Fast and Slow solutions in General Relativity: The Initialization Procedure

We apply recent results in the theory of PDE, specifically in problems with two different time scales, on Einstein's equations near their Newtonian limit. The results imply a justification to Postnewtonian approximations when initialization procedures to different orders are made on the initial data. We determine up to what order initialization is needed in order to detect the contribution to the quadrupole moment due to the slow motion of a massive body as distinct from initial data contributions to fast solutions and prove that such initialization is compatible with the constraint equations. Using the results mentioned the first Postnewtonian equations and their solutions in terms of Green functions are presented in order to indicate how to proceed in calculations with this approach.Comment: 14 pages, Late

The Initial-Boundary Value Problem in General Relativity

In this article we summarize what is known about the initial-boundary value problem for general relativity and discuss present problems related to it.Comment: 11 pages, 2 figures. Contribution to a special volume for Mario Castagnino's seventy fifth birthda

Strongly hyperbolic second order Einstein's evolution equations

BSSN-type evolution equations are discussed. The name refers to the Baumgarte, Shapiro, Shibata, and Nakamura version of the Einstein evolution equations, without introducing the conformal-traceless decomposition but keeping the three connection functions and including a densitized lapse. It is proved that a pseudo-differential first order reduction of these equations is strongly hyperbolic. In the same way, densitized Arnowitt-Deser-Misner evolution equations are found to be weakly hyperbolic. In both cases, the positive densitized lapse function and the spacelike shift vector are arbitrary given fields. This first order pseudodifferential reduction adds no extra equations to the system and so no extra constraints.Comment: LaTeX, 16 pages, uses revtex4. Referee corections and new appendix added. English grammar improved; typos correcte

Geometrically motivated hyperbolic coordinate conditions for numerical relativity: Analysis, issues and implementations

We study the implications of adopting hyperbolic driver coordinate conditions motivated by geometrical considerations. In particular, conditions that minimize the rate of change of the metric variables. We analyze the properties of the resulting system of equations and their effect when implementing excision techniques. We find that commonly used coordinate conditions lead to a characteristic structure at the excision surface where some modes are not of outflow-type with respect to any excision boundary chosen inside the horizon. Thus, boundary conditions are required for these modes. Unfortunately, the specification of these conditions is a delicate issue as the outflow modes involve both gauge and main variables. As an alternative to these driver equations, we examine conditions derived from extremizing a scalar constructed from Killing's equation and present specific numerical examples.Comment: 9 figure

Einstein's equations in Ashtekar's variables constitute a symmetric hyperbolic system

We show that the 3+1 vacuum Einstein field equations in Ashtekar's variables constitutes a first order symmetric hyperbolic system for arbitrary but fixed lapse and shift fields, by suitable adding to the system terms proportional to the constraint equations.Comment: 4 pages, revte

Numerical stability of a new conformal-traceless 3+1 formulation of the Einstein equation

There is strong evidence indicating that the particular form used to recast the Einstein equation as a 3+1 set of evolution equations has a fundamental impact on the stability properties of numerical evolutions involving black holes and/or neutron stars. Presently, the longest lived evolutions have been obtained using a parametrized hyperbolic system developed by Kidder, Scheel and Teukolsky or a conformal-traceless system introduced by Baumgarte, Shapiro, Shibata and Nakamura. We present a new conformal-traceless system. While this new system has some elements in common with the Baumgarte-Shapiro-Shibata-Nakamura system, it differs in both the type of conformal transformations and how the non-linear terms involving the extrinsic curvature are handled. We show results from 3D numerical evolutions of a single, non-rotating black hole in which we demonstrate that this new system yields a significant improvement in the life-time of the simulations.Comment: 7 pages, 2 figure

Exponential Decay for Small Non-Linear Perturbations of Expanding Flat Homogeneous Cosmologies

It is shown that during expanding phases of flat homogeneous cosmologies all small enough non-linear perturbations decay exponentially. This result holds for a large class of perfect fluid equations of state, but notably not for very ``stiff'' fluids as the pure radiation case