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Recent results on solutions of the Einstein equations with matter are surveyed and a number of open questions are stated. The first group of results presented concern asymptotically flat spacetimes, both stationary and dynamical. Then there…
In this study, we consider the numerical solution of the Neumann initial boundary value problem for the wave equation in 2D domains. Employing the Laguerre transform with respect to the temporal variable, we effectively transform this…
Many applications in science call for the numerical simulation of systems on manifolds with spherical topology. Through use of integer spin weighted spherical harmonics we present a method which allows for the implementation of arbitrary…
In Einstein's general relativity, with its nonlinear field equations, the discoveries and analyzes of various specific explicit solutions made a great impact on understanding many of the unforeseen features of the theory. Some solutions…
The drift method, introduced by the second author, provides a new formulation of the Einstein constraint equations, either in vacuum or with matter fields. The natural of the geometry underlying this method compensates for its slightly…
We study modified $f(R,(\nabla R)^2,\square R)$ gravity and show in detail how it can be reduced to Einstein gravity with a few scalar fields and then represented in the form of chiral self-gravitating model of the special type. In further…
From Einstein's theory we know that besides the electromagnetic spectrum, objects like quasars, active galactic nuclei, pulsars and black holes also generate a physical signal of purely gravitational nature. The actual form of the signal is…
A common approach for the numerical simulation of wave propagation on a spatially unbounded domain is to truncate the domain via an artificial boundary, thus forming a finite computational domain with an outer boundary. Absorbing boundary…
The hypothesis of an alternative way of obtaining gravitational waves is the physical motivation of this article. Using the linear field approximation and a symmetry transformation of the affine connection, new field equations and new gauge…
Finite difference method and pseudo-spectral method have been widely used in the numerical relativity to solve the Einstein equations. As the third major category method to solve partial differential equations, finite element method is much…
Numerical relativity describes a discrete initial value problem for general relativity. A choice of gauge involves slicing space-time into space-like hypersurfaces. This introduces past and future gauge relative to the hypersurface of…
The spinorial version of the conformal vacuum Einstein field equations are used to construct a system of quasilinear wave equations for the various conformal fields. As a part of the analysis we also show how to construct a subsidiary…
An unsteady problem is considered for a space-fractional diffusion equation in a bounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary…
The aim of this work is to obtain new analitical solutions for Einstein equations in the anisotropical domain. This will be done via the minimal geometric deformation (MGD) approach, which is a simple and systematical method that allow us…
We discuss various properties of the conformal field equations and their consequences for the asymptotic structure of space-times.
Numerical relativity is the most promising tool for theoretically modeling the inspiral and coalescence of neutron star and black hole binaries, which, in turn, are among the most promising sources of gravitational radiation for future…
We develop a variational technique for some wide classes of nonlinear evolutions. The novelty here is that we derive the main information directly from the corresponding Euler-Lagrange equations. In particular, we prove that not only the…
We study some exact and approximate solutions of Einstein's equations that can be used to describe the gravitational field of astrophysical compact objects in the limiting case of slow rotation and slight deformation. First, we show that…
We adapt Luk's analysis of the characteristic initial value problem in General Relativity to the asymptotic characteristic problem for the conformal Einstein field equations to demonstrate the local existence of solutions in a neighbourhood…
We present two new solutions to Einstein's equations in (1+5)-spacetime with a positive bulk cosmological constant. One solution has increasing and another solution decreasing bounded scale function without singularities in the range from…