Related papers: Geometrical Well Posed Systems for the Einstein Eq…
We show that Einstein equations are compatible with the presence of massive point particle idealization and find the corresponding two parameter family of solutions. They are complete defined by the bare mechanical mass $M>0$ and the…
We reduce the equations governing the spherically symmetric perturbations of static spherically symmetric solutions of the Einstein-Vlasov system (with either massive or massless particles) to a single stratified wave equation…
Einstein's equations for general relativity, when viewed as a dynamical system for evolving initial data, have a serious flaw: they cannot be proven to be well-posed (except in special coordinates). That is, they do not produce unique…
This article is the second of two in which we develop a geometric framework for analysing silent and anisotropic big bang singularities. In the present article, we record geometric conclusions obtained by combining the geometric framework…
The classification of certain class of static solutions for the Einstein-Gauss-Bonnet theory in vacuum is presented. The spacelike section of the class of metrics under consideration is a warped product of the real line with a nontrivial…
We show that in an L-annularly linearly connected, N-doubling, complete metric space, any n points lie on a K-quasi-circle, where K depends only on L, N and n. This implies, for example, that if G is a hyperbolic group that does not split…
The generalized harmonic representation of Einstein's equation is manifestly hyperbolic for a large class of gauge conditions. Unfortunately most of the useful gauges developed over the past several decades by the numerical relativity…
We test the validity of Isaacson's formula which states that high frequency and low amplitude gravitational waves behave as a radiation fluid on average. For this purpose, we numerically construct a solution of the vacuum Einstein equations…
Assuming as starting point the validity of the Einstein-Rosen metric, we study the hyperbolic system of P.D.E. to which the Einstein's field's equations can be reduced. We prove using the implicit function theorem in Banach spaces, the…
Given a smooth globally hyperbolic $(3+1)$-dimensional spacetime $(M,g)$ satisfying the Einstein vacuum equations (possibly with cosmological constant) and an inextendible timelike geodesic $\mathcal{C}$, we constructed in Part I a family…
In this paper we construct negatively curved Einstein spaces describing gravitational waves having a solvegeometry wave-front (i.e., the wave-fronts are solvable Lie groups equipped with a left-invariant metric). Using the Einstein…
Vacuum solutions to the Einstein equations can be viewed as the interplay between the geometry and the gravitational wave energy content. The constraints on initial data reflect this interaction. We assume we are looking at cosmological…
The principle part of Einstein equations in the harmonic gauge consists of a constrained system of 10 curved space wave equations for the components of the space-time metric. A well-posed initial boundary value problem based upon a new…
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 investigate how the accuracy and stability of numerical relativity simulations of 1D colliding plane waves depends on choices of equation formulations, gauge conditions, boundary conditions, and numerical methods, all in the context of a…
In this paper, we establish a relation between two seemingly unrelated concepts for solving first-order hyperbolic quasilinear systems of partial differential equations in many dimensions. These concepts are based on a variant of the…
In the harmonic description of general relativity, the principle part of Einstein equations reduces to a constrained system of 10 curved space wave equations for the components of the space-time metric. We use the pseudo-differential theory…
We establish a variant, which has the advantage of introducing only physical characteristics, of the symmetric quasi linear first order system given by H.\ Friedrich for the evolution equations of gravitating fluid bodies in General…
Spherically symmetric solutions of the SU(N) Einstein-Yang-Mills-Higgs system are constructed using the harmonic map ansatz. The problem reduces to solving a set of ordinary differential equations for the appropriate profile functions. In…
Most approaches towards a quantum theory of gravitation indicate the existence of a minimal length scale of the order of the Planck length. Quantum mechanical models incorporating such an intrinsic length scale call for a deformation of…