Relativistic Positioning Systems in Flat Space-Time with Inertial, Hyperbolic and Rotating Emitters

1. Introduction

A Relativistic Positioning System (RPS) is essentially a set of four clocks broadcasting their time by means of electromagnetic signals. Although the time of every clock may be any one, for simplicity we suppose they broadcast their proper times. The set of such four times imprinted on the signals converging at every space-time event constitutes a (four-dimensional) physical coordinate system for the region reached by the signals. Such coordinates are called emission coordinates [1]. Analogous Newtonian emission coordinate systems have been constructed from sonic signals and classified in [2].

In this work we are mainly concerned with specific RPSs in Minkowski space-time. The affine geometry and light cone structure of flat space-time make viable analytical constructions of RPSs with both inertial and non-inertial emitters. At first glance, these constructions may seem somewhat academic for practical purposes. However, concrete RPS examples contribute to develop current RPS theory and future applications. Furthermore, the study of Minkowskian RPSs pave the way for realistic constructions in a known gravitational field, modelled as a curved Lorentzian geometry, in which the use of the Ruse-Synge world function for the null geodesics [3,4] or the application of distance geometry concepts [5,6] may result mandatory.

The main aim of RPS theory is to give a precise physical frame for the location of events in a gravitationally unknown region in order to make relativistic gravimetry, as has been explained in [6,7,8]. The starting ideas of the theory of RPSs are that, in such an unknown region, the first object to be constructed is a physical coordinate system allowing to identify its events with the points of the mathematical models in use; that such physical coordinate system has to be constructed generically by means of electromagnetic signals and with a protocol independent of the particular space-time; and that the over-determined interlacement of these signals contains important information (chronometry in arbitrary directions) on the relativistic gravitational field in which we are immersed.

In [9] we have analysed the most elemental example of a RPS construction: four static emitters in flat space-time. In the present work, this construction is extended to RPSs including inertial, accelerated and rotating emitters. These constructions present significant novelties and differences with respect to the static situation, and may provide insights on fictitious gravitational potentials attached to non-inertial effects.

The location problem in Minkowski space-time has been dealt with in [9,10,11,12]. Developments on the subject are found in [13,14,15,16,17], including numerical treatment and prospects. Refs. [18,19,20] offer relevant notions to develop the RPS theory, presenting examples in two-dimensional Lorentzian spaces. General concepts and results in 3D and 4D were communicated in [21].

In this work we develop RPS examples in Minkowski space-time with four emitters following different motions. In all cases considered we carry out the same procedure. The satellite trajectories in space-time are written and the emission/reception conditions are stated, the satellite trajectories on the grid of emission coordinates are determined and the RPS solution is interpreted by representing the characteristic regions and working out a simplified positioning example. We have used the software Mathematica to represent the characteristic regions of the RPS solutions and perform numerical calculations.

This article is organized as follows: The terminology of a RPS is introduced in Section 2. Section 3 starts with a positioning example involving one inertial and three static emitters, Section 4 considers a RPS with one hyperbolic and three static emitters, Section 5 analyses the example of one static and three uniformly rotating emitters. A brief summary and conclusions are presented in Section 6. Appendix A summarizes the notation used.

2. RPS Terminology and Summary of Results in Flat Space-Time

Following [10,11], we present here a compendium of the essential RPS terminology and results used to develop this work in Minkowski space-time.

Relativistic positioning system: set of four emitters A ($A=1,2,3,4$), of world-lines ${\gamma }_A\left({\tau }^A\right)$, broadcasting their respective proper times ${\tau }^A$ by means of electromagnetic signals.

Emission coordinates of an event: the four times $\left\{{\tau }^A\right\}$ which are received at each space-time event x reached by the emitted signals.

Grid of a RPS: The four space $\mathcal{T}\equiv \left[{\tau }^1\right]\times \left[{\tau }^2\right]\times \left[{\tau }^3\right]\times \left[{\tau }^4\right]\approx {\mathbb{R}}^4$.

Configuration of the emitters for an event x: set of four events $\left\{{\gamma }_A\left({\tau }^A\right)\right\}$ of the emitters at the emission times $\left\{{\tau }^A\right\}$ received at x.

In Minkowski space-time ${\gamma }_A\equiv O{\gamma }_A\left({\tau }^A\right)$ denotes the position four-vector of emitter A with respect to the origin O of an inertial coordinate system $\left\{x^{\alpha }\right\},A=1,2,3,4$.

Configuration vector χ: vector informing of the emitter configuration at the emission times $\left\{{\tau }^A\right\}$ received at x,

$$\chi =*(e_1\wedge e_2\wedge e_3),$$

(1)

with $e_a={\gamma }_a-{\gamma }_4$ ($a=1,2,3$) the relative positions of emitters 1, 2 and 3 with respect to emitter 4, and where * stands for the Hodge dual operator and ∧ is the exterior product (see Appendix A for transcription into index notation). An emitter configuration is regular iff $\chi \ne 0$.

Null propagation equations: The following system of non-linear equations:

$${(x-{\gamma }_A)}^2=0,A=1,2,3,4,$$

(2)

with x the user position four-vector with respect to O. The solution to these equations is the coordinate transformation from emission to inertial coordinates $x^{\alpha }\left({\tau }^A\right)$.

Shadow of emitter A to emitter B: Space-time events x for which the future-directed null vectors $m_A=x-{\gamma }_A$ and $m_B=x-{\gamma }_B$ become collinear, $m_A·m_B=0$, lying on the future light cone with vertex at the intersection of $m_A$ (or $m_B$) with ${\gamma }_A$.

For four emitters, there are 12 shadows in total, defining the 12 faces of the shadow-dodecahedron $\mathcal{D}$ in the grid of emission coordinates.

Emission/reception conditions: conditions expressing that the null vectors $m_A=x-{\gamma }_A$, $A=1,2,3,4$, are all either future or past oriented, $m_A·m_B<0$, $A\ne B$:

$${\left(e_a\right)}^2>0,{(e_a-e_b)}^2>0,a,b=1,2,3.$$

(3)

Emission region: set $\mathcal{R}$ of events reached by the four signals broadcast by the positioning system. Every $x\in \mathcal{R}$ is labelled with the corresponding emission coordinates $\left\{{\tau }^A\right\}$.

Characteristic emission function: map $\Theta$ that to every $x\in \mathcal{R}$ associates its emission coordinates, that is $\Theta \left(x\right)=\left({\tau }^A\right)$.

Emission coordinate region: subset $\mathcal{C}$ of the emission region $\mathcal{R}$ where the gradients $d{\tau }^A$ are well defined and linearly independent.

Orientation of a relativistic positioning system at the event x: orientation of its emission coordinates at x or, what is the same, the orientation of the tetrad of 1-forms $\{d{\tau }^1,d{\tau }^2,d{\tau }^3,d{\tau }^4\}$. It is given by the sign $\widehat{ϵ}$ of the Jacobian determitant $j_{\Theta }\left(x\right)$ of $\Theta$ at x, $\widehat{ϵ}\equiv \mathit{sgn}{j}_{\Theta }\left(x\right)$. In terms of the gradients of the emission coordinates, one has

$$\widehat{ϵ}=\mathit{sgn}[*(d{\tau }^1\wedge d{\tau }^2\wedge d{\tau }^3\wedge d{\tau }^4)].$$

(4)

The orientation $\widehat{ϵ}$ of a RPS at an event x is, by definition, the orientation of its emission coordinates at x.

For regular configurations, the region where the vector $\chi$ is timelike or null, ${\chi }^2\le 0$, is called the central region of the RPS. In it, the orientation is uniform and is given by $\widehat{ϵ}=\mathrm{sgn}(u·\chi )$ for any observer u. But out of this region it is not possible to determine $\widehat{ϵ}$ solely from the world-lines ${\gamma }_A\left({\tau }^A\right)$ [10], additional information is necessary [11].

Solution to the null propagation equations: The solution to (2) or, equivalently, the coordinate transformation from emission to inertial coordinates is [10]

$$\begin{array}{c}\hfill x={\gamma }_4+y_{*}+\lambda \chi ,\end{array}$$

(5)

with $y_{*}$ the particular solution, found by bringing in a subsidiary vector $\xi$ satisfying the transversality condition $\xi ·\chi \ne 0$:

$$y_{*}={\displaystyle \frac{1}{\xi ·\chi }}i\left(\xi \right)H,$$

(6)

where H is the configuration bivector,

$$H={\Omega }_1{E}^1+{\Omega }_2{E}^2+{\Omega }_3{E}^3,$$

(7)

with

$$\begin{array}{c}\hfill E^1=*(e_2\wedge e_3),E^2=*(e_3\wedge e_1),E^3=*(e_1\wedge e_2),\end{array}$$

(8)

and

$$\begin{array}{c}\hfill \lambda =-{\displaystyle \frac{y_{*}^2}{(y_{*}·\chi )+\widehat{ϵ}\sqrt{\Delta }}},\Delta ={(y_{*}·\chi )}^2-y_{*}^2{\chi }^2.\end{array}$$

(9)

The following example with one inertial and three static emitters is the logical extension of the example analysed in [9] involving four static emitters spatially forming an orthogonal tetrahedron. In this example, however, the inertial emitter moves along the spatial bisectrix in such a way that, at zero coordinate time, the four emitters form a regular tetrahedron.

3. RPS with One Inertial and Three Static Emitters

In this example in Minkowski space-time, we consider emitters $2,3$ and 4 static with respect to an inertial observer u ($u^2=-1$). Emitter 1 is moving with constant velocity $\beta$ with respect to u along the spatial main bisectrix, in such a way that at proper time ${\tau }^1=0$ the four emitters are spatially forming a regular tetrahedron of vertices $(1,1,1)$, $(1,-1,-1)$, $(-1,1,-1)$ and $(-1,-1,1)$. As ${\tau }^1\to \pm \infty$, its spatial position of emitter 1 tends to $\pm \infty$ along the main bisectrix $(1,1,1)$.

3.1. Emitters’ World-Lines and Emission/Reception Conditions

Thus, the world-lines referred to the inertial obeserver u are written as:1

$$\begin{array}{cccc}\hfill {\gamma }_1=& \gamma {\tau }^1\left[u+{\displaystyle \frac{\beta }{\sqrt{3}}}(1,1,1)\right]+(1,1,1),\hfill & \hfill {\gamma }_2& ={\tau }^{2}u+(1,-1,-1),\hfill \\ \hfill {\gamma }_3=& {\tau }^{3}u+(-1,1,-1),\hfill & \hfill {\gamma }_4& ={\tau }^{4}u+(-1,-1,1),\hfill \end{array}$$

where $\gamma ={(1-{\beta }^2)}^{-{\displaystyle \frac{1}{2}}}$. The velocity $\beta$ is expressed as a fraction of the speed of light in vacuum c ($\beta <1$). We consider that the proper times of emitters 2, 3 and 4 are synchronised at their common origin ${\tau }^2={\tau }^3={\tau }^4=0$ and that the proper time of emitter 1 is synchronised with theirs at ${\tau }^1=0$. The distance $d_1$ covered by emitter 1 in one second of coordinate time $\Delta x^0=1s$ ($\Delta {\tau }^1={\gamma }^{-1}\Delta x^0={\gamma }^{-1}s$) is $d_1=\beta ls$. 2

Defining3

$$\begin{array}{cc}\hfill q_1=& \gamma {\tau }^1-{\tau }^4,q_2={\tau }^2-{\tau }^4,q_3={\tau }^3-{\tau }^4,\hfill \\ \hfill \Gamma \left({\tau }^1\right)=& {\displaystyle \frac{\gamma \beta }{\sqrt{3}}}{\tau }^1,\hfill \end{array}$$

one obtains the following position vectors with respect to the fourth emitter, $e_a={\gamma }_a-{\gamma }_4$, $a=1,2,3$:

$$\begin{array}{c}\hfill e_1=q_{1}u+(\Gamma +2,\Gamma +2,\Gamma ),e_2=q_{2}u+(2,0,-2),e_3=q_{3}u+(0,2,-2),\end{array}$$

(10)

and the following world-functions ${\Omega }_a=\frac{1}{2}{\left(e_a\right)}^2$, $a=1,2,3$:

$$\begin{array}{c}\hfill {\Omega }_1=\frac{1}{2}(2{(\Gamma +2)}^2+{\Gamma }^2-q_1^2),{\Omega }_2=\frac{1}{2}(8-q_2^2),{\Omega }_3=\frac{1}{2}(8-q_3^2).\end{array}$$

(11)

A regular emitter configuration is an emission/reception configuration (see [9,10]) iff all relative emitter positions are space-like, $e_a^2>0$ and ${(e_a-e_b)}^2>0$, $a,b=1,2,3$, that is, when the following emission/reception conditions are satisfied:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |q_1|& <\sqrt{2{(\Gamma +2)}^2+{\Gamma }^2},|q_2|<\sqrt{8},\left|q_3\right|<\sqrt{8},\hfill \\ \hfill |q_1-q_2|<& \sqrt{2{(\Gamma +2)}^2+{\Gamma }^2},|q_1-q_3|<\sqrt{2{(\Gamma +2)}^2+{\Gamma }^2},\left|q_2-q_3\right|<\sqrt{8}.\hfill \end{array}\end{array}$$

(12)

Notice that the family of parabolas

$$\begin{array}{c}\hfill y_{\gamma }\left({\tau }^1\right)\equiv 2{(\Gamma +2)}^2+{\Gamma }^2=({\gamma }^2-1){\left({\tau }^1\right)}^2+{\displaystyle \frac{8}{\sqrt{3}}}\sqrt{{\gamma }^2-1}{\tau }^1+8\end{array}$$

(13)

shares the same vertex height, which does not depend on $\gamma$, $y_{\gamma }\left({\tau }_v^1\right)={\displaystyle \frac{8}{3}}$, with ${\tau }_v^1=-{\displaystyle \frac{4}{\sqrt{3}\gamma \beta }}$.

3.2. Emitters’ Trajectories on the Grid $\mathcal{T}$

The grid coordinate ${\tau }^B$ of the trajectory ${\gamma }_A$ on the grid for $B\ne A$ is computed from the intersection of the past null cone of ${\gamma }_A$ with the world-line ${\gamma }_B$, by solving the light-cone equation ${({\gamma }_A-{\gamma }_B)}^2=0$ for ${\tau }^B$ and selecting the solution that corresponds to an emission event on ${\gamma }_B$. The grid coordinate ${\tau }^A$ of the trajectory ${\gamma }_A$ is simply ${\tau }^A$. Since all four grid coordinates of the trajectory ${\gamma }_A$ depend on ${\tau }^A$, considered the parameter of the trajectory, we omit the superscript.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_1\left(\tau \right)& :\left(\tau ,\gamma \tau -\sqrt{y_{\gamma }\left(\tau \right)},\gamma \tau -\sqrt{y_{\gamma }\left(\tau \right)},\gamma \tau -\sqrt{y_{\gamma }\left(\tau \right)}\right),\hfill \\ \hfill S_2\left(\tau \right)& :\left(\gamma \left[\tau +\frac{4\beta }{\sqrt{3}}\right]-\sqrt{z_{\beta }\left(\tau \right)},\tau ,\tau -\sqrt{8},\tau -\sqrt{8}\right),\hfill \\ \hfill S_3\left(\tau \right)& :\left(\gamma \left[\tau +\frac{4\beta }{\sqrt{3}}\right]-\sqrt{z_{\beta }\left(\tau \right)},\tau -\sqrt{8},\tau ,\tau -\sqrt{8}\right),\hfill \\ \hfill S_4\left(\tau \right)& :\left(\gamma \left[\tau +\frac{4\beta }{\sqrt{3}}\right]-\sqrt{z_{\beta }\left(\tau \right)},\tau -\sqrt{8},\tau -\sqrt{8},\tau \right),\hfill \end{array}\end{array}$$

(14)

with

$$z_{\beta }\left(\tau \right)\equiv {\gamma }^2{[\tau +\frac{4\beta }{\sqrt{3}}]}^2-{\tau }^2+8.$$

This family of parabolas shares the same discriminant with $y_{\gamma }\left(\tau \right)$, ${\Delta }_z=-\frac{32}{3}{\gamma }^2{\beta }^2<0$, and the same vertex height $z_{\beta }\left({\tau }_v\right)=y_{\gamma }\left({\tau }_v^1\right)={\displaystyle \frac{8}{3}}$, with ${\tau }_v=\gamma {\tau }_v^1$.

The important property is that at ${\tau }_v^1$ emitter 1 is at the spatial location $-{\displaystyle \frac{1}{3}}(1,1,1)$, being spatially coplanar with emitters 2, 3 and 4 and situated at the circumcenter of the triangle formed by these emitters. This property becomes relevant in Section 3.3.

In this inertial example, the emitters’ trajectories on the grid are not straight lines. Then, we can rewrite trajectories (14) to compare them with the static case [9]:

$$\begin{array}{cc}\hfill S_1\left(\tau \right)& :\tau (1,\gamma ,\gamma ,\gamma )-\sqrt{8}\sqrt{1+{\displaystyle \frac{\gamma \beta }{\sqrt{3}}}\tau +{\displaystyle \frac{{\gamma }^2{\beta }^2}{8}}{\tau }^2}(0,1,1,1),\hfill \\ \hfill S_2\left(\tau \right)& :\tau (\gamma ,1,1,1)-\sqrt{8}\left(\sqrt{1-{\displaystyle \frac{{\tau }^2}{8}}+{\displaystyle \frac{{\gamma }^2}{8}}{(\tau +{\displaystyle \frac{4\beta }{\sqrt{3}}})}^2}+{\displaystyle \frac{4\beta }{\sqrt{3}}},0,1,1\right),\hfill \\ \hfill S_3\left(\tau \right)& :\tau (\gamma ,1,1,1)-\sqrt{8}\left(\sqrt{1-{\displaystyle \frac{{\tau }^2}{8}}+{\displaystyle \frac{{\gamma }^2}{8}}{(\tau +{\displaystyle \frac{4\beta }{\sqrt{3}}})}^2}+{\displaystyle \frac{4\beta }{\sqrt{3}}},1,0,1\right),\hfill \\ \hfill S_4\left(\tau \right)& :\tau (\gamma ,1,1,1)-\sqrt{8}\left(\sqrt{1-{\displaystyle \frac{{\tau }^2}{8}}+{\displaystyle \frac{{\gamma }^2}{8}}{(\tau +{\displaystyle \frac{4\beta }{\sqrt{3}}})}^2}+{\displaystyle \frac{4\beta }{\sqrt{3}}},1,1,0\right).\hfill \end{array}$$

And approximate them up to first order in $\beta$ to obtain the following straight lines:

$$\begin{array}{cc}\hfill S_1\left(\tau \right)& :\tau \left(1,1-\sqrt{\frac{2}{3}}\beta ,1-\sqrt{\frac{2}{3}}\beta ,1-\sqrt{\frac{2}{3}}\beta \right)-(0,\sqrt{8},\sqrt{8},\sqrt{8}),\hfill \\ \hfill S_1\left(\tau \right)& :\tau \left(1,1,1,1\right)-\left(\sqrt{8}-{\displaystyle \frac{4}{\sqrt{3}}}\beta ,0,\sqrt{8},\sqrt{8}\right),\hfill \\ \hfill S_1\left(\tau \right)& :\tau \left(1,1,1,1\right)-\left(\sqrt{8}-{\displaystyle \frac{4}{\sqrt{3}}}\beta ,\sqrt{8},0,\sqrt{8}\right),\hfill \\ \hfill S_1\left(\tau \right)& :\tau \left(1,1,1,1\right)-\left(\sqrt{8}-{\displaystyle \frac{4}{\sqrt{3}}}\beta ,\sqrt{8},\sqrt{8},0\right),\hfill \end{array}$$

in agreemeent with the results obtained for four static emitters.

3.3. Shadow-Dodecahedron

As can be seen from the emission/reception conditions (12), and also from the RPS solution below, this example with one inertial emitter moving with respect to three static emitters can not be fully described on the quotient grid $\mathcal{Q}\equiv \left[q_1\right]\times \left[q_2\right]\times \left[q_3\right]\approx {\mathbb{R}}^3$, essentially because of the term $\Gamma$ which depends on ${\tau }^1$. Instead, one can consider ${\tau }^1$ a parameter and work on the grid ${\mathcal{T}}_1\left({\tau }^1\right)\equiv \left[{\tau }^2\right]\times \left[{\tau }^3\right]\times \left[{\tau }^4\right]\approx {\mathbb{R}}^3$ for each value of ${\tau }^1$. In the example with four static emitters [9], on an analougous ${\mathcal{T}}_1$-grid, the shadow-dodecahedron $\mathcal{D}$ moves along the main bisectrix for each ${\tau }^1$ but its shape does not change, that is, the relative positions of its vertices remain constant for every ${\tau }^1$. In the case of one moving emitter, howewer, apart from a translation of the shadow-dodecahedron on the ${\mathcal{T}}_1$-grid for each ${\tau }^1$ along the main bisectrix, there is a longitudinal deformation along that axis since its 14 vertices depend on ${\tau }^1$ as follows:

$$\begin{array}{cc}{V}_1=(G_{+},G_{+},G_{+}),\hfill & V_2=(G_{-},G_{-},G_{-}),\hfill \\ V_3=(G_{+}-\sqrt{8},G_{+},G_{+}),\hfill & V_4=(G_{+},G_{+}-\sqrt{8},G_{+}),\hfill \\ V_5=(G_{+},G_{+},G_{+}-\sqrt{8}),\hfill & V_6=(G_{-}+\sqrt{8},G_{-},G_{-}),\hfill \\ V_7=(G_{-},G_{-}+\sqrt{8},G_{-}),\hfill & V_8=(G_{-},G_{-},G_{-}+\sqrt{8}),\hfill \\ V_9=(G_{+}-\sqrt{8},G_{+}-\sqrt{8},G_{+}),\hfill & V_{10}=(G_{+},G_{+}-\sqrt{8},G_{+}-\sqrt{8}),\hfill \\ V_{11}=(G_{+}-\sqrt{8},G_{+},G_{+}-\sqrt{8}),\hfill & V_{12}=(G_{-}+\sqrt{8},G_{-}+\sqrt{8},G_{-}),\hfill \\ V_{13}=(G_{-},G_{-}+\sqrt{8},G_{-}+\sqrt{8}),\hfill & V_{14}=(G_{-}+\sqrt{8},G_{-},G_{-}+\sqrt{8}),\hfill \end{array}$$

with $G_{\pm }\left({\tau }^1\right)\equiv \gamma {\tau }^1\pm \sqrt{y_{\gamma }\left({\tau }^1\right)}$. Of these 14 vertices, only one concides with a satellite trajectory on the grid ${\mathcal{T}}_1$, namely $V_2$ with the trajectory of satellite 1.

The dodecahedron contracts longitudinally along the main bisectrix as we approach ${\tau }_v^1=-{\displaystyle \frac{4}{\sqrt{3}\gamma \beta }}$ from left and right, where it is smallest. Figure 1 shows the shadow-dodecahedron for $\beta ={\displaystyle \frac{1}{2}}$ at the endpoints and midpoint of the interval4${\tau }^1\in [-{\displaystyle \frac{8}{\sqrt{3}\gamma \beta }},0]$, that is, for ${\tau }^1=-8$, ${\tau }^1=-4$ and ${\tau }^1=0$. At the midpoint ${\tau }_v^1=-4$ the dodecaedron is smallest.

We can study the motion of the shadow-dodecahedron along the main bisectrix through the motion of the spatial circumcenter of the tetrahedron formed by the four satellites, whose world-line ${\gamma }_c$, parametrized with the proper time of emitter 1 (${\tau }^1$), is expressed as:

$$\begin{array}{cc}\hfill {\gamma }_c=& \gamma {\tau }^{1}u+{\displaystyle \frac{{\omega }_c}{\sqrt{3}}}(1,1,1),\hfill \\ \hfill {\omega }_c=& {\displaystyle \frac{\beta \gamma {\tau }^1(6+\sqrt{3}\beta \gamma {\tau }^1)}{8+2\sqrt{3}\beta \gamma {\tau }^1}}.\hfill \end{array}$$

(15)

As said in the previous section, this world-line is degenerate at ${\tau }^1={\tau }_v^1$, when all four emitters are spatially coplanar.

We can now write the trajectory of the spatial circumcenter on the grid ${\mathcal{T}}_1$:

$$\begin{array}{c}{S}_c=(f_c,f_c,f_c),\\ \begin{array}{cc}\hfill f_c\left({\tau }^1\right)& =\gamma {\tau }^1-{\displaystyle \frac{\sqrt{{\Delta }_c}}{2\left(3\Gamma +4\right)}},\hfill \\ \hfill {\Delta }_c& =81{\Gamma }^4+(36\sqrt{3}+324){\Gamma }^3+(120\sqrt{3}+432){\Gamma }^2+(96\sqrt{3}+288)\Gamma +192.\hfill \end{array}\end{array}$$

In the ${\mathcal{T}}_1$-grid, the spatial circumcenter described by world-line ${\gamma }_c$ (15) moves along the bisectrix of the grid. We can determine whether it lies inside the shadow-dodecahedron for any ${\tau }^1$ by comparing its coordinates with those of vertices $V_1$ and $V_2$, the end vertices on the bisectrix. It lies inside the dodecahedron for any ${\tau }^1>{\tau }_b^1$ and ${\tau }^1<{\tau }_a^1$, where ${\tau }_a^1$ and ${\tau }_b^1$ are the ${\tau }^1$-coordinates of the two intersection points of $y_{\gamma }\left({\tau }^1\right)$ and $g_c\left({\tau }^1\right)\equiv {\displaystyle \frac{{\Delta }_c}{4{\left(3\Gamma +4\right)}^2}}$. In the interval $[{\tau }_a^1,{\tau }_b^1]$ the function $g_c\left({\tau }^1\right)$ diverges at ${\tau }^1={\tau }_v^1$, when the four emitters are spatially coplanar (see Section 3.2).

3.4. Interpreting the RPS Solution

We now represent the caracteristic regions of the RPS solution in Cartesian coordinates for this case, depending on the causal character of the configuration vector $\chi$. Unlike the static situation [9], these regions change with every inertial instant $x^0$ for a given velocity $\beta$. But, as in the static situation, for every $x^0$ the RPS solution is defined in the whole Euclidean space $\{x^1,x^2,x^3\}$. Figure 2 shows these regions and the region $j_{\Theta }\left(x\right)=0$, where the jacobian determinant of the transformation from inertial to emission coordinates vanishes, for $\beta ={\displaystyle \frac{1}{2}}$ and $x^0=1$.

Figure 3 includes a two dimensional slice of the emission configuration regions shown in Figure 2, through the plane containing emitters 1, 2 and 4 at $x^0=1$ ($\beta ={\displaystyle \frac{1}{2}}$). The intersection with $j_{\Theta }\left(x\right)=0$ is also represented. We can see that around the satellites the configuration regions are interlaced with each other.

To remark the essential properties of the RPS solution in this example, we consider the inertial coordinate location of those users whose emission coordinates satisfy the restrictions ${\tau }^1=-\sigma ,{\tau }^2={\tau }^3={\tau }^4=\sigma$. We start by computing the configuration covector (1) using (10):

$${\chi }_{\mu }=-4\left(\sqrt{3}\gamma \beta \sigma -4,\left(1+\gamma \right)\sigma ,\left(1+\gamma \right)\sigma ,\left(1+\gamma \right)\sigma \right).$$

(16)

To calculate the particular solution $y_{*}$, we first compute the configuration bivector H using (10), (11), (8) and (7). Choosing the transversal vector $\xi =u$, Equation (6) then yields:

$$y_{{*}_{\mu }}={\displaystyle \frac{4}{D}}\left(0,{\sigma }^2(1+\gamma )+2\sqrt{3}\gamma \beta \sigma -4,{\sigma }^2(1+\gamma )+2\sqrt{3}\gamma \beta \sigma -4,{\sigma }^2(1+\gamma )+4\right),$$

(17)

with $D=4(\sqrt{3}\gamma \beta \sigma -4)$. With this choice of $\xi$, the transversality condition restricts this solution to $\{\sigma \in \mathbb{R}|\sigma \ne -{\tau }_v^1={\displaystyle \frac{4}{\sqrt{3}\gamma \beta }}\}$. As said earlier, at ${\tau }^1={\tau }_v^1$ all four emitters are spatially coplanar, leading to a degenerate situation.5

The solution (5) reads:

$$\begin{array}{c}\hfill x^{\mu }(-\sigma ,\sigma ,\sigma ,\sigma )=(\sigma ,-1,-1,1)+y_{*}-{\displaystyle \frac{{\lambda }_1}{{\lambda }_2}}\chi ,\end{array}$$

(18)

with polynomial functions of $\sigma$

$$\begin{array}{cc}\hfill {\lambda }_1& =4\left(3{(1+\gamma )}^2{\sigma }^4+8\sqrt{3}\gamma \beta (1+\gamma ){\sigma }^3+8(3{\gamma }^2-\gamma -4){\sigma }^2-32\sqrt{3}\gamma \beta \sigma +48\right),\hfill \\ \hfill {\lambda }_2& =D\left(-12{(1+\gamma )}^2{\sigma }^3-16\sqrt{3}\gamma \beta (1+\gamma ){\sigma }^2+16(1+\gamma )\sigma +\widehat{ϵ}D\sqrt{{\Delta }_{\sigma }}\right),\hfill \\ \hfill {\Delta }_{\sigma }& ={\displaystyle \frac{{\lambda }_1}{4}}-8{(1+\gamma )}^2{\sigma }^2.\hfill \end{array}$$

In this case, $q_1=-\sigma (\gamma +1)$, $q_2=q_3=0$, and the emission-reception conditions (12) are reduced to:

$$\left|\sigma \right|<{\displaystyle \frac{1}{\gamma +1}}\sqrt{({\gamma }^2-1){\sigma }^2-{\displaystyle \frac{8\gamma \beta }{\sqrt{3}}}\sigma +8},$$

(19)

where we have taken into account (13) and which is equivalently written as

$$P\left(\sigma \right)\equiv {\sigma }^2+{\displaystyle \frac{4}{\sqrt{3}}}\sqrt{{\displaystyle \frac{\gamma -1}{\gamma +1}}}\sigma -{\displaystyle \frac{4}{\gamma +1}}<0.$$

(20)

The roots of $P\left(\sigma \right)=0$,

$${\sigma }_{\pm }={\displaystyle \frac{2}{\sqrt{3}}}{\displaystyle \frac{1}{\sqrt{\gamma +1}}}\left(-\sqrt{\gamma -1}\pm \sqrt{\gamma +2}\right),$$

(21)

provide the limiting values for $\sigma$ imposed by the emission-reception conditions.

Table 1 shows the result setting $\beta ={\displaystyle \frac{1}{2}}$ for different values of $\sigma$. In this particular case, the emission-reception condition (19) imposes

$$-1.71\approx {\displaystyle \frac{2}{3}}(-2\sqrt{3}-\sqrt{6(\sqrt{3}-1)}+3)<\sigma <{\displaystyle \frac{2}{3}}(-2\sqrt{3}+\sqrt{6(\sqrt{3}-1)}+3)\approx 1.09.$$

(i)

For $\sigma ={\displaystyle \frac{1}{2}}$, the emitter configuration is space-like at

$x^{\mu }={\displaystyle \frac{1}{44}}\left(9\left(5\sqrt{3}+3\right),-\left(9\sqrt{3}+23\right),-\left(9\sqrt{3}+23\right),-\left(9\sqrt{3}+23\right)\right)$, which is the sole emission solution (with $\widehat{ϵ}=-1$). The solution with $\widehat{ϵ}=1$ is a reception solution.

(ii)

For $\sigma =2(1-\sqrt{3})$, the emitter configuration is light-like at

$x^{\mu }={\displaystyle \frac{1}{2}}\left(-3\left(\sqrt{3}-3\right),\left(\sqrt{3}+1\right),\left(\sqrt{3}+1\right),\left(\sqrt{3}+1\right)\right)$, which is the sole emission solution (with $\widehat{ϵ}=-1$). The solution with $\widehat{ϵ}=1$ is degenerate.

(iii)

For $\sigma =-{\displaystyle \frac{3}{2}}$, the emitter configuration is time-like at

$x_{+}^{\mu }={\displaystyle \frac{(5\sqrt{3}+9)}{4}}\left(5,3,3,3\right)$ and

$x_{-}^{\mu }={\displaystyle \frac{1}{524}}\left(49(17\sqrt{3}-9),3(49\sqrt{3}+159),3(49\sqrt{3}+159),3(49\sqrt{3}+159)\right)$, both being emission solutions (with $\widehat{ϵ}=1$ and $\widehat{ϵ}=-1$, respectively); $x_{-}\left(x_{+}\right)$ is in the front (back) emission coordinate domain.

The main results obtained in this section can be particularized by setting $\beta =0$ to describe the case where all four emitters are static and spatially forming a regular tetrahedron (see [9] for the static case in which the emitters are forming an orthogonal tetrahedron).

4. RPS with One Hyperbolic and Three Static Emitters

In this example we consider that emitters $2,3$ and 4 are static. Emitter 1 is moving in hyperbolic motion with constant proper acceleration $\alpha >0$ along the bisectrix $(1,1,1)$, in such a way that at ${\tau }^1=0$ the four emitters are spatially forming a regular tetrahedron. As ${\tau }^1\to \pm \infty$, its spatial position tends to $+\infty$ along the main bisectrix.

4.1. Emitters’ World-Lines and Emission/Reception Conditions

World-lines with respect to the inertial obeserver u:

$$\begin{array}{cccc}\hfill {\gamma }_1=& {\displaystyle \frac{\gamma \left({\tau }^1\right)\beta \left({\tau }^1\right)}{\alpha }}u+\left[{\displaystyle \frac{\gamma \left({\tau }^1\right)-1}{\sqrt{3}\alpha }}+1\right](1,1,1),\hfill & \hfill {\gamma }_2& ={\tau }^{2}u+(1,-1,-1),\hfill \\ \hfill {\gamma }_3=& {\tau }^{3}u+(-1,1,-1),\hfill & \hfill {\gamma }_4& ={\tau }^{4}u+(-1,-1,1),\hfill \end{array}$$

where $\gamma \left({\tau }^1\right)=cosh\left(\alpha {\tau }^1\right)$ and $\beta \left({\tau }^1\right)=tanh\left(\alpha {\tau }^1\right)$. Expressing the coordinate time in seconds, the proper acceleration $\alpha$ is expressed in $s^{-1}$. We consider that the proper times of emitters 2, 3 and 4 are synchronised at their common origin ${\tau }^2={\tau }^3={\tau }^4=0$ and that the proper time of emitter 1 is synchronised with theirs at ${\tau }^1=0$. The distance covered by emitter 1 in one second of coordinate time $\Delta x^0=1s$, starting from zero coordinate time when it is at rest with respect to the inertial observer ($\Delta {\tau }^1={\displaystyle \frac{arcsinh(\alpha \Delta x^0)}{\alpha }}={\displaystyle \frac{arcsinh\left(\alpha \right)}{\alpha }}$), is $d_1={\displaystyle \frac{cosh(\alpha \Delta {\tau }^1)-1}{\alpha }}ls$. 6

Defining

$$\begin{array}{cc}\hfill \Phi \left({\tau }^1\right)=& {\displaystyle \frac{\gamma \left({\tau }^1\right)\beta \left({\tau }^1\right)}{\alpha }}={\displaystyle \frac{sinh\left(\alpha {\tau }^1\right)}{\alpha }},\Sigma \left({\tau }^1\right)={\displaystyle \frac{\gamma \left({\tau }^1\right)-1}{\sqrt{3}\alpha }},\hfill \\ \hfill q_1=& \Phi -{\tau }^4,q_2={\tau }^2-{\tau }^4,q_3={\tau }^3-{\tau }^4,\hfill \end{array}$$

one obtains the following position vectors with respect to the fourth emitter $e_a={\gamma }_a-{\gamma }_4$, $a=1,2,3$:

$$\begin{array}{c}\hfill e_1=q_{1}u+(\Sigma +2,\Sigma +2,\Sigma ),e_2=q_{2}u+(2,0,-2),e_3=q_{3}u+(0,2,-2),\end{array}$$

(22)

and the following world-functions ${\Omega }_a=\frac{1}{2}{\left(e_a\right)}^2$, $a=1,2,3$:

$$\begin{array}{c}\hfill {\Omega }_1=\frac{1}{2}\left(2{(\Sigma +2)}^2+{\Sigma }^2-q_1^2\right),{\Omega }_2=\frac{1}{2}(8-q_2^2),{\Omega }_3=\frac{1}{2}(8-q_3^2).\end{array}$$

(23)

The emission/reception conditions $e_a^2>0$ and ${(e_a-e_b)}^2>0$ read:

$$\begin{array}{cc}\hfill |\Phi -{\tau }^4|& <\sqrt{2{(\Sigma +2)}^2+{\Sigma }^2},|{\tau }^2-{\tau }^4|<\sqrt{8},\left|{\tau }^3-{\tau }^4\right|<\sqrt{8},\hfill \\ \hfill |\Phi -{\tau }^2|<& \sqrt{2{(\Sigma +2)}^2+{\Sigma }^2},|\Phi -{\tau }^3|<\sqrt{2{(\Sigma +2)}^2+{\Sigma }^2},\left|{\tau }^2-{\tau }^3\right|<\sqrt{8}.\hfill \end{array}$$

4.2. Emitters’ Trajectories on the Grid $\mathcal{T}$

In this case, the equations to obtain the ${\tau }^1$-coordinate of trajectories $S_2$, $S_3$ and $S_4$ (${({\gamma }_i-{\gamma }_1)}^2=0$, $i=2,3,4$) are:

$$\begin{array}{cc}\hfill & A\sqrt{t^2+1}+B_{i}t+C_i=0,t=sinh\left(\alpha {\tau }^1\right),\hfill \\ \hfill & A={\displaystyle \frac{6}{\alpha }}-8\sqrt{3},B_i=-6{\tau }^i,C_i=3\alpha {\left({\tau }^i\right)}^2-24\alpha -A,i=2,3,4.\hfill \end{array}$$

(24)

For $\alpha ={\displaystyle \frac{\sqrt{3}}{4}}\equiv {\alpha }_0$ ($A=0$), the solution to (24) is

$$\begin{array}{c}\hfill v_0\left({\tau }^i\right)={\displaystyle \frac{4}{\sqrt{3}}}arcsinh\left({\displaystyle \frac{\sqrt{3}\left({\left({\tau }^i\right)}^2-8\right)}{8{\tau }^i}}\right),i=2,3,4.\end{array}$$

(25)

Solution $v_0$ is valid for ${\tau }^i\ne 0$. It will be an emission solution iff the corresponding inertial time at the emission event, $\Phi \left(v_0\right)$, is less than the inertial time at the reception event, ${\tau }^i$, ${\displaystyle \frac{\sqrt{3}\left({\left({\tau }^i\right)}^2-8\right)}{8{\tau }^i}}<{\tau }^i$, that is, iff ${\tau }^i>0$.

For $\alpha \ne {\alpha }_0$, the solution of (24) is

$$\begin{array}{cc}\hfill v_{\alpha }\left({\tau }^i\right)=& {\displaystyle \frac{1}{\alpha }}arcsinh\left(D_i\right),D_i={\displaystyle \frac{B_i{C}_i+\rho \sqrt{A^2(B_i^2+C_i^2-A^2)}}{A^2-B_i^2}},\rho =\pm 1,i=2,3,4.\hfill \end{array}$$

(26)

The two solutions included in (26), depending on the sign $\rho$, originate from the solution process of the square root equation (24) and therefore have to satisfy the additional condition ${\displaystyle \frac{B_i}{A}}{D}_i+{\displaystyle \frac{C_i}{A}}<0$. Additionally, solution $v_{\alpha }$ is valid for ${\tau }^i\ne \pm \left({\displaystyle \frac{1}{\alpha }}-{\displaystyle \frac{4}{\sqrt{3}}}\right)$ and will be an emission solution iff $D_i<{\tau }^i$.

Denoting by $v\left(\tau \right)$ solution (25) or (26), depending on the value of $\alpha$, and supposing the conditions for each of these solutions are satisfied, we obtain the following trajectories $S_A$ on the grid $\mathcal{T}$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_1\left(\tau \right)& :\left(\tau ,\Phi -\sqrt{2{(\Sigma +2)}^2+{\Sigma }^2},\Phi -\sqrt{2{(\Sigma +2)}^2+{\Sigma }^2},\Phi -\sqrt{2{(\Sigma +2)}^2+{\Sigma }^2}\right),\hfill \\ \hfill S_2\left(\tau \right)& :\left(v\left(\tau \right),\tau ,\tau -\sqrt{8},\tau -\sqrt{8}\right),\hfill \\ \hfill S_3\left(\tau \right)& :\left(v\left(\tau \right),\tau -\sqrt{8},\tau ,\tau -\sqrt{8}\right),\hfill \\ \hfill S_4\left(\tau \right)& :\left(v\left(\tau \right),\tau -\sqrt{8},\tau -\sqrt{8},\tau \right),\hfill \end{array}\end{array}$$

(27)

with $\Phi \left(\tau \right)={\displaystyle \frac{sinh\left(\alpha \tau \right)}{\alpha }}$ and $\Sigma \left(\tau \right)={\displaystyle \frac{\gamma \left(\tau \right)-1}{\sqrt{3}\alpha }}$.

4.3. Shadow-Dodecahedron

We can write the coordinates of the 14 vertices of the shadow-dodecahedron on the grid ${\mathcal{T}}_1\equiv \left[{\tau }^2\right]\times \left[{\tau }^3\right]\times \left[{\tau }^4\right]$, these coordinates depending on ${\tau }^1$ as follows:

$$\begin{array}{cc}{V}_1=(K_{+},K_{+},K_{+}),\hfill & V_2=(K_{-},K_{-},K_{-}),\hfill \\ V_3=(K_{+}-\sqrt{8},K_{+},K_{+}),\hfill & V_4=(K_{+},K_{+}-\sqrt{8},K_{+}),\hfill \\ V_5=(K_{+},K_{+},K_{+}-\sqrt{8}),\hfill & V_6=(K_{-}+\sqrt{8},K_{-},K_{-}),\hfill \\ V_7=(K_{-},K_{-}+\sqrt{8},K_{-}),\hfill & V_8=(K_{-},K_{-},K_{-}+\sqrt{8}),\hfill \\ V_9=(K_{+}-\sqrt{8},K_{+}-\sqrt{8},K_{+}),\hfill & V_{10}=(K_{+},K_{+}-\sqrt{8},K_{+}-\sqrt{8}),\hfill \\ V_{11}=(K_{+}-\sqrt{8},K_{+},K_{+}-\sqrt{8}),\hfill & V_{12}=(K_{-}+\sqrt{8},K_{-}+\sqrt{8},K_{-}),\hfill \\ V_{13}=(K_{-},K_{-}+\sqrt{8},K_{-}+\sqrt{8}),\hfill & V_{14}=(K_{-}+\sqrt{8},K_{-},K_{-}+\sqrt{8}),\hfill \end{array}$$

with $K_{\pm }\left({\tau }^1\right)\equiv \Phi \pm \sqrt{2{(\Sigma +2)}^2+{\Sigma }^2}$. Of these 14 vertices, only one concides with a satellite trajectory on the grid ${\mathcal{T}}_1$, namely $V_2$ with the trajectory of satellite 1.

To describe the motion of the dodecahedron on the grid ${\mathcal{T}}_1$ as ${\tau }^1\to \pm \infty$, note that vertices $V_1$ and $V_2$, the endpoints on the main bisectrix, satisfy ${lim}_{{\tau }^1\to -\infty }{V}_1=l_{\alpha }$ and ${lim}_{{\tau }^1\to \infty }{V}_2=-l_{\alpha }$, with $l_{\alpha }=(-{\displaystyle \frac{1}{\alpha }}+{\displaystyle \frac{4}{\sqrt{3}}})(1,1,1)$. The translational motion is therefore restricted by this limit: as ${\tau }^1\to -\infty$, $V_1$ does not go beyond $l_{\alpha }$ and as ${\tau }^1\to \infty$, $V_2$ does not go beyond $-l_{\alpha }$. Its longitudinal deformation, however, increases as $|{\tau }^1|\to \infty$. At ${\tau }^1=0$ the dodecahedron is smallest. Figure 4 shows the shadow-dodecahedron for $\alpha =1$ at ${\tau }^1=-2$ and ${\tau }^1=1$.

4.4. Interpreting the RPS Solution

We can represent the characteristic regions of the RPS solution in Cartesian coordinates for this example, depending on the causal character of the configuration vector $\chi$. Figure 5 shows these regions and the region where the jacobian determinant of the transformation from inertial to emission coordinates vanishes, $j_{\Theta }\left(x\right)=0$, for $\alpha =1$ and $x^0=1$. As in the previous example involving inertial motion, these regions change with every inertial instant $x^0$ for a given proper acceleration $\alpha$. However, unlike that example, for a given $x^0$ the RPS solution is not defined for the whole Euclidean space. Due to the hyperbolic motion of emitter 1, there are space-time regions for which there are no (emission) solutions to the null propagation equation ${(x-{\gamma }_1)}^2=0$.

Figure 6 includes a two dimensional slice of the emission configuration regions shown in Figure 5, through the plane containing emitters 1, 2 and 3 at $x^0=1$ ($\alpha =1$). Black points represent satellite trajectories. The intersection with $j_{\Theta }\left(x\right)=0$ is also represented.

And to remark the essential properties of the RPS solution in this hyperbolic case, we consider the inertial coordinate location of those users whose emission coordinates satisfy the restrictions $\Phi \left({\tau }^1\right)=-\sigma$, ${\tau }^2={\tau }^3={\tau }^4=\sigma$, for which

$${\chi }_{\mu }=-8\left(2+{\displaystyle \frac{\sqrt{3}}{2\alpha }}(\sqrt{{\alpha }^2{\sigma }^2+1}-1),\sigma ,\sigma ,\sigma \right),$$

(28)

where we have used (22) and (1). To calculate the particular solution $y_{*}$, we first compute H using (22), (23), (8) and (7). Then, choosing the transversal vector $\xi =u$, Equation (6) yields:

$$\begin{array}{c}{y}_{{*}_{\mu }}={\displaystyle \frac{2}{{\alpha }^{2}D}}\left(0,y_{{*}_1},y_{{*}_2},y_{{*}_3}\right),D=-4\left(4+{\displaystyle \frac{\sqrt{3}}{\alpha }}(\sqrt{{\alpha }^2{\sigma }^2+1}-1)\right),\end{array}$$

(29)

$$\begin{array}{c}\begin{array}{cc}\hfill y_{{*}_1}& =y_{{*}_2}=3{\alpha }^2{\sigma }^2-(4\sqrt{3}\alpha -2)\sqrt{{\alpha }^2{\sigma }^2+1}-8{\alpha }^2+4\sqrt{3}\alpha -2,\hfill \\ \hfill y_{{*}_3}& =3{\alpha }^2{\sigma }^2+2\sqrt{{\alpha }^2{\sigma }^2+1}+8{\alpha }^2-2.\hfill \end{array}\end{array}$$

(30)

With this choice of $\xi$, the transversality condition imposes no restriction on this solution ($D\ne 0$ for any $\{\alpha ,\sigma \}$). Taking into account (9) the solution reads:

$$\begin{array}{c}{x}^{\mu }(-\sigma ,\sigma ,\sigma ,\sigma )=(\sigma ,-1,-1,1)+y_{*}-{\displaystyle \frac{{\lambda }_1}{{\lambda }_2}}\chi ,\end{array}$$

(31)

$$\begin{array}{cc}\hfill {\lambda }_1& =27{\alpha }^4{\sigma }^4+4{\alpha }^2{\sigma }^2(-6+12{\alpha }^2+9\sqrt{{\alpha }^2{\sigma }^2+1}+4\sqrt{3}\alpha (1-3\sqrt{{\alpha }^2{\sigma }^2+1}))\hfill \\ \hfill & +8(-3+8\sqrt{3}\alpha -28{\alpha }^2+16\sqrt{3}{\alpha }^3)\sqrt{{\alpha }^2{\sigma }^2+1}+8(3-8\sqrt{3}\alpha +28{\alpha }^2-16\sqrt{3}{\alpha }^3+24{\alpha }^4),\hfill \\ \hfill {\lambda }_2& =-2{\alpha }^{2}D\left(18{\alpha }^2{\sigma }^3-4\sigma [3(1-\sqrt{{\alpha }^2{\sigma }^2+1})+4\alpha (\alpha +\sqrt{3}(\sqrt{{\alpha }^2{\sigma }^2+1}-1))]-\widehat{ϵ}{\alpha }^{2}D\sqrt{{\Delta }_{\sigma }}\right),\hfill \\ \hfill {\Delta }_{\sigma }& ={\displaystyle \frac{{\lambda }_1}{16{\alpha }^4}}-8{\sigma }^2.\hfill \end{array}$$

In this case, the emission-reception conditions are reduced to:

$$\left|\sigma \right|<{\displaystyle \frac{1}{2}}\sqrt{{\sigma }^2+{\displaystyle \frac{2}{3{\alpha }^2}}(4\sqrt{3}\alpha -3)(\sqrt{{\alpha }^2{\sigma }^2+1}-1)+8}.$$

(32)

Table 2 shows the result setting $\alpha =1$ for different values of $\sigma$. In this particular case, the previous emission-reception condition imposes

$\left|\sigma \right|<{\displaystyle \frac{2}{3}}\sqrt{{\displaystyle \frac{1}{3}}(-10\sqrt{3}+\sqrt{838-424\sqrt{3}}+32)}\approx 1.92$.

(i)

For $\tau ={\displaystyle \frac{3}{4}}$, the emitter configuration is space-like at

$x^{\mu }={\displaystyle \frac{1}{1448}}\left(1648\sqrt{3}+821,-5(103\sqrt{3}+40),-5(103\sqrt{3}+40),-5(103\sqrt{3}+40)\right)$, which is the sole emission solution (with $\widehat{ϵ}=-1$). The solution with $\widehat{ϵ}=1$ is a reception solution.

(ii)

For $\tau ={\displaystyle \frac{2}{9}}\left(-4\sqrt{3}-\sqrt{21-6\sqrt{3}}+3\right)\approx -1.60$, the emitter configuration is light-like at $x^{\mu }\approx \left(1.81,1.39,1.39,1.39\right)$, which is the sole emission solution (with $\widehat{ϵ}=-1$). The solution with $\widehat{ϵ}=1$ is degenerate.

(iii)

For $\tau =-{\displaystyle \frac{17}{10}}$, the emitter configuration is time-like at $x_{+}^{\mu }\approx (9.63,6.14,6.14,6.14)$ and $x_{-}^{\mu }\approx (1.84,1.48,1.48,1.48)$, both being emission solutions (with $\widehat{ϵ}=1$ and $\widehat{ϵ}=-1$, respectively); $x_{-}\left(x_{+}\right)$ is in the front (back) emission coordinate domain.

5. RPS with One Static and Three Rotating Emitters

In this example we consider the spatial configuration of emitters 1, 2 and 3 following uniform circular motion on the $\{x,y\}$-plane, at constant radial distance $r_0$ and equally spaced at an angle ${\displaystyle \frac{2\pi }{3}}$, and emitter 4 static at distance d on the z-axis.

5.1. Emitters’ World-Lines and Emission/Reception Conditions

If $\omega$ is the constant angular velocity with respect to the inertial observer u, we can write the world-lines as follows:

$$\begin{array}{cc}\hfill {\gamma }_1=& \gamma {\tau }^{1}u+r_0\left(cos\left(\omega \gamma {\tau }^1\right),sin\left(\omega \gamma {\tau }^1\right),0\right),\hfill \\ \hfill {\gamma }_2=& \gamma {\tau }^{2}u+r_0\left(cos(\omega \gamma {\tau }^2-{\displaystyle \frac{2\pi }{3}}),sin(\omega \gamma {\tau }^2-{\displaystyle \frac{2\pi }{3}}),0\right),\hfill \\ \hfill {\gamma }_3=& \gamma {\tau }^{3}u+r_0\left(cos(\omega \gamma {\tau }^3-{\displaystyle \frac{4\pi }{3}}),sin(\omega \gamma {\tau }^3-{\displaystyle \frac{4\pi }{3}}),0\right),{\gamma }_4={\tau }^{4}u+(0,0,d),\hfill \end{array}$$

(33)

where $\gamma ={(1-{\beta }^2)}^{-{\displaystyle \frac{1}{2}}}$ and $\beta =\omega r_0$ ($\omega r_0<1$). The velocity $\beta$ is expressed as a fraction of the speed of light in vacuum c ($\beta <1$). We consider that the proper times of emitters 1, 2 and 3 are synchronised at their common origin ${\tau }^1={\tau }^2={\tau }^3=0$ and are synchronised with emitter 4 at ${\tau }^4=0$. Expressing the coordinate time in seconds and the spatial coordinates in kilometres, the number of revolutions $s_1$ completed by emitter 1 in one second of coordinate time $\Delta x^0=1s$ ($\Delta {\tau }^1={\gamma }^{-1}\Delta x^0={\gamma }^{-1}s$) is $s_1\approx {\displaystyle \frac{\beta }{2\pi r_0}}3·{10}^5$, truncated to an integer. 7

Defining

$$\begin{array}{c}\hfill q_1={\Gamma }_1-{\tau }^4,q_2={\Gamma }_2-{\tau }^4,{\Gamma }_3-{\tau }^4,{\Gamma }_a=\gamma {\tau }^a,a=1,2,3,\end{array}$$

one obtains the following position vectors with respect to the fourth emitter, $e_a={\gamma }_a-{\gamma }_4$, $a=1,2,3$:

$$\begin{array}{cc}\hfill e_1=& q_{1}u+r_0\left(cos\left(\omega {\Gamma }_1\right),sin\left(\omega {\Gamma }_1\right),-{\displaystyle \frac{d}{r_0}}\right),\hfill \\ \hfill e_2=& q_{2}u+r_0\left(cos(\omega {\Gamma }_2-{\displaystyle \frac{2\pi }{3}}),sin(\omega {\Gamma }_2-{\displaystyle \frac{2\pi }{3}}),-{\displaystyle \frac{d}{r_0}}\right),\hfill \\ \hfill e_3=& q_{3}u+r_0\left(cos(\omega {\Gamma }_3-{\displaystyle \frac{4\pi }{3}}),sin(\omega {\Gamma }_3-{\displaystyle \frac{4\pi }{3}}),-{\displaystyle \frac{d}{r_0}}\right),\hfill \end{array}$$

(34)

and the world-functions

$$\begin{array}{c}\hfill {\Omega }_a=\frac{1}{2}(r_0^2+d^2-q_a^2),a=1,2,3.\end{array}$$

(35)

In this example, the emission/reception conditions (3) impose:

$$\begin{array}{cc}\hfill |{\Gamma }_a-{\tau }^4|<& \sqrt{r_0^2+d^2},a=1,2,3,\hfill \\ \hfill |{\Gamma }_1-{\Gamma }_2|<& \sqrt{2}{r}_0\sqrt{1+sin(\omega ({\Gamma }_1-{\Gamma }_2)+{\displaystyle \frac{\pi }{6}})},\hfill \\ \hfill |{\Gamma }_1-{\Gamma }_3|<& \sqrt{2}{r}_0\sqrt{1+sin(\omega ({\Gamma }_3-{\Gamma }_1)+{\displaystyle \frac{\pi }{6}})},\hfill \\ \hfill |{\Gamma }_2-{\Gamma }_3|<& \sqrt{2}{r}_0\sqrt{1+sin(\omega ({\Gamma }_2-{\Gamma }_3)+{\displaystyle \frac{\pi }{6}})}.\hfill \end{array}$$

5.2. Emitters’ Trajectories on the Grid $\mathcal{T}$

The emission coordinate ${\tau }^b$ of the a-emitter ($a,b=1,2,3,4$) satisfies the equation ${({\gamma }_a\left({\tau }^a\right)-{\gamma }_b\left({\tau }^b\right))}^2=0$. Due to the transcendental functions appearing in the world-lines of emitters 1, 2 and 3 (33), the ${\tau }^1$, ${\tau }^2$ and ${\tau }^3$-coordinates of the trajectories of emitters 1, 2 and 3 on the grid $\mathcal{T}$ ($S_1$, $S_2$ and $S_3$) can only be computed numerically, by solving the following equations:

$$\begin{array}{cc}\hfill {({\Gamma }_1-{\Gamma }_2)}^2=& 2r_0^2\left(1+sin(\omega ({\Gamma }_1-{\Gamma }_2)+{\displaystyle \frac{\pi }{6}})\right),\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill {({\Gamma }_3-{\Gamma }_1)}^2=& 2r_0^2\left(1+sin(\omega ({\Gamma }_3-{\Gamma }_1)+{\displaystyle \frac{\pi }{6}})\right),\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill {({\Gamma }_2-{\Gamma }_3)}^2=& 2r_0^2\left(1+sin(\omega ({\Gamma }_2-{\Gamma }_3)+{\displaystyle \frac{\pi }{6}})\right).\hfill \end{array}$$

(38)

Note that Equations (36)-(38) implicitly describe a lineal relationship between ${\tau }^a$ and ${\tau }^b$, $a,b=1,2,3$. Specifically, the solution of one coordinate in terms of the other is a straight line of unit slope, whose intercept has to be computed numerically. So, the solution for ${\tau }^2$ of Equation (36) is ${\tau }^2\left({\tau }^1\right)={\tau }^1+m$, with m the numerical solution8 to:

$$m^2={\displaystyle \frac{2r_0^2}{{\gamma }^2}}\left(1+sin(-\omega \gamma m+{\displaystyle \frac{\pi }{6}})\right).$$

(39)

For $r_0=d=1$ and $\omega ={\displaystyle \frac{1}{2}}$$(\gamma ={\displaystyle \frac{2}{\sqrt{3}}})$, $m\approx -1.732$.9

It is clear that the solution for ${\tau }^1$ of Equation (37) is ${\tau }^1\left({\tau }^3\right)={\tau }^1+m$ and the solution for ${\tau }^3$ of (38) is ${\tau }^3\left({\tau }^2\right)={\tau }^2+m$. Analogously, the solution for ${\tau }^1$ of (36) is ${\tau }^1\left({\tau }^2\right)={\tau }^2+n$, with n the (negative) numerical solution to:

$$n^2={\displaystyle \frac{2r_0^2}{{\gamma }^2}}\left(1+sin(\omega \gamma n+{\displaystyle \frac{\pi }{6}})\right).$$

(40)

For $r_0=d=1$ and $\omega ={\displaystyle \frac{1}{2}}$$(\gamma ={\displaystyle \frac{2}{\sqrt{3}}})$, $n\approx -1.140$.

Then, the solution for ${\tau }^3$ of Equation (37) is ${\tau }^3\left({\tau }^1\right)={\tau }^1+n$ and the solution for ${\tau }^2$ of (38) is ${\tau }^2\left({\tau }^3\right)={\tau }^2+n$. The following trajectories $S_A$ on the grid $\mathcal{T}$ are thus obtained:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_1\left(\tau \right)& :\left(\tau ,\tau +m,\tau +n,\gamma \tau -\sqrt{d^2+r_0^2}\right),\hfill \\ \hfill S_2\left(\tau \right)& :\left(\tau +n,\tau ,\tau +m,\gamma \tau -\sqrt{d^2+r_0^2}\right),\hfill \\ \hfill S_3\left(\tau \right)& :\left(\tau +m,\tau +n,\tau ,\gamma \tau -\sqrt{d^2+r_0^2}\right),\hfill \\ \hfill S_4\left(\tau \right)& :\left({\gamma }^{-1}\left(\tau -\sqrt{d^2+r_0^2}\right),{\gamma }^{-1}\left(\tau -\sqrt{d^2+r_0^2}\right),{\gamma }^{-1}\left(\tau -\sqrt{d^2+r_0^2}\right),\tau \right),\hfill \end{array}\end{array}$$

(41)

where m and n are obtained numerically from (39) and (40) for given values of the parametres $r_0$ and $\omega$.

5.3. Shadow-Dodecahedron

Next, we analyse the shadow-dodecahedron on the grid ${\mathcal{T}}_4\equiv \left[{\tau }^1\right]\times \left[{\tau }^2\right]\times \left[{\tau }^3\right]$ for a given ${\tau }^4$. Again, due to the trascendental functions appearing in the emission/reception conditions, the vertices of the dodecahedron have to be computed numerically. The resulting dodecahedron is composed of 3 hexagons, 3 rhomboids and 6 pentagons, leading to the following 20 vertices:

$$\begin{array}{cc}{V}_1=(L_{+},L_{+},L_{+}),\hfill & V_2=(L_{-},L_{-},L_{-}),\hfill \\ V_3=(L_{+}+n,L_{+},L_{+}),\hfill & V_4=(L_{+},L_{+}+n,L_{+}),\hfill \\ V_5=(L_{+},L_{+},L_{+}+n),\hfill & V_6=(L_{-}-n,L_{-},L_{-}),\hfill \\ V_7=(L_{-},L_{-}-n,L_{-}),\hfill & V_8=(L_{-},L_{-},L_{-}-n),\hfill \\ V_9=(L_{+}+n,L_{+},L_{+}+m),\hfill & V_{10}=(L_{+}+m,L_{+}+n,L_{+}),\hfill \\ V_{11}=(L_{+},L_{+}+m,L_{+}+n),\hfill & V_{12}=(L_{-}-n,L_{-}-m,L_{-}),\hfill \\ V_{13}=(L_{-},L_{-}-n,L_{-}-m),\hfill & V_{14}=(L_{-}-m,L_{-},L_{-}-n),\hfill \\ V_{15}=(L_{+}+n+(n-m),L_{+},L_{+}+m),\hfill & V_{16}=(L_{+}+m,L_{+}+n+(n-m),L_{+}),\hfill \\ V_{17}=(L_{+},L_{+}+m,L_{+}+n+(n-m)),\hfill & V_{18}=(L_{-}-n-(n-m),L_{-}-m,L_{-}),\hfill \\ V_{19}=(L_{-},L_{-}-n-(n-m),L_{-}-m),\hfill & V_{20}=(L_{-}-m,L_{-},L_{-}-n-(n-m)),\hfill \end{array}$$

with $L_{\pm }\left({\tau }^4\right)\equiv {\gamma }^{-1}({\tau }^4\pm \sqrt{r_0^2+d^2})$ and m and n given by (39) and (40). The coordinates of the vertices $V_a,a=3,4,...,20$, are obtained by adding or substracting the distances m and n and their difference to the coordinates of the end vertices $V_1$ and $V_2$ on the main bisectrix. In the shadow-dodecahedrons of the previous examples, with fourteen vertices forming six squares and six rhomboids, only one distance10 is added or substracted to the coordinates of the end vertices $V_1$ and $V_2$, hence we can say that in previous examples m and n coincide. Note that, unlike in previous examples, in this case of three rotating and one inertial emitter, the shadow-dodecahedron is described on the ${\mathcal{T}}_4$-grid, where the $\tau -$coordinate not forming part of the grid11 is the coordinate of the static and not of the moving emitter.

For different values of ${\tau }^4$, the shadow-dodecahedron moves along the main bisectrix of the ${\mathcal{T}}_4$-grid without changing its shape (the relative positions of its vertices remain constant for every ${\tau }^4$). Figure 7 shows the shadow-dodecahedron for $r_0=d=1$ and $\omega ={\displaystyle \frac{1}{2}}$$(\gamma ={\displaystyle \frac{2}{\sqrt{3}}})$ at ${\tau }^4=-2$, ${\tau }^4=2$ and ${\tau }^4=6$:

5.4. Interpreting the RPS Solution

In this example, representing the characteristic regions of the RPS solution in Cartesian coordinates for a given inertial time is not straightforward. Since the null propagation equations (2) can not be solved analitically for ${\tau }^A$ (the characteristic emission function $\Theta$, ${\tau }^A={\Theta }^A\left(x\right)$, can not be obtained), we can not directly express the configuration vector $\chi$ in Cartesian coordinates for a given coordinate time $x^0$. Instead, we proceed numerically, first classifying a sufficient number of quads of emission coordinates $\{{\tau }^1,{\tau }^2,{\tau }^3,{\tau }^4\}$ according to the configuration vector’s causal character associated to each of them. Then, we transform each of these quads to inertial coordinates using (5) and label the space-time points thus obtained with the same color scheme as in previous examples. For each characteristic region we compute a large number of points and select those with a given (common) inertial time $x^0$. This process unavoidably leads to approximations, for which reason the following representations have to be considered qualitatively.12 Figure 8 illustrates these regions for $x^0=5$ ($r_0=d=1$ and $\omega ={\displaystyle \frac{1}{2}}$$(\gamma ={\displaystyle \frac{2}{\sqrt{3}}})$).

Figure 9 includes two two-dimensional slices of the emission configuration regions illustrated in Figure 8, one through the plane $x^3=0$ containing emitters 1, 2 and 3 and the other through the plane $x^3=2$, both at $x^0=5$ with $r_0=d=1$ and $\omega ={\displaystyle \frac{1}{2}}$$(\gamma ={\displaystyle \frac{2}{\sqrt{3}}})$.

And to remark the essential properties of the RPS solution in this example, we consider the inertial coordinate location of those users whose emission coordinates satisfy the restrictions ${\tau }^1=\sigma$,${\tau }^2=-\sigma ,{\tau }^3=-\sigma$ and ${\tau }^4=\gamma \sigma \equiv {\Gamma }_4\left(\sigma \right)$. Using (34) and (1) we calculate the configuration vector:

$$\begin{array}{c}{\chi }_{\mu }=\sqrt{3}{r}_0\left({\chi }_0,{\chi }_1,{\chi }_2,{\chi }_3\right),\end{array}$$

(42)

$$\begin{array}{c}\begin{array}{cccc}& {\chi }_0=-{\displaystyle \frac{d{r}_0}{2}}(1+2cos\left(2\omega {\Gamma }_4\right)),\hfill & & {\chi }_1=2d{\Gamma }_{4}cos\left(\omega {\Gamma }_4\right),\hfill \\ & {\chi }_2=-2d{\Gamma }_{4}sin\left(\omega {\Gamma }_4\right),\hfill & & {\chi }_3=2r_0{\Gamma }_{4}cos\left(2\omega {\Gamma }_4\right).\hfill \end{array}\end{array}$$

(43)

To calculate the particular solution $y_{*}$, the configuration bivector H is first computed using (34), (35), (8) and (7). Choosing the transversal vector $\xi =u$, Equation (6) yields:

$$\begin{array}{c}{y}_{*\mu }={\displaystyle \frac{\sqrt{3}{r}_0}{D}}\left(0,y_{{*}_1},y_{{*}_2},y_{{*}_3}\right),D=-{\displaystyle \frac{\sqrt{3}}{2}}d{r}_0^2\left(1+2cos\left(2\omega {\Gamma }_4\right)\right),\end{array}$$

(44)

$$\begin{array}{c}\begin{array}{cc}& y_{{*}_1}=-2d{\Gamma }_4^{2}cos\left(\omega {\Gamma }_4\right),y_{{*}_2}=2d{\Gamma }_4^{2}sin\left(\omega {\Gamma }_4\right),\hfill \\ & y_{{*}_3}={\displaystyle \frac{r_0}{4}}\left(d^2+r_0^2+2(d^2+r_0^2-4{\Gamma }_4^2)cos\left(2\omega {\Gamma }_4\right)\right).\hfill \end{array}\end{array}$$

(45)

With this choice of $\xi$, the transversality condition restricts this solution to

$$\{\sigma \in \mathbb{R}|\sigma \ne {\sigma }_t\equiv \pm {\displaystyle \frac{\pi }{\gamma \omega }}({\displaystyle \frac{1}{3}}+n),n\in {\mathbb{Z}}^{+}\}.$$

At $\sigma ={\sigma }_t$ emitters 1 and 3 are at the same spatial location $(-{\displaystyle \frac{1}{2}},{\displaystyle \frac{\sqrt{3}}{2}},0)$, leading to a degenerate situation. The solution (5) reads:

$$\begin{array}{c}{x}^{\alpha }(\sigma ,-\sigma ,-\sigma ,\gamma \sigma )=(\gamma \sigma ,0,0,d)+y_{*}-{\displaystyle \frac{{\lambda }_1}{{\lambda }_2}}\chi ,\end{array}$$

(46)

$$\begin{array}{cc}\hfill {\lambda }_1& ={\displaystyle \frac{3}{16}}{r}_0^2\left(64d^2{\Gamma }_4^4+r_0^2{\left(2\left(d^2-4{\Gamma }_4^2+r_0^2\right)cos\left(2\omega {\Gamma }_4\right)+d^2+r_0^2\right)}^2\right),\hfill \\ \hfill {\lambda }_2& =D({\displaystyle \frac{3}{2}}{r}_0^4{\Gamma }_{4}cos\left(2\omega {\Gamma }_4\right)\left(2\left(d^2-4{\Gamma }_4^2+r_0^2\right)cos\left(2\omega {\Gamma }_4\right)+d^2+r_0^2\right)-12d^2{r}_0{\Gamma }_4^3+\widehat{ϵ}D\sqrt{{\Delta }_{\sigma }}),\hfill \\ \hfill {\Delta }_{\sigma }& ={\lambda }_1-3r_0^2{\Gamma }_4^2(d^2+r_0^2).\hfill \end{array}$$

In this case, the emission-reception conditions (3) are as follows:

$$|{\Gamma }_4|<{\displaystyle \frac{1}{2}}{r}_0\sqrt{2(1+sin(\pm 2\omega {\Gamma }_4+{\displaystyle \frac{\pi }{6}})},\left|{\Gamma }_4\right|<{\displaystyle \frac{1}{2}}\sqrt{r_0^2+d^2}.$$

(47)

Table 3 shows the result setting $r_0=d=1$ and $\omega ={\displaystyle \frac{1}{2}}$$(\gamma ={\displaystyle \frac{2}{\sqrt{3}}})$ for different values of $\sigma$. In this particular case, the previous emission-reception conditions impose $-{\sigma }_c<\sigma <{\sigma }_c$, with ${\sigma }_c\approx 0.570$. However, unlike in previous examples, the "physical" requirement ${\Delta }_{\sigma }\ge 0$ further restricts the domain to $-{\sigma }_p\le \sigma \le {\sigma }_p$, with ${\sigma }_p\approx 0.453$, in this case.

(i)

For $\sigma =-{\displaystyle \frac{2}{5}}$, the emitter configuration is space-like at

$x^{\alpha }\approx \left(2.654,-1.710,-0.402,-1.573\right)$, which is the sole emission solution (with $\widehat{ϵ}=-1$). The solution with $\widehat{ϵ}=1$ is a reception solution.

(ii)

For $\sigma \approx 0.447$, the emitter configuration is light-like at

$x^{\alpha }\approx \left(3.764,2.746,-0.726,2.470\right)$, which is the sole emission solution (with $\widehat{ϵ}=-1$). The solution with $\widehat{ϵ}=1$ is degenerate.

(iii)

For $\sigma \approx 0.450$, the emitter configuration is time-like at

$x_{+}^{\alpha }\approx (20.506,15.055,-4.002,13.522)$ and

$x_{-}^{\alpha }\approx (2.992,2.197,-0.584,1.973)$, both being emission solutions (with $\widehat{ϵ}=1$ and $\widehat{ϵ}=-1$, respectively); $x_{-}\left(x_{+}\right)$ is in the front (back) emission coordinate domain.

6. Summary and Conclusions

A basic construction of a RPS involving four static emitters in flat space-time was presented in [9]. The same procedure has been extended in the present work by including inertial, accelerated and rotating emitters. All these constructions provide clarifying examples of the notions involved in RPS theory (null propagation equations, bifurcation of solutions, grid space of emission coordinates, emission/reception conditions, shadows reciprocally cast by pairs of emitters, shadow-dodecahedron, emission coordinate region, central, front, and back regions, orientation, etc.). The examples analyzed offer a method to explore more involved constructions in flat and curved space-times. Despite the inherent differences with more complicated constructions, these elemental examples play a crucial role in RPS theory, similar to the one played in plant biology by the model organism Arabidopsis thaliana. Just as the model plant is essential to the molecular-biological study of more complex traits, the RPS constructions considered here contain valuable information for developing RPSs in a (known or unknown) gravitational field geometry. Furthermore, our incipient constructions of the emission coordinate regions attached to a specific RPS can be complemented with additional theoretical information, such as as the IDEAL (Intrinsic, Deductive, Explicit, ALgoritmic) characterization of gravitational fields developed in [22,23,24,25], which is necessary in order to make gravimetry (determine the true gravitational potentials) in a unknown space-time region (see [6] for details concerning this idea).