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  • 1.
    Andréasson, Håkan
    et al.
    Chalmers.
    Ringström, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Proof of the cosmic no-hair conjecture in the T3-Gowdy symmetric Einstein-Vlasov setting2016In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 18, no 7, p. 1565-1650Article in journal (Refereed)
  • 2. Heinzle, J. Mark
    et al.
    Ringström, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Future asymptotics of vacuum Bianchi type VI0 solutions2009In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 26, no 14Article in journal (Refereed)
    Abstract [en]

    In this paper, we present a thorough analysis of the future asymptotic dynamics of spatially homogeneous cosmological models of Bianchi type VI0. Each of these models converges to a flat Kasner solution (Taub solution) for late times; we give detailed asymptotic expansions describing this convergence. In particular, we prove that the future asymptotics of Bianchi type VI0 solutions cannot be approximated in any way by Bianchi type II solutions, which is in contrast to Bianchi type VIII and IX models (in the direction toward the singularity). The paper contains an extensive introduction where we put the results into a broader context. The core of these considerations consists in the fact that there exist regions in the phase space of Bianchi type VIII models where solutions can be approximated, to a high degree of accuracy, by type VI0 solutions. The behavior of solutions in these regions is essential for the question of 'locality', i.e., whether particle horizons form or not. Since Bianchi type VIII models are conjectured to be important role models for generic cosmological singularities, our understanding of Bianchi type VI0 dynamics might thus be crucial to help to shed some light on the important question of whether to expect generic singularities to be local or not.

  • 3.
    Ringstrom, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Cosmic Censorship for Gowdy Spacetimes2010In: Living Reviews in Relativity, ISSN 1433-8351, E-ISSN 1433-8351, Vol. 13, p. 2-Article in journal (Refereed)
    Abstract [en]

    Due to the complexity of Einstein's equations, it is often natural to study a question of interest in the framework of a restricted class of solutions. One way to impose a restriction is to consider solutions satisfying a given symmetry condition. There are many possible choices, but the present article is concerned with one particular choice, which we shall refer to as Gowdy symmetry. We begin by explaining the origin and meaning of this symmetry type, which has been used as a simplifying assumption in various contexts, some of which we shall mention. Nevertheless, the subject of interest here is strong cosmic censorship. Consequently, after having described what the Gowdy class of spacetimes is, we describe, as seen from the perspective of a mathematician, what is meant by strong cosmic censorship. The existing results on cosmic censorship are based on a detailed analysis of the asymptotic behavior of solutions. This analysis is in part motivated by conjectures, such as the BKL conjecture, which we shall therefore briefly describe. However, the emphasis of the article is on the mathematical analysis of the asymptotics, due to its central importance in the proof and in the hope that it might be of relevance more generally. The article ends with a description of the results that have been obtained concerning strong cosmic censorship in the class of Gowdy spacetimes.

  • 4.
    Ringström, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    A Unified Approach to the Klein-Gordon Equation on Bianchi Backgrounds2019In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 372, no 2, p. 599-656Article in journal (Refereed)
    Abstract [en]

    In this paper, we study solutions to the Klein-Gordon equation on Bianchi backgrounds. In particular, we are interested in the asymptotic behaviour of solutions in the direction of silent singularities. The main conclusion is that, for a given solution u to the Klein-Gordon equation, there are smooth functions u(i), i = 0, 1, on the Lie group under consideration, such that u(sigma) (. , sigma) - u(1) and u(. , sigma) - u(1)sigma - u(0) asymptotically converge to zero in the direction of the singularity (where s is a geometrically defined time coordinate such that the singularity corresponds to sigma -> -infinity). Here u(i), i = 0, 1, should be thought of as data on the singularity. Interestingly, it is possible to prove that the asymptotics are of this form for a large class of Bianchi spacetimes. Moreover, the conclusion applies for singularities that arematter dominated; singularities that are vacuum dominated; and even when the asymptotics of the underlying Bianchi spacetime are oscillatory. To summarise, there seems to be a universality as far as the asymptotics in the direction of silent singularities are concerned. In fact, it is tempting to conjecture that as long as the singularity of the underlying Bianchi spacetime is silent, then the asymptotics of solutions are as described above. In order to contrast the above asymptotics with the non-silent setting, we, by appealing to known results, provide a complete asymptotic characterisation of solutions to the Klein-Gordon equation on a flat Kasner background. In that setting, us does, generically, not converge.

  • 5. Ringström, Hans
    Asymptotic expansions close to the singularity in Gowdy spacetimes2004In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 21, no 3, p. S305-S322Article in journal (Refereed)
    Abstract [en]

    We consider Gowdy spacetimes under the assumption that the spatial hypersurfaces are diffeomorphic to the torus. The relevant equations are then wave map equations with the hyperbolic space as a target. In a paper by Grubisic and Moncrief, a formal expansion of solutions in the direction towards the singularity was proposed. Later, Kichenassamy and Rendall constructed a family of real analytic solutions with the maximum number of free functions and the desired asymptotics at the singularity. The condition of real analyticity was subsequently removed by Rendall. In a previous paper, we proved that one can put a condition on initial data that leads to asymptotic expansions. However, control of up to and including three derivatives in L-2 was necessary, and the condition was rather technical. The main point of the present paper is to demonstrate the existence of certain monotone quantities and to illustrate how these can be used to weaken the assumptions to one derivative in the sup norm. Furthermore, we demonstrate that the false spikes do not appear in the disc model. Finally, we show that knowledge concerning the behaviour of the solution (as time tends to the singularity) for one fixed spatial point in some situations can be used to conclude that there are smooth expansions in the neighbourhood of that spatial point.

  • 6.
    Ringström, Hans
    KTH, Superseded Departments, Mathematics.
    Curvature blow up in Bianchi VIII and IX vacuum spacetimes2000In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 17, no 4, p. 713-731Article in journal (Refereed)
    Abstract [en]

    The maximal globally hyperbolic development of non-Taub-NUT Bianchi IX vacuum initial data and of non-NUT Bianchi VIII vacuum initial data is C-2-inextendible. Furthermore, a curvature invariant is unbounded in the incomplete directions of inextendible causal geodesics.

  • 7.
    Ringström, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Curvature blow up on a dense subset of the singularity in T-3-Gowdy2005In: Journal of Hyperbolic Differential Equations, ISSN 0219-8916, Vol. 2, no 2, p. 547-564Article in journal (Refereed)
    Abstract [en]

    This paper is concerned with the Einstein vacuum equations under the additional assumption of T-3-Gowdy symmetry. We prove that there is a generic set of initial data such that the corresponding solutions exhibit curvature blow up on a dense subset of the singularity. By generic, we mean a countable intersection of open sets (i.e. a G(delta) set) which is also dense. Furthermore, the set of initial data is given the C-infinity topology. This result was presented at a conference in Miami 2004. Recently, we have obtained a stronger result, but the argument to prove it is different and much longer. Therefore, we here wish to present the original argument. Finally, combining the results presented here with a paper by Chrusciel and Lake, one obtains strong cosmic censorship for (T)3-Gowdy spacetimes.

  • 8.
    Ringström, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Data at the moment of infinite expansion for polarized Gowdy2005In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 22, no 9, p. 1647-1653Article in journal (Refereed)
    Abstract [en]

    In a recent paper by Thomas Jurke, it was proved that the asymptotic behaviour of a solution to the polarized Gowdy equation in the expanding direction is of the form alpha ln t + beta + t(-1/2)v + O(t(-3/2)), where alpha and beta are constants and v is a solution to the standard wave equation with zero mean value. Furthermore, it was proved that alpha, beta and v uniquely determine the solution. Here we wish to point out that given alpha, beta and v with the above properties, one can construct a solution to the polarized Gowdy equation with the above asymptotics. In other words, we show that alpha, beta and v constitute data at the moment of infinite expansion. We then use this fact to make the observation that there are polarized Gowdy spacetimes such that in the areal time coordinate, the quotient of the maximum and the minimum of the mean curvature on a constant time hypersurface is unbounded as time tends to infinity.

  • 9.
    Ringström, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Existence of an asymptotic velocity and implications for the asymptotic behavior in the direction of the singularity in T-3-Gowdy2006In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 59, no 7, p. 977-1041Article in journal (Refereed)
    Abstract [en]

    This is the first of two papers that together prove strong cosmic censorship in T-3-Gowdy space-times. In the end, we prove that there is a set of initial data, open with respect to the C-2 X C-1 topology and dense with respect to the C-infinity topology, such that the corresponding space-times have the following properties: Given an inextendible causal geodesic, one direction is complete and the other is incomplete; the Kretschmann scalar, i.e., the Riemann tensor contracted with itself, blows up in the incomplete direction. In fact, it is possible to give a very detailed description of the asymptotic behavior in the direction of the singularity for the generic solutions. In this paper, we shall, however, focus on the concept of asymptotic velocity. Under the symmetry assumptions made here, Einstein's equations reduce to a wave map equation with a constraint. The target of the wave map is the hyperbolic plane. There is a natural concept of kinetic and potential energy density; perhaps the most important result of this paper is that the limit of the potential energy as one lets time tend to the singularity for a fixed spatial point is 0 and that the limit exists for the kinetic energy. We define the asymptotic velocity v(infinity) to be the nonnegative square root of the limit of the kinetic energy density. The asymptotic velocity has some very important properties. In particular, curvature blowup and the existence of smooth expansions of the solutions close to the singularity can be characterized by the behavior of v(infinity). It also has properties such that if 0 < v(infinity)(theta(0)) < 1, then v(infinity) is smooth in a neighborhood of theta(0). Furthermore, if v(infinity)(theta(0)) > 1 and v(infinity) is continuous at theta(0), then v(infinity) is smooth in a neighborhood of theta(0). Finally, we show that the map from initial data to the asymptotic velocity is continuous under certain circumstances and that what will in the end constitute the generic set of solutions is an open set with respect to the C-2 X C-1 topology on initial data.

  • 10. Ringström, Hans
    Future asymptotic expansions of Bianchi VIII vacuum metrics2003In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 20, no 11, p. 1943-1989Article in journal (Refereed)
    Abstract [en]

    Bianchi VIII vacuum solutions to Einstein's equations are causally geodesically complete to the future, given an appropriate time orientation, and the objective of this paper is to analyse the asymptotic behaviour of solutions in this time direction. For the Bianchi class A spacetimes, there is a formulation of the field equations that was presented in a paper by Wainwright and Hsu, and in a previous paper we analysed the asymptotic behaviour of solutions in these variables. One objective of this paper is to give an asymptotic expansion for the metric. Furthermore, we relate this expansion to the topology of the compactified spatial hypersurfaces of homogeneity. The compactified spatial hypersurfaces have the topology of Seifert fibred spaces, and we prove that in the case of NUT Bianchi VIII spacetimes, the length of a circle fibre converges to a positive constant but that in the case of general Bianchi VIII solutions, the length tends to infinity at a rate we determine. Finally, we give asymptotic expansions for general Bianchi VII0 metrics.

  • 11.
    Ringström, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Future stability of the Einstein-non-linear scalar field system2008In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 173, no 1, p. 123-208Article in journal (Refereed)
    Abstract [en]

    We consider the question of future global non-linear stability in the case of Einstein's equations coupled to a non-linear scalar field. The class of potentials V to which our results apply is defined by the conditions V(0)> 0, V'(0)=0 and V ''(0)> 0. Thus Einstein's equations with a positive cosmological constant represents a special case, obtained by demanding that the scalar field be zero. In that context, there are stability results due to Helmut Friedrich, the methods of which are, however, not so easy to adapt to the presence of matter. The goal of the present paper is to develop methods that are more easily adaptable. Due to the extreme nature of the causal structure in models of this type, it is possible to prove a stability result which only makes local assumptions concerning the initial data and yields global conclusions in time. To be more specific, we make assumptions in a set of the form B4r(0) (p) for some r(0)> 0 on the initial hypersurface, and obtain the conclusion that all causal geodesics in the maximal globally hyperbolic development that start in Br-0 (p) are future complete. Furthermore, we derive expansions for the unknowns in a set that contains the future of Br-0 (p). The advantage of such a result is that it can be applied regardless of the global topology of the initial hypersurface. As an application, we prove future global non-linear stability of a large class of spatially locally homogeneous spacetimes with compact spatial topology.

  • 12.
    Ringström, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Instability of Spatially Homogeneous Solutions in the Class of T-2-Symmetric Solutions to Einstein's Vacuum Equations2015In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 334, no 3, p. 1299-1375Article in journal (Refereed)
    Abstract [en]

    In the subject of cosmology, spatially homogeneous solutions are often used to model the universe. It is therefore of interest to ask what happens when perturbing into the spatially inhomogeneous regime. To this end, we, in the present paper, study the future asymptotics of solutions to Einstein's vacuum equations in the case of T-2-symmetry. It turns out that in this setting, whether the solution is spatially homogeneous or not can be characterized in terms of the asymptotics of one variable appearing in the equations; there is a monotonic function such that if its limit is finite, then the solution is spatially homogeneous and if the limit is infinite, then the solution is spatially inhomogeneous. In particular, regardless of how small the initial perturbation away from spatial homogeneity is, the resulting asymptotics are very different. Using spatially homogeneous solutions as models is therefore, in this class, hard to justify.

  • 13. Ringström, Hans
    On a wave map equation arising in general relativity2004In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 57, no 5, p. 657-703Article in journal (Refereed)
    Abstract [en]

    We consider a class of space-times for which the essential part of Einstein's equations can be written as a wave map equation. The domain is not the standard one, but the target is hyperbolic space. One ends up with a 1 + 1 nonlinear wave equation, where the space variable belongs to the circle and the time variable belongs to the positive real numbers. The main objective of this paper is to analyze the asymptotics of solutions to these equations as t --> infinity. For each point in time, the solution defines a loop in hyperbolic space, and the first result is that the length of this loop tends to 0 as t(-1/2) as t --> infinity. In other words, the solution in some sense becomes spatially homogeneous. However, the asymptotic behavior need not be similar to that of spatially homogeneous solutions to the equations. The orbits of such solutions are either a point or a geodesic in the hyperbolic plane. In the nonhomogeneous case, one gets the following asymptotic behavior in the upper half-plane (after applying an isometry of hyperbolic space if necessary): (1) The solution converges to a point. (2) The solution converges to the origin on the boundary along a straight line (which need not be perpendicular to the boundary). (3) The solution goes to infinity along a curve y = const. (4) The solution oscillates around a circle inside the upper half-plane. Thus, even though the solutions become spatially homogeneous in the sense that the spatial variations die out, the asymptotic behavior may be radically different from anything observed for spatially homogeneous solutions of the equations. This analysis can then be applied to draw conclusions concerning the associated class of space-times. For instance, one obtains the leading-order behavior of the functions appearing in the metric, and one can conclude future causal geodesic completeness.

  • 14. Ringström, Hans
    On curvature decay in expanding cosmological models2006In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 264, no 3, p. 613-630Article in journal (Refereed)
    Abstract [en]

    Consider a globally hyperbolic cosmological spacetime. Topologically, the spacetime is then a compact 3-manifold in cartesian product with an interval. Assuming that there is an expanding direction, is there any relation between the topology of the 3-manifold and the asymptotics? In fact, there is a result by Michael Anderson, where he obtains relations between the long-time evolution in General Relativity and the geometrization of 3-manifolds. In order to obtain conclusions however, he makes assumptions concerning the rate of decay of the curvature as proper time tends to infinity. It is thus of interest to find out if such curvature decay conditions are always fulfilled. We consider here the Gowdy spacetimes, for which we prove that the decay condition holds. However, we observe that for general Bianchi VIII spacetimes, the curvature decay condition does not hold, but that some aspects of the expected asymptotic behaviour are still true.

  • 15. Ringström, Hans
    On Gowdy vacuum spacetimes2004In: Mathematical proceedings of the Cambridge Philosophical Society (Print), ISSN 0305-0041, E-ISSN 1469-8064, Vol. 136, p. 485-512Article in journal (Refereed)
    Abstract [en]

    Using Fuchsian techniques, a large family of Gowdy vacuum spacetimes have been constructed for which one has detailed control over asymptotic behaviour. In this paper we formulate a, condition on initial data yielding the same form of asymptotics.

  • 16. Ringström, Hans
    On the T-3-Gowdy symmetric Einstein-Maxwell equations2006In: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 7, no 1, p. 1-20Article in journal (Refereed)
    Abstract [en]

    Recently, progress has been made in the analysis of the expanding direction of Gowdy spacetimes. The purpose of the present paper is to point out that some of the techniques used in the analysis can be applied to other problems. The essential equations in the case of the Gowdy spacetimes can be considered as a special case of a wider class of variational problems. Here we are interested in the asymptotic behaviour of solutions to this class of equations. Two particular members arise when considering the T-3-Gowdy symmetric Einstein-Maxwell equations and when considering T-3-Gowdy symmetric IIB superstring cosmology. The main result concerns the rate of decay of a naturally defined energy. A subclass of the variational problems can be interpreted as wave map equations, and in that case one gets the following picture. The non-linear wave equations one ends up with have as a domain the positive real line in Cartesian product with the circle. For each point in time, the wave map can thus be seen as a loop in some Riemannian manifold. As a consequence of the decay of the energy mentioned above, the length of the loop converges to zero at a specific rate.

  • 17.
    Ringström, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the Topology and Future Stability of the Universe2013 (ed. 1)Book (Refereed)
    Abstract [en]

    The subject of the book is the topology and future stability of models of the universe. In standard cosmology, the universe is assumed to be spatially homogeneous and isotropic. However, it is of interest to know whether perturbations of the corresponding initial data lead to similar solutions or not. This is the question of stability. It is also of interest to know what the limitations on the global topology imposed by observational constraints are. These are the topics addressed in the book. The theory underlying the discussion is the general theory of relativity. Moreover, in the book, matter is modelled using kinetic theory. As background material, the general theory of the Cauchy problem for the Einstein–Vlasov equations is therefore developed.

  • 18.
    Ringström, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Origins and development of the Cauchy problem in general relativity2015In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 32, no 12, article id 124003Article in journal (Refereed)
    Abstract [en]

    The seminal work of Yvonne Choquet-Bruhat published in 1952 demonstrates that it is possible to formulate Einstein's equations as an initial value problem. The purpose of this article is to describe the background to and impact of this achievement, as well as the result itself. In some respects, the idea of viewing the field equations of general relativity as a system of evolution equations goes back to Einstein himself; in an argument justifying that gravitational waves propagate at the speed of light, Einstein used a special choice of coordinates to derive a system of wave equations for the linear perturbations on a Minkowski background. Over the following decades, Hilbert, de Donder, Lanczos, Darmois and many others worked to put Einstein's ideas on a more solid footing. In fact, the issue of local uniqueness (giving a rigorous justification for the statement that the speed of propagation of the gravitational field is bounded by that of light) was already settled in the 1930s by the work of Stellmacher. However, the first person to demonstrate both local existence and uniqueness in a setting in which the notion of finite speed of propagation makes sense was Yvonne Choquet-Bruhat. In this sense, her work lays the foundation for the formulation of Einstein's equations as an initial value problem. Following a description of the results of Choquet-Bruhat, we discuss the development of three research topics that have their origin in her work. The first one is local existence. One reason for addressing it is that it is at the heart of the original paper. Moreover, it is still an active and important research field, connected to the problem of characterizing the asymptotic behaviour of solutions that blow up in finite time. As a second topic, we turn to the questions of global uniqueness and strong cosmic censorship. These questions are of fundamental importance to anyone interested in justifying that the Cauchy problem makes sense globally. They are also closely related to the issue of singularities in general relativity. Finally, we discuss the topic of stability of solutions to Einstein's equations. This is not only an important and active area of research, it is also one that only became meaningful thanks to the work of Yvonne Choquet-Bruhat.

  • 19.
    Ringström, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Power Law Inflation2009In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 290, no 1, p. 155-218Article in journal (Refereed)
    Abstract [en]

    The subject of this paper is Einstein's equations coupled to a non-linear scalar field with an exponential potential. The problem we consider is that of proving future global non-linear stability of a class of spatially locally homogeneous solutions to the equations. There are solutions on R(+)xR(n) with accelerated expansion of power law type. We prove a result stating that if we have initial data that are close enough to those of such a solution on a ball of a certain radius, say B-4R0 (p), then all causal geodesics starting in B-R0 (p) are complete to the future in the maximal globally hyperbolic development of the data we started with. In other words, we only make local assumptions in space and obtain global conclusions in time. We also obtain asymptotic expansions in the region over which we have control. As a consequence of this result and the fact that one can analyze the asymptotic behaviour in most of the spatially homogeneous cases, we obtain quite a general stability statement in the spatially locally homogeneous setting.

  • 20.
    Ringström, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Strong cosmic censorship in T-3-Gowdy spacetimes2009In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 170, no 3, p. 1181-1240Article in journal (Refereed)
    Abstract [en]

    Einstein's vacuum equations can be viewed as an initial value problem, and given initial data there is one part of spacetime, the so-called maximal globally hyperbolic development (MGHD), which is uniquely determined up to isometry. Unfortunately, it is sometimes possible to extend the spacetime beyond the MGHD in inequivalent ways. Consequently, the initial data do not uniquely determine the spacetime, and in this sense the theory is not deterministic. It is then natural to make the strong cosmic censorship conjecture, which states that for generic initial data, the MGHD is inextendible. Since it is unrealistic to hope to prove this conjecture in all generality, it is natural to make the same conjecture within a class of spacetimes satisfying some symmetry condition. Here, we prove strong cosmic censorship in the class of T-3-Gowdy spacetimes. In a previous paper, we introduced a set G(i,c) of smooth initial data and proved that it is open in the C-1 x C-0-topology. The solutions corresponding to initial data in G(i,c) have the following properties. First, the MGHD is C-2-inextendible. Second, following a causal geodesic in a given time direction, it is either complete, or a curvature invariant, the Kretschmann scalar, is unbounded along it (in fact the Kretschmann scalar is unbounded along any causal curve that ends on the singularity). The purpose of the present paper is to prove that G(i,c) is dense in the C-infinity-topology.

  • 21.
    Ringström, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Strong cosmic censorship in the case of T(3)-Gowdy vacuum spacetimes2008In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 25, no 11, p. 114010-Article in journal (Refereed)
    Abstract [en]

    In 1952, Yvonne Choquet-Bruhat demonstrated that it makes sense to consider Einstein's vacuum equations from an initial value point of view; given initial data, there is a globally hyperbolic development. Since there are many developments, one does, however, not obtain uniqueness. This was remedied in 1969 when Choquet-Bruhat and Robert Geroch demonstrated that there is a unique maximal globally hyperbolic development (MGHD). Unfortunately, there are examples of initial data for which the MGHD is extendible, and, what is worse, extendible in inequivalent ways. Thus it is not possible to predict what spacetime one is in simply by looking at initial data and, in this sense, Einstein's equations are not deterministic. Since the examples exhibiting this behaviour are rather special, it is natural to conjecture that for generic initial data, the MGHD is inextendible. This conjecture is referred to as the strong cosmic censorship conjecture and is of central importance in mathematical relativity. In this paper, we shall describe this conjecture in detail, as well as its resolution in the special case of T(3)-Gowdy spacetimes.

  • 22. Ringström, Hans
    The Bianchi IX attractor2001In: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 2, no 3, p. 405-500Article in journal (Refereed)
    Abstract [en]

    We consider the asymptotic behaviour of spatially homogeneous space-times of Bianchi type IX close to the singularity (we also consider some of the other Bianchi types, e.g. Bianchi VIII in the stiff fluid case). The matter content is assumed to be an orthogonal perfect fluid with linear equation of state and zero cosmological constant. In terms of the variables of Wainwright and Hsu, we have the following results. In the stiff fluid case, the solution converges to a point for all the Bianchi class A types. For the other matter models we consider, the Bianchi IX solutions generically converge to an attractor consisting of the closure of the vacuum type II orbits. Furthermore, we observe that for all the Bianchi class A spacetimes, except those of vacuum Taub type, a curvature invariant is unbounded in the incomplete directions of inextendible causal geodesics.

  • 23.
    Ringström, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The Cauchy problem in general relativity2009 (ed. 1)Book (Refereed)
  • 24.
    Ringström, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The Cauchy problem in general relativity2013In: Acta Physica Polonica B, ISSN 0587-4254, E-ISSN 1509-5770, Vol. 44, no 12, p. 2621-2641Article in journal (Refereed)
    Abstract [en]

    After a brief introduction to classical relativity, we describe how to solve the Cauchy problem in general relativity. In particular, we introduce the notion of gauge source functions and explain how they can be used in order to reduce the problem to that of solving a system of hyperbolic partial differential equations. We then go on to explain how the initial value problem is formulated for the so-called Einstein-Vlasov system and describe a recent future global non-linear stability result in this setting. In particular, this result applies to models of the universe which are consistent with observations.

  • 25. Ringström, Hans
    The future asymptotics of Bianchi VIII vacuum solutions2001In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 18, no 18, p. 3791-3823Article in journal (Refereed)
    Abstract [en]

    Bianchi VIII vacuum solutions to Einstein's equations are causally geodesically complete to the future, given an appropriate time orientation, and the objective of this paper is to analyse the asymptotic behaviour of solutions in this time direction. For the Bianchi class A spacetimes, there is a formulation of the field equations that was presented in an article by Wainwright and Hsu, and we will analyse the asymptotic behaviour of solutions in these variables. We also try to give the analytic results a geometric interpretation by analysing how a normalized version of the Riemannian metric on the spatial hypersurfaces of homogeneity evolves.

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