Related papers: Solving Einstein's Equation Numerically on Manifol…
This paper develops a method for solving Einstein's equation numerically on multi-cube representations of manifolds with arbitrary spatial topologies. This method is designed to provide a set of flexible, easy to use computational…
Numerical solutions to the Einstein constraint equations are constructed on a selection of compact orientable three-dimensional manifolds with non-trivial topologies. A simple constant mean curvature solution and a somewhat more complicated…
This paper initiates the study of the Einstein equation on homogeneous supermanifolds. First, we produce explicit curvature formulas for graded Riemannian metrics on these spaces. Next, we present a construction of homogeneous…
The effectiveness of the hyperbolic relaxation method for solving the Einstein constraint equations numerically is studied here on a variety of compact orientable three-manifolds. Convergent numerical solutions are found using this method…
We construct solutions with prescribed asymptotics to the Einstein constraint equations using a cut-off technique. Moreover, we give various examples of vacuum asymptotically flat manifolds whose center of mass and angular momentum are…
We give a general survey of the solution of the Einstein constraints by the conformal method on n dimensional compact manifolds. We prove some new results about solutions with low regularity (solutions in $H_{2}$ when n=3), and solutions…
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:…
We investigate the Einstein equation with a positive cosmological constant for $4n+4$-dimensional metrics on bundles over Quaternionic K\"ahler base manifolds whose fibers are 4-dimensional Bianchi IX manifolds. The Einstein equations are…
The Ernst equation is formulated on an arbitrary Riemann surface. Analytically, the problem reduces to finding solutions of the ordinary Ernst equation which are periodic along the symmetry axis. The family of (punctured) Riemann surfaces…
The demand to obtain answers to highly complex problems within strong-field gravity has been met with significant progress in the numerical solution of Einstein's equations - along with some spectacular results - in various setups. We…
This article begins with a brief introduction to numerical relativity aimed at readers who have a background in applied mathematics but not necessarily in general relativity. I then introduce and summarise my work on the problem of treating…
In this paper, we investigate the geometry of Einstein-type equation on a Riemannian manifold, unifying various particular geometric structures recently studied in the literature, such as critical point equation and vacuum static equation.…
We study the constraint equations for the Einstein-scalar field system on compact manifolds. Using the conformal method we reformulate these equations as a determined system of nonlinear partial differential equations. By introducing a new…
It is an important problem in differential geometry to find non-naturally reductive homogeneous Einstein metrics on homogeneous manifolds. In this paper, we consider this problem for some coset spaces of compact simple Lie groups. A new…
A nonstatic and circularly symmetric exact solution of the Einstein equations (with a cosmological constant $\Lambda$ and null fluid) in $2+1$ dimensions is given. This is a nonstatic generalization of the uncharged spinless BTZ metric. For…
We study a numerical solution to Einstein's equation for a compact object composed of null particles. The solution avoids quantum scale regimes and hence neither relies upon nor ignores the interaction of quantum mechanics and gravitation.…
We construct exact static inhomogeneous solutions to Einstein's equations with counter flow of particle fluid and a positive cosmological constant by using the Sasaki metrics on three-dimensional spaces. The solutions, which admit an…
The aim of this article is to construct initial data for the Einstein equations on manifolds of the form R n+1 x T m , which are asymptotically flat at infinity, without assuming any symmetry condition in the compact direction. We use the…
We study isometric embeddings of some solutions of the Einstein equations with suffciently high symmetries into a flat ambient space. We briefly describe a method for constructing surfaces with a given symmetry. We discuss all minimal…
We construct new homogeneous Einstein spaces with negative Ricci curvature in two ways: First, we give a method for classifying and constructing a class of rank one Einstein solvmanifolds whose derived algebras are two-step nilpotent. As an…