure 15). Obviously, the other correspondences and changes of projective frames can be deduced in the same way for the three other primary events. Then, we obtain four corresponding pairs of time coordinates for in the four celestial hemispheres (Table 11).

Then, the change of projective frame on the celestial hemisphere gives the following relation:where , and where the determinant of the square matrix on the l.h.s. of this equality must be nonvanishing. Then, we deduce thatand we can take in addition .

Obviously, we obtain from the three other echoing systems three other similar systems of equations for the other six unknown time coordinates given in Table 11 for .

Now, besides, we have necessarily the relations:Indeed, if one of these four precedent equalities is not satisfied, then it means that if the event , the worldlines of the main emitters, and the ancillary one are fixed, then, at least one time stamp among the eight can vary. And then, one of the eight angles on the four celestial hemispheres necessarily can vary as well. But then, it would mean that the position of the event seen on the celestial hemispheres of the four main emitters can vary arbitrarily whenever is fixed. In other words, might have more than one corresponding “bright” point on each celestial hemisphere; and, in particular, because we have continuous functions, then it might correspond to , in particular, a connected “bright” line on one of the four celestial hemispheres. This would involve necessarily the existence of more than one and only one horismotic relation “.” This situation can be encountered in the case of the existence of conjugate points for light-like geodesics for instance in Riemannian manifolds. Then, considering only one horismos, the relations (20) must be satisfied.

Then, we obtain the following.

Lemma 20. *Let be the 4-torus. Then, the RLS provides a mapwhich is an automorphism.*

*Proof. *This lemma can be easily proved simply by solving systems of equations like (19) but we indicate interesting intermediate homogeneous equations in the computations. From the relations (20) and the equations at each primary event such as (19) at , we deduce that there are four linear relations between “” and “.” And then, it can be shown that we obtain four Möbius relations linking the four ’s to the four time coordinates , , , and of generalizing the situation encountered in the precedent -dimensional case.

More precisely, considering the primary event , we obtain (19). At the other primary event , we obtain the similar following relations ():where again we can impose the relations . Then, from now and throughout, we setAnd then, it can be shown that the relations (19) and (22), those depending explicitly on the time stamps, can be put in the following forms ( and ):wherewhere and are nonvanishing arbitrary coefficients and . Hence, we have four homogeneous algebraic equations linking the vectors and . Obviously, we have also four other similar homogeneous equations for and another deduced from the echoing systems at the other two primary events and :where and are nonvanishing arbitrary coefficients. And then, because is determined completely from (24a)-(24b), (27a)-(27b) are linearly depending on (24a)-(24b) which involves that we have linear relations between the two sets of “angles” and . Hence, taking linear combinations of the systems of equations (27a)-(27b) and (24a)-(24b) and taking into account also the remaining equations in (19) and (22) not depending on the time stamps we can deduce a system of four homogeneous equations linking and :where , and do not intervene anymore. And (28) determines univocally up to the time coordinate , that is, . Moreover, (28) is another expression for Möbius transformations between each given angle and a linear combination of , hence the result for an automorphism on the 4-torus.

*Remark 21. *We can notice from Lemma 20 that we obtain the time coordinates for from only two echoing systems, for example, at and with the four “angles” , or from the four echoing systems at , , , and with the four “angles” .

Theorem 22. *The localization and the positioning protocols or systems in a -dimensional spacetime are consistent.*

*Proof. *The proof is obvious because the RLS in the -dimensional case has a causal structure which can be decomposed in four causal substructures each equivalent to the one given for the RLS in the -dimensional case and we need only the “angles” to localize in each of these subsystems of localization.

##### 5.5. The Local Projective Structure

*Definition 23. *We call(i)*Emission grid* the Euclidean space of the positioned events ;(ii)*Localization* (or* pentametric*)* grid* the Euclidean space of the localized events where is provided by the ancillary emitter by* identification* from the horismotic relation or the “*message function*” [11] ; that is, the time stamp broadcast by at is such that ;(iii)*Anisotropic localization* (or* pentametric*)* grid* the Euclidean space of events .

*Definition 24. *We denote by the bijective map such that . And we denote by the submersion such that .

Let be an element of such that . And thus, acts linearly on . Then, the action of on and defines homographies (i.e., conformal transformations):where , , and . Therefore, we obtain the following.

Theorem 25. *The -dimensional spacetime manifold has a local 4-dimensional projective structure inherited from its causal structure.*

*Proof. *The proof is similar with the proof of Theorem 16 but with the systems of homogeneous equations (24a)-(24b) or (27a)-(27b) or (28) instead of the system (14).

*Remark 26. *The map defined* locally* by on is the so-called “*soldering map*” of Ehresmann defined on to the spacetime manifold :And the set of homogeneous equations (24a)-(24b) or (27a)-(27b) or (28) defines leaves in the trivial bundle . After reduction of the bundle to this projective bundle, the* projective Cartan connection* in the sense of* Ehresmann* [17] is defined as the differential with respect to the vertical variables