Related papers: Mathematical Issues in a Fully-Constrained Formula…
We will present a complete set of equations, in the form of an Einstein-Bianchi system, that describe the evolution of generic smooth lattices in spacetime. All 20 independent Riemann curvatures will be evolved in parallel with the…
Numerical relativity is finally approaching a state where the evolution of rather general (3+1)-dimensional data sets can be computed in order to solve the Einstein equations. After a general introduction, three topics of current interest…
Uniqueness problems in the elliptic sector of constrained formulations of Einstein equations have a dramatic effect on the physical validity of some numerical solutions, for instance when calculating the spacetime of very compact stars or…
Excision techniques are used in order to deal with black holes in numerical simulations of Einstein equations and consist in removing a topological sphere containing the physical singularity from the numerical domain, applying instead…
The Einstein equations have proven surprisingly difficult to solve numerically. A standard diagnostic of the problems which plague the field is the failure of computational schemes to satisfy the constraints, which are known to be…
In hyperbolic reductions of the Einstein equations the evolution of gauge conditions or constraint quantities is controlled by subsidiary systems. We point out a class of non-linearities in these systems which may have the potential of…
For axially symmetric solutions of Einstein equations there exists a gauge which has the remarkable property that the total mass can be written as a conserved, positive definite, integral on the spacelike slices. The mass integral provides…
$[n+1]$-dimensional ($n\geq 3$) smooth Einsteinian spaces of Euclidean and Lorentzian signature are considered. The base manifold $M$ is supposed to be smoothly foliated by a two-parameter family of codimension-two-surfaces which are…
We present an analysis of well-posedness of constrained evolution of 3+1 formulations of GR. In this analysis we explicitly take into account the energy and momentum constraints as well as possible algebraic constraints on the evolution of…
Hyperbolic formulations of the equations of motion are essential technique for proving the well-posedness of the Cauchy problem of a system, and are also helpful for implementing stable long time evolution in numerical applications. We,…
The numerical evolution of Einstein's field equations in a generic background has the potential to answer a variety of important questions in physics: from applications to the gauge-gravity duality, to modelling black hole production in TeV…
We present a set of inner boundary conditions for the numerical construction of dynamical black hole space-times, when employing a 3+1 constrained evolution scheme and an excision technique. These inner boundary conditions are heuristically…
In this work numerical methods for solving Einstein's equations are developed and applied to the study of inhomogeneous cosmological models. A two-dimensional computer code is described which implements two advanced numerical methods:…
A general covariant extension of Einstein's field equations is considered with a view to Numerical Relativity applications. The basic variables are taken to be the metric tensor and an additional four-vector. The extended field equations,…
Shibata, Ury\=u and Friedman recently suggested a new decomposition of Einstein's equations that is useful for constructing initial data. In contrast to previous decompositions, the conformal metric is no longer treated as a…
We analyse the mathematical underpinnings of a mixed hyperbolic-elliptic form of the Einstein equations of motion in which the lapse function is determined by specified mean curvature and the shift is arbitrary. We also examine a new…
The tetrad-based equations for vacuum gravity published by Estabrook, Robinson, and Wahlquist are simplified and adapted for numerical relativity. We show that the evolution equations as partial differential equations for the Ricci rotation…
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…
We review in a pedagogical fashion the 3+1-split which serves to put Einstein's equations into the form of a dynamical system with constraints. We then discuss the constraint equations under the simplifying assumption of time-symmetry.…
Various aspects of the Cauchy problem for the Einstein equations are surveyed, with the emphasis on local solutions of the evolution equations. Particular attention is payed to giving a clear explanation of conceptual issues which arise in…