878 research outputs found
On the Leibniz bracket, the Schouten bracket and the Laplacian
The Leibniz bracket of an operator on a (graded) algebra is defined and some
of its properties are studied. A basic theorem relating the Leibniz bracket of
the commutator of two operators to the Leibniz bracket of them, is obtained.
Under some natural conditions, the Leibniz bracket gives rise to a (graded) Lie
algebra structure. In particular, those algebras generated by the Leibniz
bracket of the divergence and the Laplacian operators on the exterior algebra
are considered, and the expression of the Laplacian for the product of two
functions is generalized for arbitrary exterior forms
Positioning with stationary emitters in a two-dimensional space-time
The basic elements of the relativistic positioning systems in a
two-dimensional space-time have been introduced in a previous work [Phys. Rev.
D {\bf 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 {\em
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 {\em 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.Comment: 14 pages; 5 figure
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
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