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.     

Symmetries of cosmological Cauchy horizons

We consider analytic vacuum and electrovacuum spacetimes which contain a compact null hypersurface ruled byclosed null generators. We prove that each such spacetime has a non-trivial Killing symmetry. We distinguish two classes of null surfaces, degenerate and non-degenerate ones, characterized by the zero or non-zero value of a constant analogous to the “surface gravity” of stationary black holes. We show that the non-degenerate null surfaces are always Cauchy horizons across which the Killing fields change from spacelike (in the globally hyperbolic regions) to timelike (in the acausal, analytic extensions). For the special case of a null surface diffeomorphic toT ³ we characterize the degenerate vacuum solutions completely. These consist of an infinite dimensional family of “plane wave” spacetimes which are entirely foliated by compact null surfaces. Previous work by one of us has shown that, when one dimensional Killing symmetries are allowed, then infinite dimensional families of non-degenerate, vacuum solutions exist. We recall these results for the case of Cauchy horizons diffeomorphic toT ³ and prove the generality of the previously constructed non-degenerate solutions. We briefly discuss the possibility of removing the assumptions of closed generators and analyticity and proving an appropriate generalization of our main results. Such a generalization would provide strong support for the cosmic censorship conjecture by showing that causality violating, cosmological solutions of Einstein's equations are essentially an artefact of symmetry.     

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.     

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.     

On characteristic initial-value and mixed problems

Existence and uniqueness are proved for certain initial-value problems for hyperbolic systems of second-order differential equations, each having the same principal partg ^ab _{a b} (whereg ^ab is indefinite). The initial data are given on two intersecting hypersurfaces H₁ andH ₂ one of which-sayH ₁-is a characteristic surface. The other surface,H ₂, is permitted to be spacelike, timelike, or characteristic. For Einstein's vacuum field equations we restrict ourselves to anH ₂ that is characteristic. Unlike the Cauchy problem, the data have to be necessarily of a considerably higher differentiability class (Sobolev classW ^2m–1) than the solution (Sobolev classW ^m ). On the other hand, in the mixed problem (where one of the surfaces is spacelike) corner conditions have to be fulfilled. The occurrence of constraint equations for Einstein's metric field and for harmonic coordinates can be prevented by solving certain ordinary differential propagation equations.     

Left-flat spaces and their associated space-times

It is already known that for an asymptotically flat space-time the metric coefficients and the other Newman-Penrose variables (in a suitable frame) can be constructed, in principle, by specifying certain initial data at conformal null infinity (and one further function on another null hypersurface), integrating the Newman-Penrose equations in the conformally rescaled “unphysical” space, and then transforming the results back to the physical space-time. If this is done approximately near ?⁺, for vacuum, the well-known Newman-Unti expansion is obtained. In this paper, after complexifying null infinity ?⁺ we generate, in a similar fashion, a left-flat spaceH using as much of the initial data of a given asymptotically flat space-timeM as possible, and show that the left-flat spaceH thus constructed is, in fact, the H-space corresponding toM. The advantage of our method is that it allows a reversal of procedure. Under suitable conditions we can generate from a given left-flat spaceH a class of physical space-times whose H-space is precisely the given left-flat spaceH. We shall see that the formal procedure requires only the local but not the global properties of ?⁺.     

The space of (generalized) Taub-Nut spacetimes

In this paper we show how to construct all analytic solutions of the vacuum Einstein equations having a compact Cauchy horizon diffeomorphic to S³ and ruled by closed null generators which fiber the horizon in the sense of Hopf. The set of (inequivalent) solutions is infinite dimensional, contains the two parameter Taub-NUT family as a special case, and may be uniquely parameterized by a pair of arbitrary, real analytic functions on S² (modulo an action of the conformal group of S²). The horizon of each such solution is necessarily a Killing horizon (as proven recently by Isenberg and the author) and is shown here always to be a «crushingå horizon in the sense of Eardley and Smarr. Some recent results of Gerhardt are used to show that a neighborhood of the horizon (in the globally hyperbolic region) is always foliated by constant mean curvature hypersurfaces.The possible isometry groups of the solutions considered are characterized in terms of isometries of the determining «Cauchy dataå which is specified on the horizons themselves.     

Geodesic focusing and space-time topology

Given a space-timeM and a pointp inM, it is shown that, if the locus of first conjugate points ofp along future-directed null geodesics consists of a single point, thenM admits a compact (S ³) spacelike hypersurface. If in addition the null geodesics do not intersect before focusing, then, in a simply connected space-time, the spacelike hypersurface is a partial Cauchy surface.     

The Taylor Expansion at Past Time-like Infinity

We construct initial data for the conformal vacuum field equations on a cone ${{\mathcal{N}}_p}$ with vertex p so that for the prospective vacuum solution, the point p will represent past time-like infinity i ^?, the set ${{\mathcal{N}}_p {\setminus}\{p\}}$ will represent past null infinity ${{\mathcal{J}}^-}$ , and the freely prescribed (suitably smooth) data will acquire the meaning of the incoming radiation field. It is shown that: (i) On some coordinate neighbourhood of p there exist smooth fields which satisfy at the point p the conformal vacuum field equations at all orders and induce the given data at all orders. The Taylor coefficients of these fields at p are uniquely determined by the free data. (ii) On the cone ${{\mathcal{N}}_p}$ there exists a unique set of fields which induce the given free data and satisfy the transport equations and the inner constraints induced on ${{\mathcal{N}}_p}$ by the conformal field equations. These fields are smooth at p in the sense that they coincide there at all orders with the fields which are obtained by restricting to ${{\mathcal{N}}_p}$ the functions considered in (i) and they are smooth on the smooth three-manifold ${{\mathcal{N}}_p {\setminus}\{p\}}$ in the standard sense.     

Neighborhoods of Cauchy horizons in cosmological spacetimes with one killing field

In this paper we show how to construct an infinite dimensional family of analytic, vacuum spacetimes which each have (i) T³ × R topology, (ii) a smooth, compact Cauchy horizon, and (iii) a single Killing vector field which is spacelike in the globally hyperbolic region, null on the horizon and timelike in the (acausal) extension. The key idea is to use the horizons themselves as initial data surfaces and to prove the local existence of solutions using a version of the Cauchy-Kowalewski theorem. Factoring by the action of analytic, horizon preserving diffeomorphisms we define a “space of extendible vacuum spacetimes” of the given symmetry type and show (modulo certain smoothness estimates which we do not attempt to derive) that this space defines a Lagrangian submanifold of the usual phase space for Einstein's equations. We also study the linear perturbations of a class of the extendible spacetimes and show that the generic such perturbation blows up near the background solution's Cauchy horizon. This result, though limited by the linearity of the approximation, conforms to the usual picture of unstable Cauchy horizons demanded by the strong cosmic censorship conjecture.     

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     

Radiation damping as an initial value problem

Using a simple, exactly soluble model for the interaction of one particle and a scalar field Φ, we discuss the problem of radiation reaction in terms of the initial value solution. We show that if the Cauchy data of the field fall off at spatial infinity in such a way that the field has finite energy, the particle motion is damped for t → ∞. Further, we point out that no solutions with finite field energy exist for the boundary conditions Φ^out = 0 and Φⁱⁿ + Φ^out = 0. For Φⁱⁿ = 0, nontrivial solutions exist only if it is assumed that the system has been open in the past of some initial hypersurface.     

Null cut loci of spacelike surfaces

The null cut locus of a spacelike submanifold of codimension 2 in a space-time is defined. In globally hyperbolic space-times, it is shown that the future (past) null cut locusc _n ⁺ (H) [c _n ^- (H)] of a compact, acausal, spacelike submanifoldH of codimension 2 is a closed subset of the space-time, and each pointx c _n ⁺ (H) is either a focal point ofH along some future-directed null geodesic meetingH orthogonally or there exist at least two null geodesics fromH tox, realizing the distance betweenH andx or both. Also, it can be shown that the assumptions of the Penrose's singularity theorem for open globally hyperbolic space-times may be weakened to the space-times which are conformal to an open subset of an open globally hyperbolic space-time.This study is based on Chapter 3 of the author's Ph.D. thesis.     

A plane symmetric solution of Einstein's field equations of general relativity containing zero-rest-mass scalar fields

An exact static solution of Einstein's field equations of general relativity in the presence of zero-rest-mass scalar fields has been obtained when both the metric tensor gijand the zero-rest-mass scalar field φexhibit plane symmetry in the sense of Taub [9]. Our solution generalizes the empty space-time solution with plane symmetry previously obtained by Taub to the situation when static zero-rest-mass scalar fields are present. The static plane symmetric solutoins of Einstein's field equations in the presence of massive scalar fields, and the difference between the massless and non-massless scalar fields are being investigated, and will be published separately later on. We also hope to discuss non-static plane symmetric solutions of Einstein's field equations in the presence of scalar fields in future.     

Positive mass theorems for black holes

We extend Witten's proof of the positive mass theorem at spacelike infinity to show that the mass is positive for initial data on an asymptotically flat spatial hypersurface Σ which is regular outside an apparent horizonH. In addition, we prove that if a black hole has electromagnetic charge, then the mass is greater than the modulus of the charge. These results are also valid for the Bondi mass at null infinity. Finally, in the case of the Einstein equation with a negative cosmological constant, we show that a suitably defined mass is positive for data on an asymptotically anti-de Sitter surface Σ which is regular outside an apparent horizon.     

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.     

On the stability of AdS black strings

We explore via linearized perturbation theory the Gregory–Laflamme instability of the black string solutions of Einstein's equations with negative cosmological constant recently discussed in literature. Our results indicate that the black strings whose conformal infinity is the product of time and Sd−3×S¹$S^{d-3}\times S^1$ are stable for large enough values of the event horizon radius. All topological black strings are also classically stable. We argue that this provides an explicit realization of the Gubser–Mitra conjecture.     

A method of reduction of Einstein's equations of evolution and a natural symplectic structure on the space of gravitational degrees of freedom

A program is outlined which addresses the problem of thereduction of Einstein's equations, namely, that of writing Einstein's vacuum equations in (3+1)-dimensions as anunconstrained dynamical system where the variables are thetrue degrees of freedom of the gravitational field. Our analysis is applicable for globally hyperbolic Ricci-flat spacetimes that admit constant mean curvature compact orientable spacelike Cauchy hypersurfaces M with degM=0 andM not diffeomorphic toF ₆, the underlying manifold of a certain compact orientable flat affine 3-manifold. We find that for these spacetimes, modulo the extended Poincaré conjecture and the use of local cross-sections rather than a global cross-section, (3+1)-reduction can be completed much as in the (2+1)-dimensional case. In both cases, one gets as the reduced phase space the cotangent bundleT ^* T _M of theTeichmüller space T _M of conformal structures onM, whereM is a given initial constant mean curvature compact orientable spacelike Cauchy hypersurface in a spacetime (V, g _V ), and one gets reduction of the full classical non-reduced Hamiltonian system with constraints to a reduced Hamiltonian system without constraints onT ^* T _M . For these reduced systems, the time parameter is the parameter of a family of monotonically increasing constant mean curvature compact orientable spacelike Cauchy hypersurfaces in a neighborhood of a given initial one. In the (2+1)-dimensional case, the Hamiltonian is the area functional of these hypersurfaces, and in the (3+1)-dimensional case, the Hamiltonian is the volume functional of these hypersurfaces.     

Classical and quantum models of strong cosmic censorship

The cosmic censorship conjecture states that naked singularities should not evolve from regular initial conditions in general relativity. In its strong form the conjecture asserts that space-times with Cauchy horizons must always be unstable and thus that thegeneric solution of Einstein's equations must be inextendible beyond its maximal Cauchy development. In this paper we shall show that one can construct an infinite-dimensional family ofextendible cosmological solutions similar to Taub-NUT space-time. However, we shall also show that each of these solutions is unstable in precisely the way demanded by strong cosmic censorship. Finally we show that quantum fluctuations in the metric always provide (though in an unexpectedly subtle way) the “generic perturbations” which destroy the Cauchy horizons in these models.     

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.