Non trivial static geodesically complete space times with a negative cosmological constant II n
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Non-trivial, static, geodesically complete space-times with a negative cosmological constant II. n ≥ 5 Michael T. Anderson? Piotr T. Chrusciel† Erwann Delay‡ Abstract We show that the recent work of Lee [24] implies existence of a large class of new singularity-free strictly static Lorentzian vacuum solutions of the Einstein equations with a negative cosmological constant. This holds in all space-time dimensions greater than or equal to four, and leads both to strictly static solutions and to black hole solutions. The construction allows in principle for metrics (whether black hole or not) with Yang-Mills- dilaton fields interacting with gravity through a Kaluza-Klein coupling. 1 Introduction In recent work [3] we have constructed a large class of non-trivial static, geodesi- cally complete, four-dimensional vacuum space-times with a negative cosmo- logical constant. The object of this paper is to establish existence of higher dimensional analogues of the above. More precisely, we wish to show that for ? < 0 and n ≥ 4 there exist n– dimensional strictly static1 solutions (M , g) of the vacuum Einstein equations with the following properties: 1. (M , g) is diffeomorphic to R??, for some (n? 1)–dimensional spacelike Cauchy surface ?, with the R factor corresponding to the action of the isometry group.
Voir
- riemannian einstein
- trace-free symmetric
- field
- strictly static
- properties near ∂m
- pact riemannian
- killing vector
- metrics
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Non-trivial, static, geodesically complete space-times with a negative cosmological constant II.n≥5 Michael T. Anderson∗ Piotr T. Chru´sciel† Erwann Delay‡
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Abstract
We show that the recent work of Lee [24] implies existence of a large class of new singularity-free strictly static Lorentzian vacuum solutions of the Einstein equations with a negative cosmological constant. This holds in all space-time dimensions greater than or equal to four, and leads both to strictly static solutions and to black hole solutions. The construction allows in principle for metrics (whether black hole or not) with Yang-Mills-dilaton fields interacting with gravity through a Kaluza-Klein coupling.
Introduction
In recent work [3] we have constructed a large class of non-trivial static, geodesi-cally complete, four-dimensional vacuum space-times with a negative cosmo-logical constant. The object of this paper is to establish existence of higher dimensional analogues of the above. More precisely, we wish to show that for Λ<0 andn≥4 there existn– dimensionalstrictly static1solutions (M,g) of the vacuum Einstein equations with the following properties:
1. (M,g) is diffeomorphic toR×Σ, for some (n−1)–dimensional spacelike Cauchy surface Σ, with theRfactor corresponding to the action of the isometry group.
2. (Σ, gΣ), wheregΣis the metric induced bygon Σ, is a complete Rieman-nian manifold. ∗of Mathematics, S.U.N.Y. at Stony Brook, Stony Brook, N.Y. 11794-3651.Department Partially supported by an NSF Grant DMS 0305865; emailanderson@math.sunysb.edu †ecneicSsede´tlucFas,ueiqatemh´at,soTru2700,t3FdmonGranrcdes,PamenedtMeDe´aptr France. Partially supported by a Polish Research Committee grant 2 P03B 073 24; email piotr@gargan.math.univ-tours.fr, URLwww.phys.univ-tours.fr/∼piotr ‡F84000AvPasteur,-isSde´eltcuFas,uesiuoLeur,secneicrtem´epaDtaqi´hmeMetanedt gnon, France. Partially supported by the ACI program of the French Ministry of Research; emaildelay@gargan.math.univ-tours.fr 1We shall say that a space-time isstrictly staticif it contains a globally timelike hypersur-face orthogonal Killing vector field.
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3. (M,g) is geodesically complete.
4. All polynomial invariants ofgconstructed using the curvature tensor and its derivatives up to any finite order are bounded onM.
5. (M,g) admits a globally hyperbolic (in the sense of manifolds with bound-ary) conformal completion with a timelikeI completion is smooth. The ifnis even, and is of differentiability class at leastCn−2ifnis odd.
6. (Σ, gΣ) is a conformally compactifiable manifold, with the same differen-tiabilities as in point 5.
7. The connected component of the group of isometries of (M,g) is exactly R, with an associated Killing vectorXbeing timelike throughoutM.
8. There exist no local solutions of the Killing equation other than the (glob-ally defined) timelike Killing vector fieldX.
An example of a manifold satisfying points 1-6 above is of coursen-dimensional anti-de Sitter solution. Clearly itdoes notsatisfy points 7 and 8. We expect that there exist stationaryand not staticsolutions as above, which can be constructed by solving an asymptotic Dirichlet problem for the Einstein equations in a conformally compactifiable setting. We are planning to study this question in the future. In a black hole context we have an obvious variation of the above, we discuss this in more detail in Section 2.3. Throughout this work we restrict attention to dimensionn≥4. Our approach is, in some sense, opposite to that in [25,28], where techniques previously used in general relativity have been employed to obtain uniqueness results in a Riemannian setting. Here we start with Riemannian Einstein met-rics and obtain Lorentzian ones by “Wick rotation”, as follows: Suppose that (M, g) is ann-dimensional conformally compactifiable Einstein manifold of the formM= Σ×S1, and thatS1acts onMby rotations of theS1factor while preserving the metric. Denote byX=∂τthe associated Killing vector field, and assume thatXis orthogonal to the sets Σ× {exp(iτ)}, where exp(iτ)∈S1⊂C. Then the metricgcan be (globally) written in the form g=u2dτ2+gΣ,LXu=LXgΣ=gΣ(X,∙) = 0.(1.1) It is straightforward to check that the space-time (M:=R×Σ,g), with g=−u2dt2+gΣ(1.2) is a static solution of the vacuum Einstein equations with negative cosmological constant, with Killing vector field∂t. In order to continue, some definitions are in order: LetMbe the interior of a smooth, compact,n-dimensional manifold-with-boundaryM. A Riemannian manifold (M, g) will be said to beconformally compact at infinity, orconfor-mally compact, if − g=x2g¯,
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for a smooth functionxonMsuch thatxvanishes precisely on the boundary ∂MofM, with non-vanishing gradient there. ¯ Furthergis a Riemannian metric which is regular up-to-boundary onM; the differentiability properties near∂M of a conformally compact metricgwill always refer to those ofg operator¯. The
P:= ΔL+ 2(n−1), where ΔLis the Lichnerowicz Laplacian (cf., e.g.,[24]) associated withg, plays an important role in the study of such metrics. An Einstein metricgwith scalar curvature−n(n−1) will be saidnon-degenerateifPhas trivialL2kernel on the space of trace-free symmetric 2-tensors. We prove the following openness theorem around static metrics for which the Killing vector field has no zeros (see Section 3.1 for terminology):
Theorem1.1Let(M, g)be a non-degenerate, strictly globally static confor-mally compact Riemannian Einstein metric, with conformal infinityγ:= [g¯|∂M] (conformal equivalence class), then any small static perturbation ofγis the con-formal infinity of a strictly globally static Riemannian Einstein metric on M.
Theorem 1.1 is established by the arguments presented at the beginning of Section 2.2, compare [3] for a more detailed treatment. In Section 2.2 we also describe a subclass of Riemannian metrics given by Theorem 1.1 that leads to Lorentzian Einstein metrics with the properties 1-8 listed above. In particu-lar we show that our construction provides non-trivial solutions near the AdS solution in all dimensions. By an abuse of terminology, Riemannian solutions for which the set of zeros of the Killing vector fieldXcontains ann−2 dimensional surfaceN≡Nn−2 will be referred to asblack hole solutions;Nwill be called2the horizon. The corresponding openness result here reads:
Theorem1.2Let(M, g)be a non-degenerate, globally static conformally com-pact Riemannian Einstein metric, with strictly globally static conformal infinity γ that. Suppose 1. eitherM=N×R2, with the action ofS1being by rotations ofR2, or
2.Hn−3(M) ={0}, and the zero set ofXis a smooth(n−2)–dimensional submanifold with trivial normal bundle.
Then any small globally static perturbation ofγis the conformal infinity of a static black hole solution with horizonN.
The proof of Theorem 1.2 is given at the end of Section 2.3. This paper is organised as follows: In Section 2.1 we review some results concerning the Riemannian equivalent of the problem at hand. In Section 2.2 we sketch the construction of the new solutions, and we show that our results in the remaining sections prove existence of non-trivial solutions which are near
2the corresponding Lorentzian solution the setIn Nwill be the pointwise equivalent of the black hole bifurcation surface,i.e., the intersection of the past and future event horizons.
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