Change search
Link to record
Permanent link

Direct link
BETA
Alternative names
Publications (10 of 24) Show all publications
Ringström, H. (2019). A Unified Approach to the Klein-Gordon Equation on Bianchi Backgrounds. Communications in Mathematical Physics, 372(2), 599-656
Open this publication in new window or tab >>A Unified Approach to the Klein-Gordon Equation on Bianchi Backgrounds
2019 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 372, no 2, p. 599-656Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
SPRINGER, 2019
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-265874 (URN)10.1007/s00220-019-03325-7 (DOI)000500284800007 ()2-s2.0-85061726584 (Scopus ID)
Note

QC 20200103

Available from: 2020-01-03 Created: 2020-01-03 Last updated: 2020-01-13Bibliographically approved
Ringström, H. (2015). Instability of Spatially Homogeneous Solutions in the Class of T-2-Symmetric Solutions to Einstein's Vacuum Equations. Communications in Mathematical Physics, 334(3), 1299-1375
Open this publication in new window or tab >>Instability of Spatially Homogeneous Solutions in the Class of T-2-Symmetric Solutions to Einstein's Vacuum Equations
2015 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 334, no 3, p. 1299-1375Article in journal (Refereed) Published
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.

National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-162944 (URN)10.1007/s00220-014-2258-8 (DOI)000350030500005 ()2-s2.0-84925511738 (Scopus ID)
Funder
Swedish Research CouncilGöran Gustafsson Foundation for promotion of scientific research at Uppala University and Royal Institute of Technology
Note

QC 20150402

Available from: 2015-04-02 Created: 2015-03-26 Last updated: 2017-12-04Bibliographically approved
Ringström, H. (2015). Origins and development of the Cauchy problem in general relativity. Classical and quantum gravity, 32(12), Article ID 124003.
Open this publication in new window or tab >>Origins and development of the Cauchy problem in general relativity
2015 (English)In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 32, no 12, article id 124003Article in journal (Refereed) Published
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.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-170202 (URN)10.1088/0264-9381/32/12/124003 (DOI)000356631500004 ()2-s2.0-84930671860 (Scopus ID)
Note

QC 20150630

Available from: 2015-06-30 Created: 2015-06-29 Last updated: 2019-10-02Bibliographically approved
Ringström, H. (2013). On the Topology and Future Stability of the Universe (1ed.). Oxford: Oxford University Press
Open this publication in new window or tab >>On the Topology and Future Stability of the Universe
2013 (English)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.

Place, publisher, year, edition, pages
Oxford: Oxford University Press, 2013. p. xiv, 718 Edition: 1
Series
Oxford Mathematical Monographs
Keywords
cosmology, stability, universe, topology, kinetic theory, general relativity, cauchy problem
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-136372 (URN)10.1093/acprof:oso/9780199680290.001.0001 (DOI)978-0-19-968029-0 (ISBN)
Note

QC 20140305

Available from: 2013-12-04 Created: 2013-12-04 Last updated: 2014-03-05Bibliographically approved
Ringström, H. (2013). The Cauchy problem in general relativity. Paper presented at 53rd Conference of the Cracow-School-of-Theoretical-Physics on Conformal Symmetry and Perspectives in Quantum and Mathematical Gravity, June 28-July 07, 2013, Zakopane, Poland. Acta Physica Polonica B, 44(12), 2621-2641
Open this publication in new window or tab >>The Cauchy problem in general relativity
2013 (English)In: Acta Physica Polonica B, ISSN 0587-4254, E-ISSN 1509-5770, Vol. 44, no 12, p. 2621-2641Article in journal (Refereed) Published
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.

Keywords
Cauchy problems, General Relativity, Hyperbolic partial differential equation, Non-linear stabilities, Source functions
National Category
Other Physics Topics
Identifiers
urn:nbn:se:kth:diva-142897 (URN)10.5506/APhysPolB.44.2621 (DOI)000331371900008 ()2-s2.0-84893506629 (Scopus ID)
Conference
53rd Conference of the Cracow-School-of-Theoretical-Physics on Conformal Symmetry and Perspectives in Quantum and Mathematical Gravity, June 28-July 07, 2013, Zakopane, Poland
Note

QC 20140314

Available from: 2014-03-14 Created: 2014-03-13 Last updated: 2017-12-05Bibliographically approved
Ringstrom, H. (2010). Cosmic Censorship for Gowdy Spacetimes. Living Reviews in Relativity, 13, 2
Open this publication in new window or tab >>Cosmic Censorship for Gowdy Spacetimes
2010 (English)In: Living Reviews in Relativity, ISSN 1433-8351, E-ISSN 1433-8351, Vol. 13, p. 2-Article in journal (Refereed) Published
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.

National Category
Subatomic Physics
Identifiers
urn:nbn:se:kth:diva-26878 (URN)10.12942/lrr-2010-2 (DOI)000280764100001 ()2-s2.0-77953115645 (Scopus ID)
Note
QC 20101130Available from: 2010-11-30 Created: 2010-11-29 Last updated: 2020-03-05Bibliographically approved
Heinzle, J. M. & Ringström, H. (2009). Future asymptotics of vacuum Bianchi type VI0 solutions. Classical and quantum gravity, 26(14)
Open this publication in new window or tab >>Future asymptotics of vacuum Bianchi type VI0 solutions
2009 (English)In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 26, no 14Article in journal (Refereed) Published
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.

Keywords
relativistic cosmology, attractor
Identifiers
urn:nbn:se:kth:diva-18571 (URN)10.1088/0264-9381/26/14/145001 (DOI)000267656300002 ()2-s2.0-70349495298 (Scopus ID)
Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved
Ringström, H. (2009). Power Law Inflation. Communications in Mathematical Physics, 290(1), 155-218
Open this publication in new window or tab >>Power Law Inflation
2009 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 290, no 1, p. 155-218Article in journal (Refereed) Published
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.

Keywords
scalar field system, equations, existence, spacetimes, manifolds, stability
Identifiers
urn:nbn:se:kth:diva-18548 (URN)10.1007/s00220-009-0812-6 (DOI)000267391700007 ()2-s2.0-70349684754 (Scopus ID)
Note
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved
Ringström, H. (2009). Strong cosmic censorship in T-3-Gowdy spacetimes. Annals of Mathematics, 170(3), 1181-1240
Open this publication in new window or tab >>Strong cosmic censorship in T-3-Gowdy spacetimes
2009 (English)In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 170, no 3, p. 1181-1240Article in journal (Refereed) Published
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.

Keywords
gowdy spacetimes, asymptotic-behavior, general-relativity, vacuum, spacetimes, singularity, times
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-19124 (URN)10.4007/annals.2009.170.1181 (DOI)000273791400004 ()2-s2.0-71649093596 (Scopus ID)
Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved
Ringström, H. (2009). The Cauchy problem in general relativity (1ed.). Zürich: European Mathematical Society Publishing House
Open this publication in new window or tab >>The Cauchy problem in general relativity
2009 (English)Book (Refereed)
Place, publisher, year, edition, pages
Zürich: European Mathematical Society Publishing House, 2009. p. 294 Edition: 1
Keywords
General relativity, Cauchy problem, strong cosmic censorship, non-linear wave equations
National Category
Other Mathematics
Identifiers
urn:nbn:se:kth:diva-66675 (URN)978-3-03719-053-1 (ISBN)
Note
QC 20120127Available from: 2012-01-27 Created: 2012-01-27 Last updated: 2012-01-27Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0001-9383-0748

Search in DiVA

Show all publications