Related papers: Cauchy-Characteristic Matching In General Relativi…
After a short review of prominent properties of gravitational waves and the newly born gravitational astronomy, we focus on theoretical aspects. Analytic approximation methods in general relativity have played a crucial role in the recent…
We study fine global properties of nonnegative solutions to the Cauchy Problem for the fast $p$-Laplacian evolution equation $u_t=\Delta_p u$ on the whole Euclidean space, in the so-called "good fast diffusion range" $\tfrac{2N}{N+1}<p<2$.…
A system of equations consisting of an infinite string coupled to a nonlinear oscillator is considered. The Cauchy problem for the system with the periodic initial data is studied. The main goal is to prove the convergence of the solutions…
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…
Utilizing various gauges of the radial coordinate, we give a General Relativistic (GR) description of static spherically symmetric spacetimes with a massive point source and vacuum outside this singularity. We show that in GR there exists a…
This revision includes clarified exposition and simplified analysis. Solutions of the Einstein equations which are periodic and have standing gravitational waves are valuable approximations to more physically realistic solutions with…
A new numerical scheme to solve the Einstein field equations based upon the generalized harmonic decomposition of the Ricci tensor is introduced. The source functions driving the wave equations that define generalized harmonic coordinates…
Recently, Choquet-Bruhat and York and Abrahams, Anderson, Choquet-Bruhat, and York (AACY) have cast the 3+1 evolution equations of general relativity in gauge-covariant and causal ``first-order symmetric hyperbolic form,'' thereby cleanly…
Spacetime is foliated by spatial hypersurfaces in the 3+1 split of General Relativity. The initial value problem then consists of specifying initial data for all relevant fields on one such a spatial hypersurface. These fields are the…
We show how to solve the Cauchy problem for the Einstein-Vlasov equations with geometric initial data on a light-cone.
Inspiralling and coalescing binary black holes are promising sources of gravitational radiation. The orbital motion and gravitational-wave emission of such system can be modelled using a variety of approximation schemes and numerical…
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…
We consider the strategy of realizing the solution of a Cauchy problem with radial data as a limit of radial solutions to initial-boundary value problems posed on the exterior of vanishing balls centered at the origin. The goal is to gauge…
In this thesis we consider several aspects of general relativity relating to exact solutions of the Einstein equations. In the first part gravitational plane waves in the Rosen form are investigated, and we develop a formalism for writing…
We study the plane-symmetric collision of two gravitational waves and describe the global spacetime geometry generated by this collision. To this end, we formulate the characteristic initial value problem for the Einstein equations, when…
It is argued that, in order for the gravitational field to be propagated as a wave, it is necessary for it to satisfy a further set of field equations, in addition to those of Einstein and Hilbert, and these equations mean there is a…
In the mathematical physics literature, there are heuristic arguments, going back three decades, suggesting that for an open set of initially smooth solutions to the Einstein-vacuum equations in high dimensions, stable, approximately…
The Einstein equations are non-linear and the particles of which the gravitational effect is described by these equations are lastly unknown. If renormalizable fields are assumed, then results are obtained only in the case of a at space.…
The use of the sine-Gordon equation as a model of magnetic flux propagation in Josephson junctions motivates studying the initial-value problem for this equation in the semiclassical limit in which the dispersion parameter $\e$ tends to…
For a one-dimensional mildly quasilinear wave equation given in the upper half-plane, we consider the Cauchy problem. The initial conditions have discontinuity of the first kind at one point. We construct the solution using the method of…