Gravitational waves in general relativity: XIV. Bondi expansions and the ``polyhomogeneity'' of \Scri

The structure of polyhomogeneous space-times (i.e., space-times with metrics which admit an expansion in terms of $r^{-j}\log^i r$) constructed by a Bondi--Sachs type method is analysed. The occurrence of some log terms in an asymptotic expansion of the metric is related to the non--vanishing of the Weyl tensor at Scri. Various quantities of interest, including the Bondi mass loss formula, the peeling--off of the Riemann tensor and the Newman--Penrose constants of motion are re-examined in this context.Comment: LaTeX, 28pp, CMA-MR14-9

Local and Global Analytic Solutions for a Class of Characteristic Problems of the Einstein Vacuum Equations in the "Double Null Foliation Gauge"

The main goal of this work consists in showing that the analytic solutions for a class of characteristic problems for the Einstein vacuum equations have an existence region larger than the one provided by the Cauchy-Kowalevski theorem due to the intrinsic hyperbolicity of the Einstein equations. To prove this result we first describe a geometric way of writing the vacuum Einstein equations for the characteristic problems we are considering, in a gauge characterized by the introduction of a double null cone foliation of the spacetime. Then we prove that the existence region for the analytic solutions can be extended to a larger region which depends only on the validity of the apriori estimates for the Weyl equations, associated to the "Bel-Robinson norms". In particular if the initial data are sufficiently small we show that the analytic solution is global. Before showing how to extend the existence region we describe the same result in the case of the Burger equation, which, even if much simpler, nevertheless requires analogous logical steps required for the general proof. Due to length of this work, in this paper we mainly concentrate on the definition of the gauge we use and on writing in a "geometric" way the Einstein equations, then we show how the Cauchy-Kowalevski theorem is adapted to the characteristic problem for the Einstein equations and we describe how the existence region can be extended in the case of the Burger equation. Finally we describe the structure of the extension proof in the case of the Einstein equations. The technical parts of this last result is the content of a second paper.Comment: 68 page

Einstein equations in the null quasi-spherical gauge III: numerical algorithms

We describe numerical techniques used in the construction of our 4th order evolution for the full Einstein equations, and assess the accuracy of representative solutions. The code is based on a null gauge with a quasi-spherical radial coordinate, and simulates the interaction of a single black hole with gravitational radiation. Techniques used include spherical harmonic representations, convolution spline interpolation and filtering, and an RK4 "method of lines" evolution. For sample initial data of "intermediate" size (gravitational field with 19% of the black hole mass), the code is accurate to 1 part in 10^5, until null time z=55 when the coordinate condition breaks down.Comment: Latex, 38 pages, 29 figures (360Kb compressed

On the existence of Killing vector fields

In covariant metric theories of coupled gravity-matter systems the necessary and sufficient conditions ensuring the existence of a Killing vector field are investigated. It is shown that the symmetries of initial data sets are preserved by the evolution of hyperbolic systems.Comment: 9 pages, no figure, to appear in Class. Quant. Gra

Ghost points in inverse scattering constructions of stationary Einstein metrics

We prove a removable singularities theorem for stationary Einstein equations, with useful implications for constructions of stationary solutions using soliton methods

Collapse Dynamics of a Star of Dark Matter and Dark Energy

In this work, we study the collapse dynamics of an inhomogeneous spherically symmetric star made of dark matter (DM) and dark energy (DE). The dark matter is taken in the form of a dust cloud while anisotropic fluid is chosen as the candidate for dark energy. It is investigated how dark energy modifies the collapsing process and is examined whether dark energy has any effect on the Cosmic Censorship Conjecture. The collapsing star is assumed to be of finite radius and the space time is divided into three distinct regions $\Sigma$ and $V^{\pm}$, where $\Sigma$ represents the boundary of the star and $V^{-}(V^{+})$ denotes the interior (exterior) of the star. The junction conditions for matching $V^{\pm}$ over $\Sigma$ are specified. Role of Dark energy in the formation of apparent horizon is studied and central singularity is analyzed.Comment: 13 page

Uniqueness Theorem for Static Black Hole Solutions of sigma-models in Higher Dimensions

We prove the uniqueness theorem for self-gravitating non-linear sigma-models in higher dimensional spacetime. Applying the positive mass theorem we show that Schwarzschild-Tagherlini spacetime is the only maximally extended, static asymptotically flat solution with non-rotating regular event horizon with a constant mapping.Comment: 5 peges, Revtex, to be published in Class.Quantum Gra

Uniqueness properties of the Kerr metric

We obtain a geometrical condition on vacuum, stationary, asymptotically flat spacetimes which is necessary and sufficient for the spacetime to be locally isometric to Kerr. Namely, we prove a theorem stating that an asymptotically flat, stationary, vacuum spacetime such that the so-called Killing form is an eigenvector of the self-dual Weyl tensor must be locally isometric to Kerr. Asymptotic flatness is a fundamental hypothesis of the theorem, as we demonstrate by writing down the family of metrics obtained when this requirement is dropped. This result indicates why the Kerr metric plays such an important role in general relativity. It may also be of interest in order to extend the uniqueness theorems of black holes to the non-connected and to the non-analytic case.Comment: 30 pages, LaTeX, submitted to Classical and Quantum Gravit

A spacetime characterization of the Kerr metric

We obtain a characterization of the Kerr metric among stationary, asymptotically flat, vacuum spacetimes, which extends the characterization in terms of the Simon tensor (defined only in the manifold of trajectories) to the whole spacetime. More precisely, we define a three index tensor on any spacetime with a Killing field, which vanishes identically for Kerr and which coincides in the strictly stationary region with the Simon tensor when projected down into the manifold of trajectories. We prove that a stationary asymptotically flat vacuum spacetime with vanishing spacetime Simon tensor is locally isometric to Kerr. A geometrical interpretation of this characterization in terms of the Weyl tensor is also given. Namely, a stationary, asymptotically flat vacuum spacetime such that each principal null direction of the Killing form is a repeated principal null direction of the Weyl tensor is locally isometric to Kerr.Comment: 23 pages, No figures, LaTeX, to appear in Classical and Quantum Gravit

Static self-gravitating many-body systems in Einstein gravity

We consider the problem of constructing static, elastic, many-body systems in Einstein gravity. The solutions constructed are deformations of static many-body configurations in Newtonian gravity. No symmetry assumptions are made.Comment: 15 page