Piotr T. Chrusciel

Existence of non-trivial, vacuum, asymptotically simple spacetimes

We construct non-trivial vacuum space-times with a global Scri. The construction proceeds by proving extension results across compact boundaries for initial data sets, adapting the gluing arguments of Corvino and Schoen. Another application of the extension results is existence of initial data which are exactly Schwarzschild both near infinity and near each of the connected component of the apparent horizon.A method is described of programming a memory array on a single integrated circuit so that a portion of each data word is characterized as CAM, with the remaining portion of each data word functioning as RAM. The programmable memory array is partitioned into CAM and RAM subfields by disabling the comparators in each memory cell in selected columns of CAM cells to create RAM-functioning cells. Said partitioning may be re-programmed to enable the comparators in said RAM-functioning cells to be re-enabled, so that said cells may participate in subsequent comparisons to a search word. The described memory array permits direct retrieval and storage of associated information in RAM-functioning cells corresponding to data words which are determined to match a given search word. This direct retrieval and storage process can efficiently be utilized without computing or decoding an address for the associated information.

Gravitational waves in general relativity XIV. Bondi expansions and the ‘polyhomogeneity’ of ℐ

The structure of polyhomogeneous space-times (i.e. space-times with metrics which admit an expansion in terms of r-j logi 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 ℐ. The validity in this more general context of various results from the standard treatment of ℐ, including the Bondi mass loss formula, the peeling-off of the Riemann tensor and the Newman-Penrose constants of motion, is considered.

On the topology of stationary black holes

We prove that the domain of outer communication of a stationary, globally hyperbolic spacetime satisfying the null energy condition must be simply connected. Under suitable additional hypotheses, this implies, in particular, that each connected component of a cross section of the event horizon of a stationary black hole must have spherical topology.

Non-trivial, static, geodesically complete, vacuum space-times with a negative cosmological constant

We construct a large class of new singularity-free static lorentzian four-dimensional solutions of the vacuum Einstein equations with a negative cosmological constant. The new families of metrics contain space-times with, or without, black hole regions. Two uniqueness results are also established.

Maximal hypersurfaces in stationary asymptotically flat spacetimes

Existence of maximal hypersurfaces and of foliations by maximal hypersurfaces is proven in two classes of asymptotically flat spacetimes which possess a one parameter group of isometries whose orbits are timelike “near infinity.”. The first class consists of strongly causal asymptotically flat spacetimes which contain no “black hole or white hole” (but may contain ”ergoregions” where the Killing orbits fail to be timelike). The second class of spacetimes possess a black hole and a white hole, with the black and white hole horizons intersecting in a compact 2-surfaceS.

On the global structure of Robinson-Trautman space-times

The structure of polyhomogeneous space–times (i.e., space–times with metrics which admit an expansion in terms of r log 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.

THE CLASSIFICATION OF STATIC VACUUM SPACETIMES CONTAINING AN ASYMPTOTICALLY FLAT SPACELIKE HYPERSURFACE WITH COMPACT INTERIOR

We prove non-existence of static, vacuum, appropriately regular, asymptotically flat black hole spacetimes with degenerate (not necessarily connected) components of the event horizon. This finishes the classification of static, vacuum, asymptotically flat domains of outer communication in an appropriate class of spacetimes, showing that the domains of outer communication of the Schwarzschild black holes exhaust the space of appropriately regular black hole exteriors.

Global solutions of the Einstein-Maxwell equations in higher dimensions

We consider the Einstein–Maxwell equations in space-dimension n. We point out that the Lindblad–Rodnianski stability proof applies to those equations whatever the space-dimension n ≥ 3. In even spacetime dimension n + 1 ≥ 6, we use the standard conformal method on a Minkowski background to give a simple proof that the maximal globally hyperbolic development of initial data sets which are sufficiently close to the data for Minkowski spacetime and which are Schwarzschildian outside of a compact set lead to geodesically complete spacetimes, with a complete Scri, with a smooth conformal structure, and with the gravitational field approaching the Minkowski metric along null directions at least as fast as r−(n−1)/2.

Hyperboloidal Cauchy data for vacuum Einstein equations and obstructions to smoothness of null infinity.

Various works have suggested that the Bondi-Sachs-Penrose decay conditions on the gravitational field at null infinity are not generally representative of asymptotically flat spacetimes. We have made a detailed analysis of the constraint equations for «asymptotically hyperboloidal» initial data and find that log terms arise generically in asymptotic expansions. These terms are absent in the corresponding Bondi-Sachs-Penrose expansions, and can be related to explicit geometric quantities. We have nevertheless shown that there exists a large class of «nongeneric» solutions of the constraint equations, the evolution of which leads to spacetimes satisfying the Bondi-Sachs-Penrose smoothness conditions

The Hamiltonian mass of asymptotically anti-de Sitter space-times

We give a definition of mass for spacelike hypersurfaces in space-times with metrics which are asymptotic to the anti-de Sitter one, or to a class of generalizations thereof. We present the results of gr-qc/0110014 which show that our definition provides a geometric invariant for a spacelike hypersurface embedded in a space-time. Some further global invariants are also given.We give a Hamiltonian definition of mass for spacelike hypersurfaces in spacetimes with metrics which are asymptotic to the anti-de Sitter one, or to a class of