Spacelike Infinity Research Articles

Static black holes with a negative cosmological constant: Deformed horizon and anti–de Sitter boundaries

Using perturbative techniques, we investigate the existence and properties of a static solution for the Einstein equation with a negative cosmological constant, which we call the deformed black hole. We derive a solution for a static and axisymmetric perturbation of the Schwarzschild--anti-de Sitter black hole that is regular in the range from the horizon to spacelike infinity. The key result is that this perturbation simultaneously deforms the two boundary surfaces---i.e., both the horizon and spacelike two-surface at infinity. Then we discuss the Abbott-Deser mass and the Ashtekar-Magnon one for the deformed black hole, and according to the Ashtekar-Magnon definition, we construct the thermodynamic first law of the deformed black hole. The first law has a correction term which can be interpreted as the work term that is necessary for the deformation of the boundary surfaces. Because the work term is negative, the horizon area of the deformed black hole becomes larger than that of the Schwarzschild--anti-de Sitter black hole, if compared under the same mass, indicating that the quasistatic deformation of the Schwarzschild--anti-de Sitter black hole may be compatible with the thermodynamic second law (i.e., the area theorem).

Gravitational and electromagnetic fields near an anti–de Sitter–like infinity

We analyze asymptotic structure of general gravitational and electromagnetic fields near an anti-de Sitter-like conformal infinity. Dependence of the radiative component of the fields on a null direction along which the infinity is approached is obtained. The directional pattern of outgoing and ingoing radiation, which supplements standard peeling property, is determined by the algebraic (Petrov) type of the fields and also by orientation of principal null directions with respect to the timelike infinity. The dependence on the orientation is a new feature if compared to spacelike infinity.

Plane waves and spacelike infinity

In an earlier paper, we showed that the causal boundary of any homogeneous plane wave satisfying the null convergence condition consists of a single null curve. In Einstein–Hilbert gravity, this would include any homogeneous plane wave satisfying the weak null energy condition. For conformally flat plane waves such as the Penrose limit of AdS5 × S5, all spacelike curves that reach infinity also end on this boundary and the completion is Hausdorff. However, the more generic case (including, e.g., the Penrose limits of AdS4 × S7 and AdS7 × S4) is more complicated. In one natural topology, not all spacelike curves have limit points in the causal completion, indicating the need to introduce additional points at ‘spacelike infinity’—the endpoints of spacelike curves. We classify the distinct ways in which spacelike curves can approach infinity, finding a two-dimensional set of distinct limits. The dimensionality of the set of points at spacelike infinity is not, however, fixed from this argument. In an alternative topology, the causal completion is already compact, but the completion is non-Hausdorff.

Scaling Algebras for Charged Fields and Short-Distance Analysis for Localizable and Topological Charges

The method of scaling algebras, which has been introduced earlier as a means for analyzing the short-distance behaviour of quantum field theories in the setting of the model-independent, operator-algebraic approach, is extended to the case of fields carrying superselection charges. In doing so, consideration will be given to strictly localizable charges ("DHR-type" superselection charges) as well as to charges which can only be localized in regions extending to spacelike infinity ("BF-type" superselection charges). A criterion for the preservance of superselection charges in the short-distance scaling limit is proposed. Consequences of this preservance of superselection charges are studied. The conjugate charge of a preserved charge is also preserved, and for charges of DHR-type, the preservance of all charges of a quantum field theory in the scaling limit leads to equivalence of local and global intertwiners between superselection sectors.

Spin-2 fields on Minkowski space near spacelike and null infinity

We show that the spin-2 equations on Minkowski space in the gauge of the ‘regular finite initial value problem at spacelike infinity’ imply estimates which, together with the transport equations on the cylinder at spacelike infinity, allow us to obtain for a large class of initial data information on the smoothness of the solution near spacelike and null infinity of any desired precision.

COLLAPSING AND EXPANDING CYLINDRICALLY SYMMETRIC FIELDS WITH LIGHTLIKE WAVE-FRONTS IN GENERAL RELATIVITY

The dynamics of collapsing and expanding cylindrically symmetric gravitational and matter fields with lightlike wave-fronts is studied in General Relativity, using the Barrabés–Israel method. As an application of the general formulae developed, the collapse of a matter field that satisfies the condition RABgAB =0, (A,B = z, φ), in an otherwise flat spacetime background is studied. In particular, it is found that the gravitational collapse of a purely gravitational wave or a null dust fluid cannot be realized in a flat spacetime background. The studies are further specified to the collapse of purely gravitational waves and the general conditions for such collapse are found. It is shown that after the waves arrive at the axis, in general, part of them is reflected to spacelike infinity along the future light cone, and part of it is focused to form spacetime singularities on the symmetry axis. The cases where the collapse does not result in the formation of spacetime singularities are also identified.

Initial data for stationary spacetimes near spacelike infinity

We study Cauchy initial data forasymptotically flat, stationary vacuumspacetimes near spacelike infinity. Thefall-off behaviour of the intrinsic metricand the extrinsic curvature ischaracterized. We prove that they have ananalytic expansion in powers of a radialcoordinate. The coefficients of theexpansion are analytic functions of theangles. This result allow us to fill a gapin the proof found in the literature of thestatement that all asymptotically flat,vacuum stationary spacetimes admit ananalytic compactification at null infinity.Stationary initial data are physicallyimportant and highly non-trivial examplesof a large class of data with similarregularity properties at spacelikeinfinity, namely, initial data for whichthe metric and the extrinsic curvaturehave asymptotic expansion in terms ofpowers of a radial coordinate. We isolatethe property of the stationary data whichis responsible for this kind of expansion.

A C>1 Completion of the Kerr Space–Time at Spacelike Infinity Including I+ and I−

It is well known that, for asymptotically flat spacetimes, one cannot in general have a smooth differentiable structure at spacelike infinity, i0. Normally, one uses direction dependent structures, whose regularity has to match the regularity of the (rescaled) metric. The standard C>1-structure at i0 ensures sufficient regularity in spacelike directions, but examples show very low regularity on I+ and I−. The alternative C1+-structure shows that both null and spacelike directions may be treated on an equal footing, at the expense of some manageable logarithmic singularities at i0. In this paper, we show that the Kerr spacetime may be rescaled and given a structure which is C>1 in both null and spacelike directions from i0.

Asymptotic behaviour of cylindrical waves interacting withspinning strings

We consider a family of cylindrical spacetimesendowed with angular momentum that aresolutions to the vacuum Einstein equationsoutside the symmetry axis. This family wasrecently obtained by performing a completegauge fixing adapted to cylindrical symmetry.In this paper, we find boundary conditionsthat ensure that the metric arising from thisgauge fixing is well defined and that theresulting reduced system has a consistentHamiltonian dynamics. These boundaryconditions must be imposed both on thesymmetry axis and in the region far from theaxis at spacelike infinity. Employing suchconditions, we determine the asymptoticbehaviour of the metric close to and far fromthe axis. In each of these regions, theapproximate metric describes a conicalgeometry with a time dislocation. Inparticular, around the symmetry axis theeffect of the singularity consists in inducinga constant deficit angle and a timelikehelical structure. Based on these results andon the fact that the degrees of freedom inour family of metrics coincide with those ofcylindrical vacuum gravity, we argue that theanalysed set of spacetimes representcylindrical gravitational waves surrounding aspinning cosmic string. For any of thesespacetimes, a prediction of our analysis isthat the wave content increases the deficitangle at spatial infinity with respect tothat detected around theaxis.

Quantum Delocalization of the Electric Charge

The classical Maxwell–Dirac and Maxwell–Klein–Gordon theories admit solutions of the field equations where the corresponding electric current vanishes in the causal complement of some bounded region of Minkowski space. This poses the interesting question of whether states with a similarly well localized charge density also exist in quantum electrodynamics. For a large family of charged states, the dominant quantum corrections at spacelike infinity to the expectation values of local observables are computed. It turns out that certain moments of the charge density decrease no faster than the Coulomb field in spacelike directions. In contrast to the classical theory, it is therefore impossible to define the electric charge support of these states in a meaningful way.

Asymptotically Flat Initial Data¶with Prescribed Regularity at Infinity

We prove the existence of a large class of asymptotically flat initial data with non-vanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at space-like infinity in terms of powers of a radial coordinate.

Global monopoles in the Brans-Dicke theory

A gravitating global monopole produces a repulsive grativational field outside the core in addition to a solid angular deficit in the Brans-Dicke theory. As a new feature, the angular deficit is dependent on the values of \phi_{\infty} and \omega, where \phi_{\infty} is asymptotic value of scalar field in space-like infinity and \omega is the Brans-Dicke parameter.

Bondi-type systems near spacelike infinity and the calculation of the Newman–Penrose constants

We relate Bondi systems near spacelike infinity to another type of gauge conditions. While the former are based on null infinity, the latter are defined in terms of Einstein propagation, the conformal structure, and data on some Cauchy hypersurface. For a certain class of time symmetric space–times we study an expansion which allows us to determine the behavior of various fields arising in Bondi systems in the region of space–time where null infinity touches spacelike infinity. The coefficients of these expansions can be read off from the initial data. We obtain, in particular, expressions for the constants discovered by Newman and Penrose in terms of the initial data. For this purpose we calculate a certain expansion introduced by Friedrich [J. Geom. Phys. 24, 83–163 (1998)] up to third order.

The Fefferman-Graham ambiguity and AdS black holes

Asymptotically anti-de Sitter space-times in pure gravity with negative cosmological constant are described, in all space-time dimensions greater than two, by classical degrees of freedom on the conformal boundary at space-like infinity. Their effective boundary action has a conformal anomaly for even dimensions and is conformally invariant for odd ones. These degrees of freedom are encoded in traceless tensor fields in the Fefferman-Graham asymptotic metric for any choice of conformally flat boundary and generate all Schwarzschild and Kerr black holes in anti-de Sitter space-time. We argue that these fields describe components of an energy-momentum tensor of a boundary theory and show explicitly how this is realized in 2+1 dimensions. There, the Fefferman-Graham fields reduce to the generators of the Virasoro algebra and give the mass and the angular momentum of the BTZ black holes. Their local expression is the Liouville field in a general curved background.

Calculating asymptotic quantities near space-like and null infinity from Cauchy data

We discuss how asymptotic quantities, originally introduced on null infinity in terms of Bondi-type gauge conditions, can be calculated near space-like infinity to any desired precision.

On the differentiability conditions at spacelike infinity

We consider spacetimes which are asymptotically flat at spacelike infinity, . It is well known that, in general, one cannot have a smooth differentiable structure at , but rather one has to use direction-dependent structures there. Instead of the usual -differentiable structure, we suggest a weaker differential structure, a structure. The reason for this is that there do not appear to be any completions of the Schwarzschild spacetime which is in both spacelike and null directions at . In a structure all directions can be treated on an equal footing, at the expense of logarithmic singularities at . We show that, in general, the relevant part of the curvature tensor, the Weyl part, is free from these singularities, and that the (rescaled) Weyl tensor has a certain symmetry property.

Timelike infinity and asymptotic symmetry

By extending Ashtekar and Romano’s definition of spacelike infinity to the timelike direction, a new definition of asymptotic flatness at timelike infinity for an isolated system with a source is proposed. The treatment provides unit spacelike three-hyperboloid timelike infinity and avoids the introduction of the troublesome differentiability conditions which were necessary in the previous works on asymptotically flat space–times at timelike infinity. Asymptotic flatness is characterized by the falloff rate of the energy-momentum tensor at timelike infinity, which makes it easier to understand physically what space–times are investigated. The notion of the order of the asymptotic flatness is naturally introduced from the rate. The definition gives a systematized picture of hierarchy in the asymptotic structure, which was not clear in the previous works. It is found that if the energy-momentum tensor falls off at a rate faster than ∼t−2, the space–time is asymptotically flat and asymptotically stationary in the sense that the Lie derivative of the metric with respect to ∂t falls off at the rate ∼t−2. It also admits an asymptotic symmetry group similar to the Poincaré group. If the energy-momentum tensor falls off at a rate faster than ∼t−3, the four-momentum of a space–time may be defined. On the other hand, angular momentum is defined only for space–times in which the energy-momentum tensor falls off at a rate faster than ∼t−4.

Quantum properties of topological black holes

We examine quantum properties of topological black holes which are asymptotically anti--de Sitter. First, massless scalar fields and Weyl spinors which propagate in the background of an anti--de Sitter black hole are considered in an exactly soluble two--dimensional toy model. The Boulware--, Unruh--, and Hartle--Hawking vacua are defined. The latter results to coincide with the Unruh vacuum due to the boundary conditions necessary in asymptotically adS spacetimes. We show that the Hartle--Hawking vacuum represents a thermal equilibrium state with the temperature found in the Euclidean formulation. The renormalized stress tensor for this quantum state is well--defined everywhere, for any genus and for all solutions which do not have an inner Cauchy horizon, whereas in this last case it diverges on the inner horizon. The four--dimensional case is finally considered, the equilibrium states are discussed and a luminosity formula for the black hole of any genus is obtained. Since spacelike infinity in anti--de Sitter space acts like a mirror, it is pointed out how this would imply information loss in gravitational collapse. The black hole's mass spectrum according to Bekenstein's view is discussed and compared to that provided by string theory.

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.

Positive definite problem of energy density and radiative energy flux for pulse cylindrical gravitational wave

By using the general expressions of energy-momentum pseudo-tensor of the cylindrical gravitational waves (GW) given by Rosen and Virbhadra in Cartesian coordinates, the concrete forms of energy density and radiative energy flux of the pulse cylindrical GW are obtained. Their physical properties, suitable range and asymptotic behaviour are considered. It is found that: For the region in which space radial coordinates to origin are greater than the pulse width of the pulse cylindrical GW, the energy density and radiative energy flux of the outward travelling pulse cylindrical GW propagating along a light-cone are positive definite. However, for the region in which the space radial coordinates are less than the pulse width, there is no guarantee for the positive definite property of the radiative energy flux of the outward travelling wave. Moreover, we show that the asymptotic behaviour of the energy and energy flux densities of the pulse cylindrical GW and that of the Riemann curvature tensor have good self-consistancy in space-like, time-like and null infinity regions.