Conformal Structures of Static Vacuum Data

In the Cauchy problem for asymptotically flat vacuum data the solution-jets along the cylinder at space-like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future/past null infinity. The tendency of these singularities to spread along the null generators of null infinity obstructs the development of a smooth conformal structure at null infinity. For the solution-jets arising from time reflection symmetric data to extend smoothly to the critical sets it is necessary that the Cotton tensor of the initial three-metric h satisfies a certain conformally invariant condition (*) at space-like infinity, it is sufficient that h be asymptotically static at space-like infinity. The purpose of this article is to characterize the gap between these conditions. We show that with the class of metrics which satisfy condition (*) on the Cotton tensor and a certain non-degeneracy requirement is associated a one-form κ with conformally invariant differential dκ. We provide two criteria. If h is real analytic, κ is closed, and one of its integrals satisfies a certain equation then h is conformal to static data near space-like infinity. If h is smooth, κ is asymptotically closed, and one of its integrals satisfies a certain equation asymptotically then h is asymptotically conformal to static data at space-like infinity.     

Cauchy problems for the conformal vacuum field equations in general relativity

Cauchy problems for Einstein's conformal vacuum field equations are reduced to Cauchy problems for first order quasilinear symmetric hyperbolic systems. The “hyperboloidal initial value” problem, where Cauchy data are given on a spacelike hypersurface which intersects past null infinity at a spacelike two-surface, is discussed and translated into the conformally related picture. It is shown that for conformal hyperboloidal initial data of classH ^S,s≧4, there is a unique (up to questions of extensibility) development which is a solution of the conformal vacuum field equations of classH ^S. It provides a solution of Einstein's vacuum field equations which has a smooth structure at past null infinity.     

On the existence ofn-geodesically complete or future complete solutions of Einstein's field equations with smooth asymptotic structure

It is demonstrated that initial data sufficiently close to De-Sitter data develop into solutions of Einstein's equations Ric[g]=g with positive cosmological constant , which are asymptotically simple in the past as well as in the future, whence null geodesically complete. Furthermore it is shown that hyperboloidal initial data (describing hypersurfaces which intersect future null infinity in a space-like two-sphere), which are sufficiently close to Minkowskian hyperboloidal data, develop into future asymptotically simple whence null geodesically future complete solutions of Einstein's equations Ric[g]=0, for which future null infinity forms a regular cone with vertexi ⁺ that represents future time-like infinity.     

The second order spin-2 system in flat space near space-like and null-infinity

In previous work, the numerical solution of the linearized gravitational field equations near space-like and null-infinity was discussed in the form of the spin-2 zero-rest-mass equation for the perturbations of the conformal Weyl curvature. The motivation was to study the behavior of the field and properties of the numerical evolution of the system near infinity using Friedrich’s conformal representation of space-like infinity as a cylinder. It has been pointed out by H.O. Kreiss and others that the numerical evolution of a system using second order wave equations has several advantages compared to a system of first order equations. Therefore, in the present paper we derive a system of second order wave equations and prove that the solution spaces of the two systems are the same if appropriate initial and boundary data are given. We study the properties of this system of coupled wave equations in the same geometric setting and discuss the differences between the two approaches.     

On static and radiative space-times

The conformal constraint equations on space-like hypersurfaces are discussed near points which represent either time-like or spatial infinity for an asymptotically flat solution of Einstein's vacuum field equations. In the case of time-like infinity a certain radiativity condition, is derived which must be satisfied by the data at that point. The case of space-like infinity is analysed in detail for static space-times with non-vanishing mass. It is shown that the conformal structure implied here on a slice of constant Killing time, which extends analytically through infinity, satisfies at spatial infinity the radiativity condition. Thus to any static solution exists a certain radiative solution which has a smooth structure at past null infinity and is regular at past time-like infinity. A characterization of these solutions by their free data is given and non-symmetry properties are discussed.     

String theory and cosmological singularities

In general relativity space-like or null singularities are common: they imply that ‘time’ can have a beginning or end. Well-known examples are singularities inside black holes and initial or final singularities in expanding or contracting universes. In recent times, string theory is providing new perspectives of such singularities which may lead to an understanding of these in the standard framework of time evolution in quantum mechanics. In this article, we describe some of these approaches.      

A Rigidity Property of Asymptotically Simple Spacetimes Arising from Conformally Flat Data

Given a time symmetric initial data set for the vacuum Einstein field equations which is conformally flat, it is shown that the solutions to the regular finite initial value problem at spatial infinity extend smoothly through the critical sets where null infinity touches spatial infinity if and only if the initial data coincides with Schwarzschild data.     

On purely radiative space-times

The existence of space-times representing pure gravitational radiation which comes in from infinity and interacts with itself is discussed. They are characterized as solutions of Einstein's vacuum field equations possessing a smooth structure at past null infinity which forms the future null cone at past timelike infinity with complete generators. The pure radiation problem is analysed where free initial data for Einstein's field equations are prescribed on the null cone at past time-like infinity. It is demonstrated how the pure radiation problem can be formulated as a local initial value problem for the symmetric hyperbolic system of reduced conformal vacuum field equations. Its solutions are uniquely determined by the free data.Work supported by a Heisenberg-fellowship of the Deutsche Forschungsgemeinschaft     

Existence and structure of past asymptotically simple solutions of Einstein's field equations with positive cosmological constant

The initial value problem for Einstein's field equations with positive cosmological constant is analysed where data are prescribed at past conformal infinity. It is found that the data on past conformal infinity are given, up to arbitrary conformal rescalings, by a freely specifyble Riemannian metric and a trace-free, symmetric tensorfield of valence two, which satisfies a divergence equation. For each initial data set exists a unique (semi-global) past asymptotically simple solution of Einstein's equations. The case is discussed where in such a space-time exists a Killing vector field with a time-like trajectory which approaches a point p on conformal infinity. It is shown that in a neighbourhood of the trajectory near p the space-time is conformally flat.     

A Model Problem for Conformal Parameterizations of the Einstein Constraint Equations

We study the conformal and conformal thin sandwich (CTS) methods as candidates for parameterizing the set vacuum initial data for the Cauchy problem of general relativity. To this end we consider a small family of symmetric conformal data. Within this family we obtain an existence result so long as the mean curvature has constant sign. When the mean curvature changes sign we find that solutions either do not exist, or they are not unique. In some cases solutions are shown to be non-unique. Moreover, the theory for mean curvatures with changing sign is shown to be extremely sensitive with respect to the value of a coupling constant in the Einstein constraint equations.     

Vacuum Geometry of the <Emphasis Type="Italic">N</Emphasis> = 2 Wess-Zumino Model

Expansions of the gravitational field arising from the development of asymptotically Euclidean, time symmetric, conformally flat initial data are calculated in a neighbourhood of spatial and null infinities up to order 6. To this end a certain representation of spatial infinity as a cylinder is used. This setup is based on the properties of conformal geodesics. It is found that these expansions suggest that null infinity has to be non-smooth unless the Newman-Penrose constants of the spacetime, and some other higher order quantities of the spacetime vanish. As a consequence of these results it is conjectured that similar conditions occur if one were to take the expansions to even higher orders. Furthermore, the smoothness conditions obtained suggest that if time symmetric initial data which are conformally flat in a neighbourhood of spatial infinity yield a smooth null infinity, then the initial data must in fact be Schwarzschildean around spatial infinity.It will be assumed that the reader is familiar with the ideas of the so-called conformal framework to describe the properties of isolated bodies and the concept of asymptotic flatness. For a recent review, the reader is remitted to [18]     

Nonsingular black holes and degrees of freedom in quantum gravity

Spherically symmetric space-times provide many examples for interesting black hole solutions, which classically are all singular. Following a general program, space-like singularities in spherically symmetric quantum geometry, as well as other inhomogeneous models, are shown to be absent. Moreover, one sees how the classical reduction from infinitely many kinematical degrees of freedom to only one physical one, the mass, can arise, where aspects of quantum cosmology such as the problem of initial conditions play a role.     

Gravitational and electromagnetic fields near a de Sitter-like infinity

We present a characterization of general gravitational and electromagnetic fields near de Sitter-like conformal infinity which supplements the standard peeling behavior. This is based on an explicit evaluation of the dependence of the radiative component of the fields on the null direction from which infinity is approached. It is shown that the directional pattern of radiation has a universal character that is determined by the algebraic (Petrov) type of the spacetime. Specifically, the radiation field vanishes along directions opposite to principal null directions.     

Non-singular space-times with a negative cosmological constant: II. Static solutions of the Einstein–Maxwell equations

We construct infinite-dimensional families of non-singular static space-times, solutions of the vacuum Einstein–Maxwell equations with a negative cosmological constant. The families include an infinite-dimensional family of solutions with the usual AdS conformal structure at conformal infinity.     

A Well-Posed Numerical Method to Track Isolated Conformal Map Singularities in Hele-Shaw Flow

We present a new numerical method for calculating an evolving 2D Hele-Shaw interface when surface tension effects are neglected. In the case where the flow is directed from the less viscous fluid into the more viscous fluid, the motion of the interface is ill-posed; small deviations in the initial condition will produce significant changes in the ensuing motion. This situation is disastrous for numerical computation, as small roundoff errors can quickly lead to large inaccuracies in the computed solution. Our method of computation is most easily formulated using a conformal map from the fluid domain into a unit disk. The method relies on analytically continuing the initial data and equations of motion into the region exterior to the disk, where the evolution problem becomes well-posed. The equations are then numerically solved in the extended domain. The presence of singularities in the conformal map outside of the disk introduces specific structures along the fluid interface. Our method can explicitly track the location of isolated pole and branch point singularities, allowing us to draw connections between the development of interfacial patterns and the motion of singularities as they approach the unit disk. In particular, we are able to relate physical features such as finger shape, side-branch formation, and competition between fingers to the nature and location of the singularities. The usefulness of this method in studying the formation of topological singularities (self intersections of the interface) is also pointed out.     

Null Surface Quantization and Quantum Field Theory in Asymptotically Flat Space-Time

A quantum theory of the free scalar, electromagnetic and gravitational fields in a curved asymptotically flat space-time is developed. It is shown that the Penrose conformal technique makes it possible to reformulate the null infinity quantization as a problem of the quantization on the proper null surface in the corresponding Penrose space. The Schwinger dynamical principle is exploited to derive the corresponding null surface commutation relations. The general covariant and gauge-independent form of the commutation relations is also given. The existence of the asymptotic symmetry (BMS) group in the asymptotically flat space-time is used to define uniquely the “in” and “out” vacuum states. The explicit expressions for the S-matrix operator and for the S-matrix elements in the asymptotically simple space-time are given. The functional integration method is used to find the expression for the density matrix describing the observations at ∮+ in the weakly asymptotically simple space-time when the information loss due to the event horizons or the existence of bare singularities is possible. The application of the developed approach to the problem of quantum evaporation of black holes (Hawking effect) is briefly discussed.     

A class of spherically symmetric solutions with conformal killing vectors

Spherically symmetric solutions with a conformal Killing vector in the (r, t) surface allow the null geodesics to be found with relative ease. Knowledge of the null geodesics is essential to calculating the optical properties of a solution via the optical scalar equations. Solutions of this type may be useful for the treatment of the optical properties of an inhomogeneous universe. We first address the question of whether the large class of spherically symmetric solutions found by McVittie possess conformal symmetry. We also investigate the potential for using conformal Killing vectors to aid in the solution of Einstein's Field Equations.     

Nonexistence of conformally flat slices in Kerr and other stationary spacetimes

It is proved that stationary solutions to the vacuum Einstein field equations with a nonvanishing angular momentum have no Cauchy slice that is maximal, conformally flat, and nonboosted. The proof is based on results coming from a certain type of asymptotic expansion near null and spatial infinity--which also show that the development of Bowen-York-type data cannot have a development admitting a smooth null infinity--and from the fact that stationary solutions do admit a smooth null infinity.