From emission to inertial coordinates: an analytical approach

Bartolomé Coll, Joan Josep Ferrando and Juan Antonio Morales-Lladosa

In a previous work [Class. Quantum Grav. 27 (2010) 065013] we have studied relativistic positioning systems in the Minkowski space-time, and we have obtained the transformation from emission to inertial coordinates for an arbitrary configuration of the emitters. The formula giving this transformation applies in all the emission coordinate region and involves the orientation of the positioning system (the sign of the Jacobian determinant of the map that to every event associates its emission coordinates). However, there exists an inherent limitation on the applicability of this formula: only the users in the central region of a positioning system can obtain the orientation from the sole emission data. Here we present an observational procedure to determine this orientation which provides a minimum additional information allowing any user to obtain its inertial coordinates, irrespectively of its location in the emission region of the positioning system.