Positioning with stationary emitters in a two-dimensional space-time

Abstract

The basic elements of the relativistic positioning systems in a two-dimensional space-time have been introduced in a previous work [Phys. Rev. D 73, 084017 (2006)] where geodesic positioning systems, constituted by two geodesic emitters, have been considered in a flat space-time. Here, we want to show in what precise senses positioning systems allow to make relativistic gravimetry. For this purpose, we consider stationary positioning systems, constituted by two uniformly accelerated emitters separated by a constant distance, in two different situations: absence of gravitational field (Minkowski plane) and presence of a gravitational mass (Schwarzschild plane). The physical coordinate system constituted by the electromagnetic signals broadcasting the proper time of the emitters are the so called emission coordinates, and we show that, in such emission coordinates, the trajectories of the emitters in both situations, absence and presence of a gravitational field, are identical. The interesting point is that, in spite of this fact, particular additional information on the system or on the user allows not only to distinguish both space-times, but also to complete the dynamical description of emitters and user and even to measure the mass of the gravitational field. The precise information under which these dynamical and gravimetric results may be obtained is carefully pointed out.

Precision measurements of the top quark mass from the Tevatron in the pre-LHC era

“…The CDF algorithm [63] employs other variables in addition to secondary vertex information, but has not been used for top mass measurements. The DØ NN algorithm [64] was used for top mass measurements. It takes into account information from impact parameter measurements, the secondary vertex tagger, and the soft lepton tagger (SLT).…”

Mapping Cartesian Coordinates Into Emission Coordinates: Some Toy Models

“…This makes a direct comparison possible with the relations obtained in [7] and, consequently, the same properties apply (in particular the possibility of determining the parameters k, q, σ).…”

“…In this paper, starting from the results obtained in [6,7], we show how an explicit map can be obtained from the Cartesian coordinates and the emission coordinates, for suitably chosen set of emitters, whose world-lines are supposed to be known by the users. We start from the 2-dimensional cases, both in inertial and accelerated frames, and show how the results can be generalized to 4-dimensional cases, in order to deal with Minkowski 1+3 dimensional space-time and the space-time where a small inhomogeneity is introduced (i.e.…”

“…Actually, a 2-dimensional approach to this new paradigm of relativistic positioning has been thoroughly described by Coll and collaborators [6,7]: there, the definition of the coordinates domain (i.e. the space-time region where emission coordinates are well defined), the information that the data coming from a relativistic positioning system give on the space-time metric interval and the interest of these results in gravimetry, are discussed and analyzed for some prototypical situations.…”

Relativistic versus Newtonian Frames

“…Coll positioning systems are yet now quite well developed for two-dimensional space-times, see [11,17] where several results have been developed. For instance, the knowledge that the positioning system is stationary and that the space-time is created by a given mass, allows to know the accelerations of the emitters, their mutual radar distances and the space-time metric in null emission coordinates.…”