Singularities at real points of ℋ space

In the presence of gravitational radiation, there are ordinarily no shear-free slices of null infinity. A four-complex-dimensional set of shear-free slices of complexified null infinity do exist. They comprise the manifold space. In general, there are no preferred real subspaces of space associated with slices of real null infinity. However, for radiation fields possessing a twist-free axial symmetry, a two-parameter family of shear-free slices of real null infinity exist and therefore pick out a preferred two-dimensional real subspace of space. In this paper, we study the geometry of these 2-spaces for the particular case of quadrupole radiation fields for which determination of the shear-free slices reduces to the standard problem of determining orbits of a particle moving in a potential. Our principal interest is the investigation of possible singularities caused by sufficiently intense radiation fields. We find that such singularities do occur for radiation fields having the characteristic powerc ⁵/G.     

Asymptotic directional structure of radiation for fields of algebraic type D

The directional behavior of dominant components of algebraically special spin-s fields near a spacelike, timelike or null conformal infinity is studied. By extending our previous general investigations, we concentrate on fields which admit a pair of equivalent algebraically special null directions, such as the Petrov type-D gravitational fields or algebraically general electromagnetic fields. We introduce and discuss a canonical choice of the reference tetrad near infinity in all possible situations, and we present the corresponding asymptotic directional structures using the most natural parametrizations.Dedicated to Prof. Jií Horáek on the occasion of his 60th birthday     

Sources of gravitational waves in asymptotically flat Einstein-Maxwell spacetime

In this work, the dynamic of isolated systems in general relativity is described when gravitational radiation and electromagnetic fields are present. In this construction, the asymptotic fields received at null infinity together with the regularized null cone cuts equation, and the center of mass of an asymptotically flat Einstein-Maxwell spacetime are used. A set of equations are derived in the low speed regime, linking their time evolution to the emitted gravitational radiation and to the Maxwell fields received at infinity. These equations should be useful when describing the dynamic of compact sources, such as the final moments of binary coalescence and the evolution of the final black hole. Additionally, we compare our equations with those coming from a similar approach given by Newman, finding some differences in the motion of the center of mass and spin of the gravitational system.     

Radiation Fields on Schwarzschild Spacetime

In this paper we define the radiation field for the wave equation on the Schwarzschild black hole spacetime. In this context it has two components: the rescaled restriction of the time derivative of a solution to null infinity and to the event horizon. In the process, we establish some regularity properties of solutions of the wave equation on the spacetime. In particular, we prove that the regularity of the solution across the event horizon and across null infinity is determined by the regularity and decay rate of the initial data at the event horizon and at infinity. We also show that the radiation field is unitary with respect to the conserved energy and prove support theorems for each piece of the radiation field.     

Kerr–Schild metrics and gravitational radiation

We describe conditions assuring that the Kerr–Schild type solutions of Einstein's equations with pure radiation fields are asymptotically flat at future null infinity. Such metrics cannot describe “true” gravitational radiation from bounded sources—it is shown that the Bondi news function vanishes identically. We obtain formulae for the total energy and angular momentum at ℐ. As an example we consider a non-stationary generalization of the Kerr metric given by Vaidya and Patel. Angular momentum and total energy are expressed in closed form as functions of retarded time.     

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     

Gravitational radiation reaction in quasistatic,axially symmetric systems with possibly strong fields

The leading radiation reaction in quasistatic, axially symmetric systems is calculated by matched asymptotic expansions. Earlier results on inductive transfer of near-zone energy are combined via matching with a wave-zone expansion in the Regge-Wheeler gauge. Two independent measures of the damping are computed: (1) the change of the 1/Rcoefficient in a Weyl-Levi-Cività type of multipole expansion in the near zone and (2) the change in Bondi energy at future null infinity. When an averaging process is justified, such as in (nearly) periodic systems, both quantities yield a generalized version of the usual quadrupole formula, provided the radiation is outgoing at future null infinity. It is also shown that terms nonanalytic in the slow-motion parameter make a time-even contribution to the near-zone monopole coefficient. The principal advantage of this calculation over previous work is that the sources are permitted to have strong gravitational fields. Moreover, no specific model for the sources is assumed.     

Peeling of Dirac and Maxwell fields on a Schwarzschild background

We study the peeling of Dirac and Maxwell fields on a Schwarzschild background following the approach developed by the authors in Mason and Nicolas (2009) [12] for the wave equation. The method combines a conformal compactification with vector field techniques in order to work out the optimal space of initial data for a given transverse regularity of the rescaled field across null infinity. The results show that analogous decay and regularity assumptions in Minkowski and Schwarzschild produce the same regularity across null infinity. The results are valid also for the classes of asymptotically simple spacetimes constructed by Corvino–Schoen/Chru?ciel–Delay.     

Symmetries of Type D Pure Radiation Fields

The geometrical symmetries corresponding to the continuous groups of collineations and motions generated by a null vector l are considered. These symmetries have been translated into the language of Newman-Penrose formalism for pure radiation (PR) type D fields. It is seen that for such fields, conformal, special conformal and homothetic motions degenerate to motion. The concept of free curvature, matter curvature and matter affine collineations have been introduced and the conditions under which PR type D fields admit such collineations have been obtained. Moreover, it is shown that the projective collineation degenerate to matter affine, special projective, conformal, special conformal, null geodesic and special null geodesic collineations. It is also seen that type D pure radiation fields admit Maxwell collineation along the propagation vector l.     

Gravitational fields near space-like and null infinity

Near space-like infinity an initial value problem for the conformal Einstein equations is formulated such that: (i) the data and equations are regular, (ii) space-like and null infinity have a finite representation, with their structure and location known a priori, and (iii) the setting relies entirely on general properties of conformal structures.A first analysis of this problem shows that the solutions develop in general a certain type of logarithmic singularity at the set where null infinity touches space-like infinity. These singularities form an intrinsic part of the solutions' conformal structure. Conditions on the free initial data near space-like infinity are derived which ensure that for solutions developing from these data singularities of this type cannot occur.     

Ultrarelativistic black-hole encounters

A method is given for investigating the ultrarelativistic encounter of two black holes which complements the usual low-speed approximation. It depends on the fact that the ultrarelativistic limit of a moving Schwarzschild black hole is a certain plane-fronted impulsive gravitational wave. This enables linearized theory on a curved background to be used. The solution is obtained, with the help of generalized function techniques, to the first order in the background energy, and the radiation pattern at conformal null infinity is examined. The whole approach has a strong connection with the theory of twistors.     

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]     

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.     

State Independence for Tunnelling Processes Through Black Hole Horizons and Hawking Radiation

Tunnelling processes through black hole horizons have recently been investigated in the framework of WKB theory, discovering an interesting interplay with Hawking radiation. In this paper, we instead adopt the point of view proper of QFT in curved spacetime, namely, we use a suitable scaling limit towards a Killing horizon to obtain the leading order of the correlation function relevant for the tunnelling. The computation is done for a certain large class of reference quantum states for scalar fields, including Hadamard states. In the limit of sharp localization either on the external side or on opposite sides of the horizon, the quantum correlation functions appear to have thermal nature. In both cases the characteristic temperature is referred to the surface gravity associated with the Killing field and thus connected with the Hawking one. Our approach is valid for every stationary charged rotating non-extremal black hole. However, since the computation is completely local, it covers the case of a Killing horizon which just temporarily exists in some finite region, too. These results provide strong support to the idea that the Hawking radiation, which is detected at future null infinity and needs some global structures to be defined, is actually related to a local phenomenon taking place even for local geometric structures (local Killing horizons), existing just for a while.     

Neutrino fields in Einstein-Cartan theory

The spin-coefficient formalism presented elsewhere is here applied to classical neutrino fields in Einstein-Cartan theory. It is shown that the neutrino current vector is tangent to an expansion-free null geodesic congruence with constant and equal twist and shear, which vanish if and only if the congruence is a repeated principal null congruence of the gravitational field. The geodesics are both extremals and autoparallels. All exact solutions for the case of pure radiation fields are obtained, and it is shown that the only possible ghost solutions have a plane wave metric.     

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 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.     

Limitations of the concept of free fields in Einstein's theory of gravitation

It is shown explicitly that the linearized theory does not constitute any approximation to the exact solutions in the case of free fields. The only regular solution satisfying, as boundary condition, the requirement of a sufficiently rapid decrease at infinity is a flat space. The problem of conservation laws is discussed anew. The continuity equation satisfied by Einstein's pseudotensor does not guarantee the existence of global conservation laws. Solutions violating the energy conservation are interpretable as representing gravitational radiation absorbed or emitted by sources at infinity.     

Censorship,strong curvature,and asymptotic causal pathology

Necessary and sufficient conditions are established for a weakly asymptotically simple and empty, null convergent, generic space-time to be future asymptotically predictable. These conditions require that the causal structure of the space-time is well behaved near spatial infinity and future null infinity, and that there are no singularities of less than a certain finite strength in the future asymptotic limit.