NEW DEVELOPMENTS INLORENTZIAN GEOMETRY

Berlin, November 18-20, 2009

Department of Mathematics, Technische Universitat BerlinSFB 647: Space – Time – Matter

Berlin Mathematical School

M Plaue A Rendall M Scherfner

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PROGRAM

The talks will be held in room H1012, in the main building of the TU Berlin.

1st day 2nd day 3rd day09:20 – 09:30 Opening09:30 – 10:20 Y Choquet-Bruhat M Sanchez H Baum10:30 – 11:20 G Hall P LeFloch C Bar11:30 – 11:55 W C Lim S Suhr E Minguzzi12:00 – 14:00 Lunch break Lunch break Lunch break14:00 – 14:25 R Deszcz A M Candela O Muller14:30 – 14:55 W Hasse M A Javaloyes I Kath15:00 – 15:30 Coffee break Coffee break Coffee break15:30 J Smulevici R Geroch Closing

(until 15:55) (until 16:20) (until 15:45)16:00 Poster session

TALKS

C Bar: Dirac type operators on Lorentzian manifolds andquantization

We first summarize basic analytic properties of Dirac type operators onglobally hyperbolic spacetimes. Then we quantize Dirac fields in the senseof algebraic quantum field theory. It turns out that a fermionic CAR quan-tization requires much more restrictive assumptions than a bosonic CCRquantization.

H Baum: The Lorentzian conformal analogon of Calabi–Yaumanifolds

Calabi–Yau manifolds are Riemannian manifolds with holonomy group SU(m).They are Ricci-flat and Kahler and admit a 2-parameter family of parallelspinors. In the talk we will discuss the Lorentzian conformal analogon of thissituation. If on a manifold a class of conformally equivalent metrics [g] isgiven, then one can consider the holonomy group of the conformal manifold(M, [g]), which is a subgroup of O(p + 1, q + 1) if the metric g has signature(p, q). There is a close relation between algebraic properties of the conformalholonomy group and the existence of Einstein metrics in the conformal classas well as to the existence of conformal Killing spinors. In the talk we willexplain classification results for conformal holonomy groups of Lorentzianmanifolds. In particular, we will describe Lorentzian manifolds (M, g) withconformal holonomy group SU(1, m), which can be viewed as the confor-mal analogon of Calabi–Yau manifolds. Such Lorentzian metrics g, known

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as Fefferman metrics, appear on S1-bundles over strictly pseudoconvex CRspin manifolds and admit a 2-parameter family of conformal Killing spinors.

AM Candela: Geodesics in stationary spacetimes: variationaltools and geometric arguments

In the last years, an intensive research on the existence of geodesics in sta-tionary spacetimes has been carried out and, from an analytical viewpoint,variational tools and topological methods have been systematically appliedto the strongly indefinite associated action functional.Recently, some technical assumptions, which are needed for this approach,have found a natural interpretation in the conformal structure (causality)of the manifold.As a consequence, we are able to prove the existence of geodesics connectingtwo points or, more in general, two submanifolds in any globally hyperbolicstationary spacetime which admits a complete timelike Killing vector fieldand a complete Cauchy hypersurface.Moreover, some multiplicity results follow “easily” from the analytic ap-proach.

Y Choquet-Bruhat: Cauchy problem on a characteristic conefor the Einstein equations in arbitrary dimensions

We prove local existence and geometric uniqueness of a Lorentzian metricsolution of the Einstein equations in arbitrary dimensions with data theconformal class of the quadratic form induced by this metric on a character-istic cone. We use a wave-map gauge, establish the corresponding wave-mapgauge Einstein constraints, study their solutions and their use for the evo-lution problem.The talk is based on joint work with Piotr Chrusciel and Jose Marın-Garcıa.

R Deszcz: Generalized Robertson–Walker spacetimessatisfying curvature conditions of pseudosymmetry type

In this talk we present a survey of results on generalized Robertson–Walkerspacetimes (M, g), of dimension greater or equal four, for which the (0, 6)-tensors: R ·R and R ·C−C ·R are expressed by some linear combinations ofthe Tachibana tensors formed by the metric tensor g, the curvature tensor R,the Ricci tensor S, the Weyl conformal curvature tensor C and the Kulkarni–Nomizu product of g and S.

R Geroch: Faster than Light?

It is widely believed that relativity – both special and general – must requirethat no physical signal travel at a speed faster than that of light. Andthere are good arguments – both mathematical and physical – to supportthis belief. For instance, the assumption that there could be superluminalsignals in relativity gives rise to well-known paradoxes. We suggest that

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this situation is not as clear-cut as it appears at first sight. Indeed, weshall argue that relativity with superluminal signals allowed is as viable andself-consistent as a physical theory as when such signals are excluded.

G Hall: Projective structure in 4-dimensional Lorentzmanifolds

This talk is based on joint work with Dr David Lonie and will discuss the fol-lowing problem; suppose g and g′ are two Lorentz metrics on a 4-dimensionalconnected manifold M and let D and D′ be their respective Levi–Civitaconnections. Suppose also that the unparametrised geodesics of D and D′

coincide. How are D and D′ (and g and g′) related? Such a problem has re-ceived much attention recently. In the case when (M, g) is an Einstein space,the problem has been solved and, in fact, either each of g and g′ is a metricof constant curvature on M or D = D′ (and in the latter case, (M, g′) isalso an Einstein space). This lecture is concerned with the case when (M, g)is not an Einstein space. It will be shown that by a consideration of theholonomy group of (M, g) (more precisely of D) considerable progress canbe made in either establishing again the consequence D = D′ (which occursin many cases) or actually finding the metric g′ and connection D′ in termsof g and D when D and D′ are not equal.There is an important relationship between this problem and the Newton–Einstein principle of equivalence in general relativity. Thus, for the im-portant class of (non-trivial) vacuum space–times in general relativity, theunparametrised timelike space–time geodesics determine D uniquely andwith one special case, (the “pp-waves”), excepted determine g up to a con-stant conformal factor. However, for example, the situation for the FRWLcosmological metrics is a little more complicated and D and D′ need not beequal.Some brief remarks will be made on the link between this work and thestudy of projective symmetry on 4-dimensional manifolds.

W Hasse: On timelike surfaces in Lorentzian manifolds

We discuss the geometry of timelike surfaces (two-dimensional submanifolds)in a Lorentzian manifold and its interpretation in terms of general relativity.A classification of such surfaces is presented which distinguishes four casesof special algebraic properties of the second fundamental form from thegeneric case. With the physical interpretation of the timelike surface as theworldsheet of a “track”, our classification turns out to be closely relatedto the visual appearance of the track, gyroscopic transport along it andinertial forces perpendicular to it. We illustrate our general results withtimelike surfaces in the Kerr–Newman spacetime.The talk is based on joint work with Volker Perlick.

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MA Javaloyes: The bumpy metric theorem in Lorentziangeometry

We prove the Lorentzian bumpy metric theorem using equivariant varia-tional genericity. The theorem states that, on a given compact manifoldM , the set of Lorentzian metrics that admit only nondegenerate closedgeodesics is generic relatively to the Ck-topology, k = 2, . . . ,∞, in the setof Lorentzian metrics on M . A higher order genericity Riemannian resultof Klingenberg and Takens is extended to Lorentzian geometry.

I Kath: Lorentzian extrinsic symmetric spaces

A non-degenerate submanifold of a pseudo-Euclidean space is called an ex-trinsic symmetric space if it is invariant under the reflection at each ofits normal spaces. Similar to usual symmetric spaces extrinsic symmetricspaces can be characterised by curvature. They are exactly those connectedcomplete submanifolds whose second fundamental form is parallel. We de-scribe extrinsic symmetric spaces by their associated infinitesimal objects.We sketch a structure theory for these algebraic objects. As an applicationwe classify all Lorentzian extrinsic symmetric spaces in arbitrary pseudo-Euclidean spaces.

P LeFloch: Einstein spacetimes with bounded curvature

I will present recent results on Einstein spacetimes of general relativity whenthe curvature is solely assumed to be bounded and no assumption on itsderivatives is made. One such result, in a joint work with B.-L. Chen, con-cerns the optimal regularity of pointed spacetimes in which, by definition,an “observer” has been specified. Under geometric bounds on the curva-ture and injectivity radius near the observer, there exist a CMC (constantmean curvature) foliation as well as CMC–harmonic coordinates, which aredefined in geodesic balls with definite size depending only on the assumedbounds, so that the components of the Lorentzian metric has optimal regu-larity in these coordinates. The proof combines geometric estimates (Jacobifield, comparison theorems) and quantitative estimates for nonlinear ellipticequations with low regularity.

References:P.G. LeFloch, Injectivity radius and optimal regularity for Lorentzian man-ifolds with bounded curvature, Actes Semin. Theor. Spectr. Geom. (2010).B.-L. Chen and P.G. LeFloch, Local foliations and optimal regularity ofEinstein spacetimes, J. Geom. Phys. 59 (2009), 913-941.P.G. LeFloch, Local canonical foliations of Lorentzian manifolds with bound-ed curvature, Proc. Workshop on “Geometry, Topology, QFT, and Cosmol-ogy”, May 2008, Paris-Meudon Observatory, 2009.B.-L. Chen and P.G. LeFloch, Injectivity radius estimates for Lorentzianmanifolds, Commun. Math. Phys. 278 (2008), 679–713.

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WC Lim: Spiky mixmaster dynamics

It was thought that the general dynamics near a singularity are mildly in-homogeneous, i.e. spatial derivative terms are much smaller than time de-rivative terms. Recent evidence of formation of spiky structures suggestsotherwise. Exact solutions for the spikes have been found by applying asolution-generating transformation to known solutions. Numerical simula-tions indicate that general solutions converge to the spike solutions near asingularity, and that the spikes recur.

E Minguzzi: Time functions as utilities

In this talk I show that the problem of the existence of a time function orof a semi-time function in general relativity is mathematically equivalentto that of the existence of a continuous utility function for an agent inmicroeconomics. Theorems developed on the economics side can thereforebe imported to the relativistic side. With this strategy it is possible to givea new proof that stable causality is equivalent to the existence of a timefunction and also to clarify in which circumstances the causal structure canbe recovered from the set of time functions allowed by the spacetime.

O Muller: Nice foliations of globally hyperbolic manifolds

In this short talk, I present some new results about the construction oftime functions in globally hyperbolic manifolds which display additionalproperties, like bounded length of the gradient. The results presented arethose obtained in my article in the arXiv with the same title.

M Sanchez: On the structure of globally hyperbolicspacetimes and their embeddability in Lorentz–Minkowski space

Since the independent works by Greene and Clarke in 1970, it is well-knownthat classical Nash’ theorem can be extended to the indefinite case, i.e.,any semi-Riemannian manifold can be isometrically embedded in a semi-Euclidean space of sufficiently large dimension and index. However, theproblem becomes more complicated if one tries to embed a Lorentzian man-ifold (M, g) in some Lorentz–Minkowski space LN . Clarke considered alsothis case, but in his epoch the role of the so-called folk problems on smootha-bility was unclear, and was not taken into account in his proof. Our aim isto explain how the solution of the smothability problems yield also a simpleand complete answer for the embedding problem.More precisely, a direct argument shows that such a embedding exists iff(M, g) is a stably causal spacetime which admits a temporal function τ suchthat g(∇τ,∇τ) < −1. Then, we revisit the techniques for smoothability inorder to show that any globally hyperbolic spacetime admits such a function.This not only gives a self-contained solution of the embedding problem, butalso yields a global orthogonal splitting with bounded lapse for any globallyhyperbolic spacetime.

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The talk is based on a joint work with O. Muller, arXiv:0812.4439.

J Smulevici: Structure of singularities in cosmologicalspacetimes with symmetry

I will review recent results concerning the study of the global Cauchy prob-lem for the Einstein equations and the Einstein–Vlasov system under T2 orsurface symmetric assumptions. More precisely, I will focus on the asymp-totic behavior of the area element and the related issues of the structure ofsingularities and strong cosmic censorship.

S Suhr: Maximal geodesics and class A space–times

Aubry–Mather theory in several degrees of freedom is a theory about theglobal dynamical properties of minimizers of certain positive definite La-grangian systems. Here a flowline of the Euler–Lagrange flow is a minimizerif the lift to the Abelian cover minimizes the Lagrangian action of the liftedLagrangian. An introductory reference is [Ma] (see below). When attempt-ing to develop such a theory for maximizers of the geodesic flow of a compactLorentzian manifold, one is naturally led to a class of space–times here calledclass A space–times. A compact space–time (M, g) is of class A if (M, g) isvicious and the Abelian cover (M, g) is globally hyperbolic. The first partof the talk will show that the set of class A space–times is open in the fineC0-topology on Lorentzian metrics of a given compact manifold M . Theproof consists in a new application of a method due to D. Burago [Bu].The second result to be presented is a coarse-Lipschitz continuity of theLorentzian distance of (M, g) on large subsets of I+. The natural questionof Lipschitz continuity then is reduced to a conjecture related to the timelikeco-ray condition of Galloway and Horta [GH] and conditon (S) of Seifert[S].

References:[Bu] D. Yu Burago. Periodic Metrics, Advances in Soviet Mathematics,Volume 9 (1992), 205-210[GH] G. Galloway and A. Horta. Regularity of Lorentzian Busemann Func-tions, Trans. Amer. Math. Soc., 348, 5(1996), 2063-2084[Ma] J. Mather. Variational Construction of Connecting Orbits, Ann. Inst.Fourier, 43, 5(1993), 1349-1386[S] H.-J. Seifert. Global Connectivity by Timelike Geodesics. Z. Natur-forsch, 22a (1967), 1356-1360

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POSTERS

M Caballero and RM Rubio: Calabi–Bernstein problems forspacelike slices in certain generalized Robertson–Walker

spacetimes

All the entire solutions to the maximal surface differential equation in cer-tain generalized Robertson–Walker spacetimes obeying several energy con-ditions are found. Motivated by this problem we study the correspondingparametric version.

AC Coken: On higher curvatures of a strip in Lorentzian space

In this paper, we define and calculate the higher order curvature of a curvein Lorentzian space. Then we give the higher order curvatures of a strip(non-null curve–Lorentzian manifold pair) in Lorentzian space. We alsoderive the relations between the higher order curvature strip and the highercurvatures of the non-null curve in Lorentzian space.

M Rinaldelli: The stability of causality under metricperturbations of limited extension

By definition a spacetime is stably causal if it is possible to widen the lightcones all over the spacetime without spoiling causality. This condition isalso equivalent to the antisymmetry of the Seifert relation. We investi-gate by means of antisymmetric relations what happens if causality is pre-served widening the light cones in finite spacetime regions or at infinity. Inthe former case, recently added to the causal hierarchy as “compact stablecausality”, we prove that this level can be obtained as the antisymmetrycondition of a new causality relation which we identify. In the latter, weshow the corresponding relation and we prove that if the spacetime is atleast non-total imprisoning then it is also stably causal (here we use theequivalence between stable causality and K-causality). Finally, we give atopological characterization of stable causality and compact stable causalityin the space of conformal metrics equipped with the interval topology.

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LIST OF PARTICIPANTS

Altomani Andrea University of Luxembourg LuxembourgAvila Gaston Albert Einstein Institute GermanyBar Christian University of Potsdam GermanyBarbos Aneta Albert Einstein Institute GermanyBaum Helga Humboldt University of Berlin GermanyBenavides Navarro Jhon Jairo University of Florence ItalyBorn Stefan TU Berlin GermanyCaballero Magdalena University of Cordoba SpainCandela Anna Maria University of Bari ItalyCederbaum Carla Albert Einstein Institute GermanyChoquet-Bruhat Yvonne I.H.E.S. FranceCoken A.Ceylan Suleyman Demirel University TurkeyDeszcz Ryszard WrocÃlaw University of Environ-

mental and Life SciencesPoland

Dillen Franki Katholieke Universiteit Leuven BelgiumDirmeier Alexander TU Berlin GermanyFischmann Matthias Albert Einstein Institute GermanyGeroch Robert University of Chicago USAGudapati Nishanth Free University of Berlin, AEI GermanyHall Graham University of Aberdeen Scotland, UKHasse Wolfgang TU Berlin GermanyHrdina Jaroslav Brno University of Technology Czech RepublicJavaloyes Miguel Angel University of Granada SpainKath Ines EMAU Greifswald GermanyKures Miroslav Brno University of Technology Czech RepublicLampe Matthias University of Leipzig GermanyLawn Marie-Amelie University of Luxembourg LuxembourgLeFloch Philippe University of Paris 6, CNRS FranceLim Woei Chet Albert Einstein Institute GermanyMuller Olaf University of Regensburg GermanyMinguzzi Ettore University of Florence ItalyNardmann Marc University of Regensburg GermanyNikcevic Stana SANU SerbiaNungesser Ernesto Albert Einstein Institute GermanyPlaue Matthias TU Berlin GermanyRendall Alan Albert Einstein Institute GermanyRinaldelli Mauro University of Florence ItalyRubio Rafael M. University of Cordoba SpainSanchez Miguel University of Granada SpainScherfner Mike TU Berlin GermanySimon Udo TU Berlin Germany

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Smulevici Jacques Albert Einstein Institute GermanyStiller Michael University of Hamburg GermanySuhr Stefan University of Freiburg GermanyUllrich Stefan TU Berlin GermanyVasik Petr Brno University of Technology Czech Republic

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RESTAURANTS & PUBS

Damas Goethestr. 4, between Knesebeckstr. and Steinplatz. Syrianrestaurant, with a small Syrian diner next door selling falafel.The restaurant is slightly expensive. Opening at noon.

Dicke Wirtin Carmerstr. 9, near Savignyplatz. Traditional Berlin pub(Kneipe) with German cuisine. With smoking/non-smokingareas. Opening at noon.

Manjurani Knesebeckstr. 4, Indian restaurant. Opening at noon.Math cafeteria Math building, ground floor. Mostly coffee and sandwiches,

small selection of warm food at noon. Open 8 am – 6 pm.Mensa The students’ cafeteria, Hardenbergstr. 34. Very reasonably

priced food. Open 11 am – 2:30 pm.Moon Thai 2 Knesebeckstr. 15, Thai restaurant. Closed from 3 – 5 pm.Pasta e Basta Knesebeckstr. 94, Italian restaurant offering a large variety of

food – except pizza. Slightly expensive. Opening at noon.Schwarzes Cafe Kantstr. 148, near Savignyplatz. Trendy diner frequented by

students, tourists and artsy type people alike. Open 24/7.Staff cafeteria Math building (Straße des 17. Juni 136), 9th floor. Reasonably

priced food. Open 11 am – 4 pm.Telefunken cafeteria TU high-rise (Ernst-Reuter-Platz 7), top floor. Very nice view,

slightly more expensive than the staff or student cafeteria.Open 9 am – 5 pm.

There are a number of restaurants, coffee bars etc. near and around Savignyplatz,which may be found at the south end of Knesebeckstr.

CLOSE-BY PLACES OF INTEREST

Aquarium Budapester Str. 32. Next to fish and othersea creatures the Berlin Aquarium featuresamphibians, reptiles, insects and arachnids.Open 9 am – 6 pm.

Kaiser Wilhelm Memorial Church Breitscheidplatz, a memorial as a warning forfuture generations.

Kaufhaus des Westens The KaDeWe at Wittenbergplatz is the secondlargest department store in Europe. Open 10am – 8 pm.

Zoological Garden Hardenbergplatz 8, most visited zoo in Europeand most species-rich zoo in the world. Open9 am – 5 pm.

BIB Main library H Main buildingMA Math building (also math library) Mensa CafeteriaTEL TU high-rise