NEW DEVELOPMENTS IN LORENTZIAN GEOMETRY

Transcription

1 NEW DEVELOPMENTS IN LORENTZIAN GEOMETRY TALKS C Bär: Dirac type operators on Lorentzian manifolds and quantization We first summarize basic analytic properties of Dirac type operators on globally hyperbolic spacetimes. Then we quantize Dirac fields in the sense of algebraic quantum field theory. It turns out that a fermionic CAR quantization requires much more restrictive assumptions than a bosonic CCR quantization. H Baum: The Lorentzian conformal analogon of Calabi Yau manifolds Calabi Yau manifolds are Riemannian manifolds with holonomy group SU(m). They are Ricci-flat and Kähler and admit a 2-parameter family of parallel spinors. In the talk we will discuss the Lorentzian conformal analogon of this situation. If on a manifold a class of conformally equivalent metrics [g] is given, 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 conformal holonomy group and the existence of Einstein metrics in the conformal class as well as to the existence of conformal Killing spinors. In the talk we will explain classification results for conformal holonomy groups of Lorentzian manifolds. In particular, we will describe Lorentzian manifolds (M, g) with conformal holonomy group SU(1, m), which can be viewed as the conformal analogon of Calabi Yau manifolds. Such Lorentzian metrics g, known as Fefferman metrics, appear on S 1 -bundles over strictly pseudoconvex CR spin manifolds and admit a 2-parameter family of conformal Killing spinors. A M Candela: Geodesics in stationary spacetimes: variational tools and geometric arguments In the last years, an intensive research on the existence of geodesics in stationary spacetimes has been carried out and, from an analytical viewpoint, variational tools and topological methods have been systematically applied to 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 connecting two points or, more in general, two submanifolds in any globally hyperbolic Updated: November 2,

2 2 stationary spacetime which admits a complete timelike Killing vector field and a complete Cauchy hypersurface. Moreover, some multiplicity results follow easily from the analytic approach. Y Choquet-Bruhat: Cauchy problem on a characteristic cone for the Einstein equations in arbitrary dimensions We prove local existence and geometric uniqueness of a Lorentzian metric solution of the Einstein equations in arbitrary dimensions with data the conformal class of the quadratic form induced by this metric on a characteristic cone. We use a wave-map gauge, establish the corresponding wave-map gauge Einstein constraints, study their solutions and their use for the evolution problem. The talk is based on joint work with Piotr Chruściel and José Marín-García. R Deszcz: Generalized Robertson Walker spacetimes satisfying curvature conditions of pseudosymmetry type I In this talk we present a survey of results on generalized Robertson Walker spacetimes (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 of the 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 require that no physical signal travel at a speed faster than that of light. And there are good arguments both mathematical and physical to support this belief. For instance, the assumption that there could be superluminal signals in relativity gives rise to well-known paradoxes. We suggest that this situation is not as clear-cut as it appears at first sight. Indeed, we shall argue that relativity with superluminal signals allowed is as viable and self-consistent as a physical theory as when such signals are excluded. G Hall: Projective structure in 4-dimensional Lorentz manifolds This talk is based on joint work with Dr David Lonie and will discuss the following problem; suppose g and g are two Lorentz metrics on a 4-dimensional connected manifold M and let D and D be their respective Levi Civita connections. 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 received 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 metric of constant curvature on M or D = D (and in the latter case, (M, g ) is

3 also 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 the holonomy group of (M, g) (more precisely of D) considerable progress can be made in either establishing again the consequence D = D (which occurs in many cases) or actually finding the metric g and connection D in terms of 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 important class of (non-trivial) vacuum space times in general relativity, the unparametrised timelike space time geodesics determine D uniquely and with one special case, (the pp-waves ), excepted determine g up to a constant conformal factor. However, for example, the situation for the FRWL cosmological metrics is a little more complicated and D and D need not be equal. Some brief remarks will be made on the link between this work and the study 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 cases of special algebraic properties of the second fundamental form from the generic case. With the physical interpretation of the timelike surface as the worldsheet of a track, our classification turns out to be closely related to the visual appearance of the track, gyroscopic transport along it and inertial forces perpendicular to it. We illustrate our general results with timelike surfaces in the Kerr Newman spacetime. The talk is based on joint work with Volker Perlick. M A Javaloyes: The bumpy metric theorem in Lorentzian geometry We prove the Lorentzian bumpy metric theorem using equivariant variational genericity. The theorem states that, on a given compact manifold M, the set of Lorentzian metrics that admit only nondegenerate closed geodesics is generic relatively to the C k -topology, k = 2,...,, in the set of Lorentzian metrics on M. A higher order genericity Riemannian result of 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 extrinsic symmetric space if it is invariant under the reflection at each of its normal spaces. Similar to usual symmetric spaces extrinsic symmetric spaces can be characterised by curvature. They are exactly those connected 3

4 4 complete submanifolds whose second fundamental form is parallel. We describe extrinsic symmetric spaces by their associated infinitesimal objects. We sketch a structure theory for these algebraic objects. As an application we 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 when the curvature is solely assumed to be bounded and no assumption on its derivatives is made. One such result, in a joint work with B.-L. Chen, concerns the optimal regularity of pointed spacetimes in which, by definition, an observer has been specified. Under geometric bounds on the curvature and injectivity radius near the observer, there exist a CMC (constant mean curvature) foliation as well as CMC harmonic coordinates, which are defined in geodesic balls with definite size depending only on the assumed bounds, so that the components of the Lorentzian metric has optimal regularity in these coordinates. The proof combines geometric estimates (Jacobi field, comparison theorems) and quantitative estimates for nonlinear elliptic equations with low regularity. References: P.G. LeFloch, Injectivity radius and optimal regularity for Lorentzian manifolds with bounded curvature, Actes Semin. Theor. Spectr. Geom. (2010). B.-L. Chen and P.G. LeFloch, Local foliations and optimal regularity of Einstein spacetimes, J. Geom. Phys. 59 (2009), P.G. LeFloch, Local canonical foliations of Lorentzian manifolds with bounded curvature, Proc. Workshop on Geometry, Topology, QFT, and Cosmology, May 2008, Paris-Meudon Observatory, B.-L. Chen and P.G. LeFloch, Injectivity radius estimates for Lorentzian manifolds, Commun. Math. Phys. 278 (2008), W C Lim: Spiky mixmaster dynamics It was thought that the general dynamics near a singularity are mildly inhomogeneous, i.e. spatial derivative terms are much smaller than time derivative terms. Recent evidence of formation of spiky structures suggests otherwise. Exact solutions for the spikes have been found by applying a solution-generating transformation to known solutions. Numerical simulations indicate that general solutions converge to the spike solutions near a singularity, 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 or of a semi-time function in general relativity is mathematically equivalent to that of the existence of a continuous utility function for an agent in

5 microeconomics. Theorems developed on the economics side can therefore be imported to the relativistic side. With this strategy it is possible to give a new proof that stable causality is equivalent to the existence of a time function and also to clarify in which circumstances the causal structure can be recovered from the set of time functions allowed by the spacetime. O Müller: Nice foliations of globally hyperbolic manifolds In this short talk, I present some new results about the construction of time functions in globally hyperbolic manifolds which display additional properties, like bounded length of the gradient. The results presented are those obtained in my article in the arxiv with the same title. M Sánchez: On the structure of globally hyperbolic spacetimes and their embeddability in Lorentz Minkowski space Since the independent works by Greene and Clarke in 1970, it is well-known that 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, the problem becomes more complicated if one tries to embed a Lorentzian manifold (M, g) in some Lorentz Minkowski space L N. Clarke considered also this case, but in his epoch the role of the so-called folk problems on smoothability was unclear, and was not taken into account in his proof. Our aim is to explain how the solution of the smothability problems yield also a simple and 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 such that g( τ, τ) < 1. Then, we revisit the techniques for smoothability in order to show that any globally hyperbolic spacetime admits such a function. This not only gives a self-contained solution of the embedding problem, but also yields a global orthogonal splitting with bounded lapse for any globally hyperbolic spacetime. The talk is based on a joint work with O. Müller, arxiv: S Suhr: Maximal geodesics and class A space times Aubry Mather theory in several degrees of freedom is a theory about the global dynamical properties of minimizers of certain positive definite Lagrangian systems. Here a flowline of the Euler Lagrange flow is a minimizer if the lift to the Abelian cover minimizes the Lagrangian action of the lifted Lagrangian. An introductory reference is [Ma] (see below). When attempting to develop such a theory for maximizers of the geodesic flow of a compact Lorentzian manifold, one is naturally led to a class of space times here called class A space times. A compact space time (M, g) is of class A if (M, g) is vicious and the Abelian cover (M, g) is globally hyperbolic. The first part of the talk will show that the set of class A space times is open in the fine 5

6 6 C 0 -topology on Lorentzian metrics of a given compact manifold M. The proof 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 the Lorentzian distance of (M, g) on large subsets of I +. The natural question of Lipschitz continuity then is reduced to a conjecture related to the timelike co-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), [GH] G. Galloway and A. Horta. Regularity of Lorentzian Busemann Functions, Trans. Amer. Math. Soc., 348, 5(1996), [Ma] J. Mather. Variational Construction of Connecting Orbits, Ann. Inst. Fourier, 43, 5(1993), [S] H.-J. Seifert. Global Connectivity by Timelike Geodesics. Z. Naturforsch, 22a (1967),

7 7 POSTERS M Caballero and R M Rubio: Calabi Bernstein problems for spacelike slices in certain generalized Robertson Walker spacetimes All the entire solutions to the maximal surface differential equation in certain generalized Robertson Walker spacetimes obeying several energy conditions are found. Motivated by this problem we study the corresponding parametric version. A C Çöken: On higher curvatures of a strip in Lorentzian space In this paper, we define and calculate the higher order curvature of a curve in Lorentzian space. Then we give the higher order curvatures of a strip (non-null curve Lorentzian manifold pair) in Lorentzian space. We also derive the relations between the higher order curvature strip and the higher curvatures of the non-null curve in Lorentzian space. M Rinaldelli: The stability of causality under metric perturbations of limited extension By definition a spacetime is stably causal if it is possible to widen the light cones all over the spacetime without spoiling causality. This condition is also equivalent to the antisymmetry of the Seifert relation. We investigate by means of antisymmetric relations what happens if causality is preserved widening the light cones in finite spacetime regions or at infinity. In the former case, recently added to the causal hierarchy as compact stable causality, we prove that this level can be obtained as the antisymmetry condition of a new causality relation which we identify. In the latter, we show the corresponding relation and we prove that if the spacetime is at least non-total imprisoning then it is also stably causal (here we use the equivalence between stable causality and K-causality). Finally, we give a topological characterization of stable causality and compact stable causality in the space of conformal metrics equipped with the interval topology.

8 8 LIST OF PARTICIPANTS Altomani Andrea University of Luxembourg Luxembourg Avila Gaston Albert Einstein Institute Germany Bär Christian University of Potsdam Germany Baum Helga Humboldt University of Berlin Germany Benavides Navarro Jhon Jairo University of Florence Italy Born Stefan TU Berlin Germany Caballero Magdalena University of Córdoba Spain Candela Anna Maria Università di Bari Italy Cederbaum Carla Albert Einstein Institute Germany Choquet-Bruhat Yvonne I.H.É.S. France Çöken A.Ceylan Süleyman Demirel University Turkey Deszcz Ryszard Wroc law University of Environmental and Life Sciences Poland Dirmeier Alexander TU Berlin Germany Geroch Robert University of Chicago USA Gudapati Nishanth Freie Universität Berlin, AEI Germany Hall Graham University of Aberdeen UK Hasse Wolfgang TU Berlin Germany Hrdina Jaroslav Brno University of Technology Czech Republic Javaloyes Miguel Angel Universidad de Granada Spain Kath Ines EMAU Greifswald Germany Kureš Miroslav Brno University of Technology Czech Republic Lawn Marie-Amélie University of Luxembourg Luxembourg LeFloch Philippe University of Paris 6, CNRS France Lim Woei Chet Albert Einstein Institute Germany Müller Olaf Universität Regensburg Germany Minguzzi Ettore Università Degli Studi Di Firenze Italy Nardmann Marc Universität Regensburg Germany Nikcevic Stana SANU Serbia Nungesser Ernesto Albert Einstein Institute Germany Plaue Matthias TU Berlin Germany Rendall Alan Albert Einstein Institute Germany Rinaldelli Mauro Università di Firenze Italy Rubio Rafael M. University of Córdoba Spain Sánchez Miguel Universidad de Granada Spain Scherfner Mike TU Berlin Germany Simon Udo TU Berlin Germany Stiller Michael University of Hamburg Germany Suhr Stefan Universität Freiburg Germany Ullrich Stefan TU Berlin Germany Vasik Petr Brno University of Technology Czech Republic