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With the goal of taking a step toward the construction of astrophysically realistic initial data for numerical simulations of black holes, we for the first time derive a family of fully general relativistic initial data based on…
We construct examples of black hole formation from regular, one-ended asymptotically flat Cauchy data for the Einstein-Maxwell-charged scalar field system in spherical symmetry which are exactly isometric to extremal Reissner-Nordstr\"om…
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…
This work consists of two distinct parts. In the first part we present a new method for solving the initial value problem of general relativity. Given any spatial metric with a surface orthogonal Killing field and two freely specified…
We discuss the existence of asymptotically Euclidean initial data sets to the vacuum Einstein field equations which would give rise (modulo an existence result for the evolution equations near spatial infinity) to developments with a past…
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…
Construction of binary black hole initial data is a prerequisite for numerical evolutions of binary black holes. This paper reports improvements to the binary black hole initial data solver in the Spectral Einstein Code, to allow robust…
We begin a study of a multi-parameter family of Cauchy initial-value problems for the modified nonlinear Schr\"odinger equation, analyzing the solution in the semiclassical limit. We use the inverse scattering transform for this equation,…
In this paper we study the spherically symmetric characteristic initial data problem for the Einstein-Maxwell-scalar field system with a positive cosmological constant in the interior of a black hole, assuming an exponential Price law along…
The conformal formulation of the Einstein constraint equations is first reviewed, and we then consider the design, analysis, and implementation of adaptive multilevel finite element-type numerical methods for the resulting coupled nonlinear…
Initial-boundary value problems for integrable nonlinear partial differential equations have become tractable in recent years due to the development of so-called unified transform techniques. The main obstruction to applying these methods…
We formulate an initial boundary value problem (IBVP) for the vacuum Einstein equations by describing the boundary conditions of a spacetime metric in its associated gauge. This gauge is determined, equivariantly with respect to…
A classical problem in general relativity is the Cauchy problem for the linearised Einstein equation (the initial value problem for gravitational waves) on a globally hyperbolic vacuum spacetime. A well-known result is that it is uniquely…
In this work, we introduce a spectral-infinite element method for solving Einstein's constraint equations in hyperbolic form. As an application of this, we use this method for computing asymptotically flat perturbations of a Kerr black hole…
For an arbitrary strong, spherically symmetric super-horizon curvature perturbation, we present analytical solutions of the Einstein equations in terms of asymptotic expansion over the ratio of the Hubble radius to the length-scale of the…
An algebraic-hyperbolic method for solving the Hamiltonian and momentum constraints has recently been shown to be well posed for general nonlinear perturbations of the initial data for a Schwarzschild black hole. This is a new approach to…
We solve for single distorted black hole initial data using the puncture method, where the Hamiltonian constraint is written as an elliptic equation in R^3 for the nonsingular part of the metric conformal factor. With this approach we can…
We show how to prescribe the initial data of a characteristic problem satisfying the costraints, the smallness, the regularity and the asymptotic decay suitable to prove a global existence result. In this paper, the first of two, we show in…
The existence of the initial value constraints means that specifying initial data for the Einstein equations is non-trivial. The standard method of constructing initial data in the asymptotically flat case is to choose an asymptotically…
The purpose of this work is to construct asymptotically flat, time symmetric initial data with an apparent horizon of prescribed intrinsic and extrinsic geometry. To do this, we use the parabolic partial differential equation for…