29 research outputs found
Coordinates and frames from the causal point of view
Lorentzian frames may belong to one of the 199 causal classes. Of these
numerous causal classes, people are essentially aware only of two of them.
Nevertheless, other causal classes are present in some well-known solutions, or
present a strong interest in the physical construction of coordinate systems.
Here we show the unusual causal classes to which belong so familiar coordinate
systems as those of Lema{\^{\i}}tre, those of Eddington-Finkelstein, or those
of Bondi-Sachs. Also the causal classes associated to the Coll light
coordinates (four congruences of real geodetic null lines) and to the Coll
positioning systems (light signals broadcasted by four clocks) are analyzed.
The role that these results play in the comprehension and classification of
relativistic coordinate systems is emphasized.Comment: 5 pages, 1 figure, short communication in A Century of Relativity
Physics, Proceedings of the XXVIII Spanish Relativity Meeting ERE-2006, 6-10
September - Oviedo - Spain (AIP Conference Proceedings
Maximal slicings in spherical symmetry: local existence and construction
We show that any spherically symmetric spacetime locally admits a maximal
spacelike slicing and we give a procedure allowing its construction. The
construction procedure that we have designed is based on purely geometrical
arguments and, in practice, leads to solve a decoupled system of first order
quasi-linear partial differential equations. We have explicitly built up
maximal foliations in Minkowski and Friedmann spacetimes. Our approach admits
further generalizations and efficient computational implementation. As by
product, we suggest some applications of our work in the task of calibrating
Numerical Relativity complex codes, usually written in Cartesian coordinates.Comment: 25 pages, 6 figure
Spherical symmetric parabolic dust collapse: C¹ matching metric with zero intrinsic energy
The collapse of marginally bound, inhomogeneous, pressureless (dust) matter, in spherical symmetry, is considered. The starting point is not, in this case, the integration of the Einstein equations from some suitable initial conditions. Instead, starting from the corresponding general exact solution of these equations, depending on two arbitrary functions of the radial coordinate, the fulfillment of the Lichnerowicz matching conditions of the interior collapsing metric and the exterior Schwarzschild one is tentatively assumed (the continuity of the metric and its first derivatives on the time-like hypersurface describing the evolution of the spherical 2-surface boundary of the collapsing cloud), and the consequences of such a tentative assumption are explored. The whole analytical family of resulting models is obtained and some of them are picked out as physical better models on the basis of the finite and constant value of its intrinsic energy
On the uniqueness of the space-time energy in General Relativity: the illuminating case of the Schwarzschild metric
The case of asymptotic Minkowskian space-times is considered. A special class of asymptotic rectilinear coordinates at the spatial infinity, related to a spe- cific system of free falling observers, is chosen. This choice is applied in particular to the Schwarzschild metric, obtaining a vanishing energy for this space-time. This result is compared with the result of some known theorems on the uniqueness of the energy of any asymptotic Minkowskian space, showing that there is no contradiction between both results, the differences becoming from the use of coordinates with dif- ferent operational meanings. The suitability of Gauss coordinates when defining an intrinsic energy is considered and it is finally concluded that a Schwarzschild metric is a particular case of space-times with vanishing intrinsic 4-momenta
Solutions of the Einstein field equations for a bounded and finite discontinuous source, and its generalization: Metric matching conditions and jumping effects
We consider the metrics of the General Relativity, whose energy-momentum tensor has a bounded support where it is continuous except for a finite step across the corresponding boundary surface. As a consequence, the first derivative of the metric across this boundary could perhaps present a finite step too. However, we can assume that the metric is C1 class everywhere. In such a case, although the partial second derivatives of the metric exhibit finite (no Dirac δ functions) discontinuities, the Dirac δ functions will still appear in the conservation equation of the energy-momentum tensor. As a consequence, strictly speaking, the corresponding metric solutions of the Einstein field equations can only exist in the sense of distributions. Then, we assume that the metric considered is C1 class everywhere and is a solution of the Einstein field equations in this sense. We explore the consequences of these two assumptions, and in doing so we derive the general conditions that constrain the jumps in the second partial derivatives across the boundary. The example of the Oppenheimer-Snyder metric is considered and some new results are obtained on it. Then, the formalism developed in this exploration is applied to a different situation, i.e., to a given generalization of the Einstein field equations for the case where the partial second derivatives of the metric exist but are not symmetric
Cosmic primordial density fluctuations and Bell inequalities
The temperature measurements, T, of the perturbed cosmic microwave background, performed by the
cosmic background explorer satellite (COBE), are considered. A dichotomist function, f ¼ 1, is defined
such that f ¼ þ1 if δT > 0 and f ¼ −1 if δT < 0, where δT stands for the fluctuation of T. Then, it is
assumed that behind the appearance of these fluctuations there is local realism. Under this assumption,
some specific Clauser-Horne-Shimony-Holt (CHSH) inequalities are proved for these fluctuation temperatures
measured by COBE in the different sky directions. The calculation of these inequalities from the
actual temperature measurements shows that these inequalities are not violated. This result cannot be
anticipated by calculating the commutators of the cosmic density quantum operators. This must be
remarked here since, in the case of a system of two entangled spin 1
2 particles, its CHSH inequalities
violation can be inferred from the nonvanishing value of the corresponding spin measurement
commutators. The above nonviolation of the observed cosmic CHSH inequalities is compatible with
the existence of local realism behind the cosmic measurement results. Nevertheless, assuming again local
realism, some new cosmic CHSH inequalities can be derived for the case of the WMAP measurements
whose accuracy is better than the one of the above considered COBE measurements. More specifically, in
the WMAP case, some significant cross correlations between the temperature and polarization maps are
detected, and the new cosmic CHSH inequalities are the ones built with these cross correlations. Now, the
occasional violation of these CHSH inequalities would mean the failure of the assumed local realism in
accordance with the quantum origin of the primordial temperature and polarization fluctuations in the
framework of standard inflation
Stability of the intrinsic energy vanishing in the Schwarzschild metric under slow rotation
The linearized Kerr metric is considered and put in some Gauss coordinates which are further intrinsic ones. The linear and angular 4-momenta of this metric are calculated in these coordinates and the resulting value is just zero. Thus, the global vanishing previously found for the Schwarzschild metric remains linearly stable under slow rotational perturbations of this metric