5 research outputs found
Two-dimensional approach to relativistic positioning systems
A relativistic positioning system is a physical realization of a coordinate
system consisting in four clocks in arbitrary motion broadcasting their proper
times. The basic elements of the relativistic positioning systems are presented
in the two-dimensional case. This simplified approach allows to explain and to
analyze the properties and interest of these new systems. The positioning
system defined by geodesic emitters in flat metric is developed in detail. The
information that the data generated by a relativistic positioning system give
on the space-time metric interval is analyzed, and the interest of these
results in gravimetry is pointed out.Comment: 11 pages, 5 figures. v2: a brief description of the principal
bibliography has been adde
Positioning systems in Minkowski space-time: from emission to inertial coordinates
The coordinate transformation between emission coordinates and inertial
coordinates in Minkowski space-time is obtained for arbitrary configurations of
the emitters. It appears that a positioning system always generates two
different coordinate domains, namely, the front and the back emission
coordinate domains. For both domains, the corresponding covariant expression of
the transformation is explicitly given in terms of the emitter world-lines.
This task requires the notion of orientation of an emitter configuration. The
orientation is shown to be computable from the emission coordinates for the
users of a `central' region of the front emission coordinate domain. Other
space-time regions associated with the emission coordinates are also outlined.Comment: 20 pages; 1 figur
Relativistic Positioning Systems: The Emission Coordinates
This paper introduces some general properties of the gravitational metric and
the natural basis of vectors and covectors in 4-dimensional emission
coordinates. Emission coordinates are a class of space-time coordinates defined
and generated by 4 emitters (satellites) broadcasting their proper time by
means of electromagnetic signals. They are a constitutive ingredient of the
simplest conceivable relativistic positioning systems. Their study is aimed to
develop a theory of these positioning systems, based on the framework and
concepts of general relativity, as opposed to introducing `relativistic
effects' in a classical framework. In particular, we characterize the causal
character of the coordinate vectors, covectors and 2-planes, which are of an
unusual type. We obtain the inequality conditions for the contravariant metric
to be Lorentzian, and the non-trivial and unexpected identities satisfied by
the angles formed by each pair of natural vectors. We also prove that the
metric can be naturally split in such a way that there appear 2 parameters
(scalar functions) dependent exclusively on the trajectory of the emitters,
hence independent of the time broadcast, and 4 parameters, one for each
emitter, scaling linearly with the time broadcast by the corresponding
satellite, hence independent of the others.Comment: 13 pages, 3 figures. Only format changed for a new submission.
Submitted to Class. Quantum Gra
Painlev\'e-Gullstrand synchronizations in spherical symmetry
A Painlev\'e-Gullstrand synchronization is a slicing of the space-time by a
family of flat spacelike 3-surfaces. For spherically symmetric space-times, we
show that a Painlev\'e-Gullstrand synchronization only exists in the region
where , being the curvature radius of the isometry group
orbits (-spheres). This condition says that the Misner-Sharp gravitational
energy of these 2-spheres is not negative and has an intrinsic meaning in terms
of the norm of the mean extrinsic curvature vector. It also provides an
algebraic inequality involving the Weyl curvature scalar and the Ricci
eigenvalues. We prove that the energy and momentum densities associated with
the Weinberg complex of a Painlev\'e-Gullstrand slice vanish in these curvature
coordinates, and we give a new interpretation of these slices by using
semi-metric Newtonian connections. It is also outlined that, by solving the
vacuum Einstein's equations in a coordinate system adapted to a
Painlev\'e-Gullstrand synchronization, the Schwarzschild solution is directly
obtained in a whole coordinate domain that includes the horizon and both its
interior and exterior regions.Comment: 16 page