scholarly journals Solving Einstein's equations for rotating spacetimes: Evolution of relativistic star clusters

1994 ◽  
Vol 49 (10) ◽  
pp. 5153-5164 ◽  
Author(s):  
Andrew M. Abrahams ◽  
Gregory B. Cook ◽  
Stuart L. Shapiro ◽  
Saul A. Teukolsky
Keyword(s):  
Star Clusters ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
Relativistic Star

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.

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.

scholarly journals LOCAL COSMIC STRING AND C-FIELD

2006 ◽  
Vol 21 (18) ◽  
pp. 3727-3732 ◽  
Author(s):  
F. RAHAMAN ◽  
R. MONDAL ◽  
M. KALAM
Keyword(s):  
Cosmic String ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Einstein’S Equations ◽  
Einstein's Equations

We investigate a local cosmic string with a phenomenological energy–momentum tensor as prescribed by Vilenkin, in the presence of C-field. The solutions of full nonlinear Einstein's equations for exterior and interior regions of such a string are presented.

Sign in / Sign up

Export Citation Format

Share Document