Related papers: The initial value problem as it relates to numeric…
This article, written to appear as a chapter in "The Springer Handbook of Spacetime", is a review of the initial value problem for Einstein's gravitational field theory in general relativity. Designed to be accessible to graduate students…
We discuss the initial value problem of general relativity in its recently unified Lagrangian and Hamiltonian pictures and present a multi-domain pseudo-spectral collocation method to solve the resulting coupled nonlinear partial…
The conformal method for constructing initial data for Einstein's equations is presented in both the Hamiltonian and Lagrangian picture (extrinsic curvature decomposition and conformal thin sandwich formalism, respectively), and advantages…
The initial-value problem is posed by giving a conformal three-metric on each of two nearby spacelike hypersurfaces, their proper-time separation up to a multiplier to be determined, and the mean (extrinsic) curvature of one slice. The…
Initial data are the starting point for any numerical simulation. In the case of numerical relativity, Einstein's equations constrain our choices of these initial data. We will examine several of the formalisms used for specifying Cauchy…
In certain models of conformal gravity, the propagation of gravitational waves is governed by a fourth order scalar partial differential equation. We study the initial value problem for a generalization of this equation, and derive a…
We present a new numerical scheme to solve the initial value problem for black hole-neutron star binaries. This method takes advantage of the flexibility and fast convergence of a multidomain spectral representation of the initial data to…
The initial value problem is introduced after a thorough review of the essential geometry. The initial value equations are put into elliptic form using both conformal transformations and a treatment of the extrinsic curvature introduced…
The Einstein constraint equations describe the space of initial data for the evolution equations, dictating how space should curve within spacetime. Under certain assumptions, the constraints reduce to a scalar quasilinear parabolic…
We investigate the existence and the global causal structure of plane symmetric spacetimes with weak regularity when the matter consists of an irrotational perfect fluid with pressure equal to its mass-energy density. Our theory encompasses…
The initial value problem of scalar-tensor theories of gravity (STT) is analyzed in the physical (Jordan) frame using a 3+1 decomposition of spacetime. A first order strongly hyperbolic system is obtained for which the well posedness of the…
Construction of astrophysically realistic initial data remains a central problem when modelling the merger and eventual coalescence of binary black holes in numerical relativity. The objective of this paper is to provide astrophysically…
We derive, in 3+1 spacetime dimensions, two alternative systems of quasi-linear wave equations, based on Friedrich's conformal field equations. We analyse their equivalence to Einstein's vacuum field equations when appropriate constraint…
We present a new approach to the problem of binary black holes in the pre-coalescence stage, i.e. when the notion of orbit has still some meaning. Contrary to previous numerical treatments which are based on the initial value formulation of…
Many evolution problems in physics are described by partial differential equations on an infinite domain; therefore, one is interested in the solutions to such problems for a given initial dataset. A prominent example is the binary black…
We present a new scheme for constructing initial data for the Einstein field equations using the conformal thin-sandwich formulation that does not assume conformal flatness or approximate Killing vectors. This includes a method for…
We present a numerical scheme for determining hyperboloidal initial data sets for the conformal field equations by using pseudo-spectral methods. This problem is split into two parts. The first step is the determination of a suitable…
We compare the results of constructing binary black hole initial data with three different decompositions of the constraint equations of general relativity. For each decomposition we compute the initial data using a superposition of two…
Given a time symmetric initial data set for the vacuum Einstein field equations which is conformally flat near infinity, it is shown that the solutions to the regular finite initial value problem at spatial infinity extend smoothly through…
Numerical relativity codes now being developed will evolve initial data representing colliding black holes at a relatively late stage in the collision. The choice of initial data used for code development has been made on the basis of…