An exact solution of Einstein's equations for two particles falling freely in an external gravitational field

1988 ◽  
Vol 20 (6) ◽  
pp. 607-622 ◽  
Author(s):  
W. B. Bonnor
Keyword(s):  
Exact Solution ◽  
Gravitational Field ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
External Gravitational Field

THE EQUIVALENCE PROBLEM IN TELEPARALLEL GRAVITY

2002 ◽  
Vol 17 (29) ◽  
pp. 4161-4166
Author(s):  
J. FONSECA
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Equivalence Problem ◽  
Teleparallel Gravity ◽  
Alternative Formulation ◽  
Einstein’S Equations ◽  
Field Equations ◽  
Einstein's Equations ◽  
Invariant Description ◽  
Torsion Free

The teleparallel equivalent of general relativity (TEGR) is an alternative formulation of Einstein's equations in the framework of Riemann-Cartan spacetimes. The gravitational field can be described either by the curvature of the torsion-free connection of general relativity (GR) or by the torsion of the curvature-free connection of the TEGR. Both in GR and TEGR the freedom in the choice of coordinates gives rise to the equivalence problem of deciding whether two solutions of the field equations are the same. This problem is solved by means of a invariant description of the gravitational field. We investigate whether the equivalence between GR and TEGR also holds at the level of these invariant descriptions. We show that the GR description assures equivalence in TEGR only in very special situations. These results are illustrated on teleparallel spacetimes with torsion and Gödel metric.

scholarly journals The Localized Energy Distribution of Dark Energy Star Solutions

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Paul Halpern ◽  
Michael Pecorino
Keyword(s):  
Exact Solution ◽  
Dark Energy ◽  
Scalar Field ◽  
Energy Distribution ◽  
Phantom Energy ◽  
Energy Localization ◽  
Einstein’S Equations ◽  
Energy Distributions ◽  
Einstein's Equations ◽  
Energy Star

We examine the question of energy localization for an exact solution of Einstein's equations with a scalar field corresponding to the phantom energy interpretation of dark energy. We apply three different energy-momentum complexes, the Einstein, the Papapetrou, and the Møller prescriptions, to the exterior metric and determine the energy distribution for each. Comparing the results, we find that the three prescriptions yield identical energy distributions.

Black holes, star clusters, and naked singularities: numerical solution of Einstein’s equations

1992 ◽  
Vol 340 (1658) ◽  
pp. 365-390 ◽  
Keyword(s):  
Black Holes ◽  
General Relativity ◽  
Gravitational Field ◽  
Numerical Solution ◽  
Star Clusters ◽  
Particle Simulation ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
Naked Singularities ◽  
Hoop Conjecture

We describe a new method for the numerical solution of Einstein’s equations for the dynamical evolution of a collisionless gas of particles in general relativity. The gravitational field can be arbitrarily strong and particle velocities can approach the speed of light. The computational method uses the tools of numerical relativity and N -body particle simulation to follow the full nonlinear behaviour of these systems. Specifically, we solve the Vlasov equation in general relativity by particle simulation. The gravitational field is integrated by using the 3 + 1 formalism of Arnowitt, Deser and Misner. Physical applications include the stability of relativistic star clusters the binding energy criterion for stability, and the collapse of star clusters to black holes. Astrophysical issues addressed include the possible origin of quasars and active galactic nuclei via the collapse of dense star clusters to supermassive black holes. The method described here also provides a new tool for studying the cosmic censorship hypothesis and the possibility of naked singularities. The formation of a naked singularity during the collapse of a finite object would pose a serious difficulty for the theory of general relativity. The hoop conjecture suggests that this possibility will never happen provided the object is sufficiently compact (≤ M ) in all of its spatial dimensions. But what about the collapse of a long, non-rotating, prolate object to a thin spindle? Such collapse leads to a strong singularity in newtonian gravitation. Using our numerical code to evolve collisionless gas spheroids in full general relativity, we find that in all cases the spheroids collapse to singularities. When the spheroids are sufficiently compact the singularities are hidden inside black holes. However, when the spheroids are sufficiently large there are no apparent horizons. These results lend support to the hoop conjecture and appear to demonstrate that naked singularities can form in asymptotically flat space-times.

GRAVITATIONAL FIELD OF STATIC THIN PLANAR WALLS IN WEAK-FIELD APPROXIMATION

2003 ◽  
Vol 12 (08) ◽  
pp. 1385-1397 ◽  
Author(s):  
L. CAMPANELLI ◽  
P. CEA ◽  
G. L. FOGLI ◽  
L. TEDESCO
Keyword(s):  
Gravitational Field ◽  
Field Approximation ◽  
Weak Field ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
Weak Field Approximation ◽  
Planar Wall

We investigate gravitational properties of thin planar wall solutions of the Einstein's equations in the weak field approximation. We find the general metric solutions and discuss the behavior of a particle placed initially at rest to one side of the plane. Moreover we study the case of non-reflection-symmetric solutions.

A generalized Kerr-Schild solution of Einstein's equations

1974 ◽  
Vol 75 (3) ◽  
pp. 383-390 ◽  
Author(s):  
P. C. Vaidya
Keyword(s):  
Exact Solution ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
Einstein's Equation ◽  
Null Congruence

AbstractA very general exact solution of Einstein's equation Rik = σξiξk, is given in terms of the Kerr-Schild metric, gik = ηij + Hξiξk, where ξi is a shearfree geodetic null congruence. The Kerr-Schild solution for Rik = 0 is derived as a particular case.

COVARIANT CONFORMAL DECOMPOSITION OF EINSTEIN EQUATIONS

2002 ◽  
Vol 17 (20) ◽  
pp. 2762-2762
Author(s):  
E. GOURGOULHON ◽  
J. NOVAK
Keyword(s):  
Numerical Simulations ◽  
Extrinsic Curvature ◽  
Einstein Equations ◽  
Numerical Implementation ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
Tensor Density ◽  
Dirac Gauge ◽  
Ill Posed ◽  
Isolated Systems

It has been shown1,2 that the usual 3+1 form of Einstein's equations may be ill-posed. This result has been previously observed in numerical simulations3,4. We present a 3+1 type formalism inspired by these works to decompose Einstein's equations. This decomposition is motivated by the aim of stable numerical implementation and resolution of the equations. We introduce the conformal 3-"metric" (scaled by the determinant of the usual 3-metric) which is a tensor density of weight -2/3. The Einstein equations are then derived in terms of this "metric", of the conformal extrinsic curvature and in terms of the associated derivative. We also introduce a flat 3-metric (the asymptotic metric for isolated systems) and the associated derivative. Finally, the generalized Dirac gauge (introduced by Smarr and York5) is used in this formalism and some examples of formulation of Einstein's equations are shown.

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