Newtonian and relativistic emission coordinates

Emission coordinates are those generated by positioning systems. Positioning systems are physical systems constituted by four emitters broadcasting their respective times by means of sound or light signals. We analyze the incidence of the space-time causal structure on the construction of emission coordinates. The Newtonian case of four emitters at rest is analyzed and contrasted with the corresponding situation in special relativity.

Figure 1

The 199 causal classes of relativistic frames and coordinates (cf. Ref. 12).

Figure 2

At any instant $t=\mathrm{\text{const}}$, the positions ${\kappa }^A(t)\equiv A$ ($A=1$, 2, 3, 4) of the four clocks generically define the four vertices $A$, $B$, $C$, $D$ (all $\ne$) of a 3-dimensional tetrahedron. If the clocks are at rest in an inertial system, the outer open segments $s_{AB}$ and $s_{BA}$ of the straight line ${\ell }_{AB}$ containing the edge $i_{AB}$ between the vertices $A$ and $B$ represent the shadows of the signals $B$ and $A$ produced by $A$ and $B$, respectively.