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The global characteristic initial value problem for linear wave equations on globally hyperbolic Lorentzian manifolds is examined, for a class of smooth initial value hypersurfaces satisfying favourable global properties. First it is shown…
We review the Cauchy problem for f(R) theories of gravity, in metric and metric-affine for- mulations, pointing out analogies and differences with respect to General Relativity. The role of conformal transformations, effective scalar fields…
In this pedagogically structured article, we describe a generalized harmonic formulation of the Einstein equations in spherical symmetry which is regular at the origin. The generalized harmonic approach has attracted significant attention…
A new class of solutions of the Einstein field equations in spherical symmetry is found. The new solutions are mathematically described as the metrics admitting separation of variables in area-radius coordinates. Physically, they describe…
A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse problems aim at solving an elliptic partial differential…
This article is a guide to the literature on existence theorems for the Einstein equations which also draws attention to open problems in the field. The local in time Cauchy problem, which is relatively well understood, is treated first.…
In recent years, the Characteristic formulation of numerical relativity has found increasing use in the extraction of gravitational radiation from numerically generated spacetimes. In this paper, we formulate the Characteristic initial…
Utilizing various gauges of the radial coordinate we give a description of static spherically symmetric space-times with point singularity at the center and vacuum outside the singularity. We show that in general relativity (GR) there exist…
The analysis of a general multibody physical system governed by Einstein's equations in quite difficult, even if numerical methods (on a computer) are used. Some of the difficulties -- many coupled degrees of freedom, dynamic instability --…
The dynamics of gravitational waves is investigated in full 3+1 dimensional numerical relativity, emphasizing the difficulties that one might encounter in numerical evolutions, particularly those arising from non-linearities and gauge…
In this paper we shall analyse the structure of the Cauchy Problem (CP briefly) for General Relativity (GR briefly) by applying the theory of first order symmetric hyperbolic systems.
The completion of a network of advanced laser-interferometric gravitational-wave observatories around 2001 will make possible the study of the inspiral and coalescence of binary systems of compact objects (neutron stars and black holes),…
The system consisting of a self gravitating perfect fluid and scalar field is considered in detail. The scalar fields considered are the quintessence and ``tachyonic'' forms which have important application in cosmology. Mathematical…
Gravitational waves radiated during binary black hole coalescence is a perfect probe for studying the characteristics of strong gravity. Advanced techniques for creating numerical relativity substitute models for eccentric binary black hole…
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,…
The asymptotic behavior of geometry near the boundary of maximal Cauchy development is studied using a perturbative method, which at the zeroth order reduces Einstein's equations to an exactly solvable set of equations---Einstein's…
Theoretical arguments and cosmological observations suggest that Einstein's theory of general relativity needs to be modified at high energies. One of the best motivated higher-curvature extensions of general relativity is…
Motivated by the strong cosmic censorship conjecture, we study the linear scalar wave equation in the interior of subextremal strictly charged Reissner-Nordstr\"om black holes by analyzing a suitably-defined "scattering map" at $0$…
We investigate a simple variation of the Generalized Harmonic method for evolving the Einstein equations. A flat space wave equation for metric perturbations is separated from the Ricci tensor, with the rest of the Ricci tensor becoming a…
The Cauchy problem is considered for the scalar wave equation in the Schwarzschild geometry. Using an integral spectral representation we derive the exact decay rate for solutions of the Cauchy problem with spherical symmetric initial data,…