The Cauchy Problem in General Relativity
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The only known proof of the theorem that a given globally hyperbolic extension of a spacetime is the maximal such extension requires the WEC.
“The fact that the only known proof of the theorem that a given globally hyperbolic extension of a spacetime is the maximal such extension requires the WEC (Ringstr¨ om 2009), in conjunction with the fact that standard proofs of the well-posedness of the initial-value formulation for general relativity rely on the DEC, suggest that it may be the energy conditions that intervene to ensure a cogent”
From A Primer on Energy Conditions (2017) · cites this paper as Ringström 2009
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Energy conditions in general relativity and quantum field theory
Kontou, E.-A., & Sanders, K. (2020) · Classical and Quantum Gravity · doi:10.1088/1361-6382/ab8fcf · Cited by 223 · see citing works
“, energy conditions seem by no means a necessary condition for a classical field theory to have a well-posed initial value formulation, even though they may enter as a useful method in the proof of such a property [8].”
BackgroundMethodCuriel, E. (2017) · Einstein Studies · doi:10.1007/978-1-4939-3210-8_3 · Cited by 226 · see citing works
Choquet–Bruhat, Y. (2014) · doi:10.1093/oso/9780199666454.003.0010 · Cited by 205 · see citing works
The Cauchy problem in general relativity
Ringström, H. (2013) · doi:10.1093/acprof:oso/9780199680290.003.0002 · Cited by 118 · see citing works
Self-citationEuropean Journal for Philosophy of Science
(2013) · European Journal for Philosophy of Science · doi:10.1007/13194.1879-4920 · Cited by 116 · see citing works
On the Topology and Future Stability of the Universe
Ringström, H. (2013) · doi:10.1093/acprof:oso/9780199680290.001.0001 · Cited by 112 · see citing works
Self-citationFifty Million Elvis Fans Can't be Wrong
Belot, G. (2017) · Noûs · doi:10.1111/nous.12200 · Cited by 104 · see citing works
Mathematical general relativity: A sampler
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Introduction to General Relativity, Black Holes, and Cosmology
Choquet–Bruhat, Y. (2014) · doi:10.1093/oso/9780199666454.001.0001 · Cited by 75 · see citing works
Spooky Action at a Temporal Distance
Adlam, E. (2018) · Entropy · doi:10.3390/e20010041 · Cited by 73 · see citing works
Future Stability of the Einstein-Maxwell-Scalar Field System
Svedberg, C. (2011) · Annales Henri Poincaré · doi:10.1007/s00023-011-0099-y · Cited by 63 · see citing works
“6 in [10] as presented in [10].”
BackgroundMethodResultInfluentialCovariant phase space, constraints, gauge and the Peierls formula
Khavkine, I. (2014) · International Journal of Modern Physics A · doi:10.1142/s0217751x14300099 · Cited by 65 · see citing works
Adlam, E. (2022) · Foundations of Physics · doi:10.1007/s10701-022-00546-0 · Cited by 60 · see citing works
Singularity avoidance in Bianchi I quantum cosmology
Kiefer, C., Kwidzinski, N., & Piontek, D. (2019) · The European Physical Journal C · doi:10.1140/epjc/s10052-019-7193-6 · Cited by 50 · see citing works
“In [32] one finds an existence and uniqueness theorem (Theorem 8.”
BackgroundWitek, H., Gualtieri, L., & Pani, P. (2020) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.101.124055 · Cited by 48 · see citing works
Propagation of polarized gravitational waves
Andersson, L., Joudioux, J., Oancea, M. A., & Raj, A. (2021) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.103.044053 · Cited by 47 · see citing works
Global Hyperbolicity for Spacetimes with Continuous Metrics
Sämann, C. (2015) · Annales Henri Poincaré · doi:10.1007/s00023-015-0425-x · Cited by 45 · see citing works
Strong Cosmic Censorship in Vacuum Space-Times with Compact, Locally Homogeneous Cauchy Surfaces
Chruściel, P. T., & Rendall, A. D. (1995) · Annals of Physics · doi:10.1006/aphy.1995.1084 · Cited by 45 · see citing works
Cosmic Censorship for Gowdy Spacetimes
Ringström, H. (2010) · Living Reviews in Relativity · doi:10.12942/lrr-2010-2 · Cited by 44 · see citing works
“Theorem 2 is due to the work of Choquet-Bruhat and Geroch; see [13] for the original paper and [82] for a recent presentation.”
BackgroundInfluentialSelf-citationInverse problem for Einstein-scalar field equations
Kurylev, Y., Lassas, M. J., Oksanen, L., et al. (2022) · Duke Mathematical Journal · doi:10.1215/00127094-2022-0064 · Cited by 40 · see citing works
“4, 75, 83]. Above we paid also no attention to the size of the smoothness index k. For results on the Cauchy problem for the Einstein and the Eistein-scalar eld systems in low regularity settings see [20, 19, 76, 77]. Appendix C: On active measurements By an active measurement we mean a model where we can control some…”
Costa, J. L., Girão, P. M., Natário, J., et al. (2014) · Classical and Quantum Gravity · doi:10.1088/0264-9381/32/1/015017 · Cited by 38 · see citing works
“nd that point (which might continue indefinitely) cannot be uniquely determined from the initial data. ∗For the precise definitions and a modern account of the Cauchy problem in general relativity, see [35]. †Throughout this work we will simply use “Reissner-Nordstöm” to mean any of the anti-de Sitter (Λ < 0), the as…”
MethodInitial boundary value problems for Einstein’s field equations and geometric uniqueness
Friedrich, H. (2009) · General Relativity and Gravitation · doi:10.1007/s10714-009-0800-3 · Cited by 37 · see citing works
Quasilinear Hyperbolic Fuchsian Systems and AVTD Behavior in T 2-Symmetric Vacuum Spacetimes
Ames, E., Beyer, F., Isenberg, J. A., & LeFloch, P. G. (2013) · Annales Henri Poincaré · doi:10.1007/s00023-012-0228-2 · Cited by 35 · see citing works
Costa, J. L., Girão, P. M., Natário, J., & Silva, J. D. (2018) · Communications in Mathematical Physics · doi:10.1007/s00220-018-3122-z · Cited by 36 · see citing works
Proof of the cosmic no-hair conjecture in the $\mathbb T^3$-Gowdy symmetric Einstein–Vlasov setting
Andréasson, H., & Ringström, H. (2016) · Journal of the European Mathematical Society · doi:10.4171/jems/623 · Cited by 33 · see citing works
Self-citationStable Big Bang formation for Einstein’s equations: The complete sub-critical regime
Fournodavlos, G., Rodnianski, I., & Speck, J. (2023) · Journal of the American Mathematical Society · doi:10.1090/jams/1015 · Cited by 31 · see citing works
Causality for Nonlocal Phenomena
Eckstein, M., & Miller, T. (2017) · Annales Henri Poincaré · doi:10.1007/s00023-017-0566-1 · Cited by 31 · see citing works
Origins and development of the Cauchy problem in general relativity
Ringström, H. (2015) · Classical and Quantum Gravity · doi:10.1088/0264-9381/32/12/124003 · Cited by 29 · see citing works
“The Taub spacetimes are the maximal Cauchy developments of left invariant and locally rotationally symmetric vacuum initial data on SU(2); see [46] or [148] for details.”
BackgroundSelf-citationThe Light Ray Transform in Stationary and Static Lorentzian Geometries
Feizmohammadi, A., Ilmavirta, J., & Oksanen, L. (2020) · Journal of Geometric Analysis · doi:10.1007/s12220-020-00409-y · Cited by 28 · see citing works
The general relativistic constraint equations
Carlotto, A. (2021) · Living Reviews in Relativity · doi:10.1007/s41114-020-00030-z · Cited by 28 · see citing works
Penrose’s 1965 singularity theorem: from geodesic incompleteness to cosmic censorship
Landsman, K. (2022) · General Relativity and Gravitation · doi:10.1007/s10714-022-02973-w · Cited by 28 · see citing works
“(Ringström, 2009, p. 188) Conjecture 3.5 (Strong cosmic censorship).”
BackgroundInfluentialDeterminism and General Relativity
Smeenk, C., & Wüthrich, C. (2021) · Philosophy of Science · doi:10.1086/713904 · Cited by 27 · see citing works
Doboszewski, J. (2017) · European studies in philosophy of science · doi:10.1007/978-3-319-55486-0_11 · Cited by 26 · see citing works
The Initial Value Problem in General Relativity
Isenberg, J. A. (2014) · Springer eBooks · doi:10.1007/978-3-662-46035-1_16 · Cited by 26 · see citing works
Singularities, Black Holes, and Cosmic Censorship: A Tribute to Roger Penrose
Landsman, K. (2021) · Foundations of Physics · doi:10.1007/s10701-021-00432-1 · Cited by 36 · see citing works
Logarithmic Local Energy Decay for Scalar Waves on a General Class of Asymptotically Flat Spacetimes
Moschidis, G. (2016) · Annals of PDE · doi:10.1007/s40818-016-0010-8 · Cited by 26 · see citing works
Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations
Kurylev, Y., Lassas, M. J., & Uhlmann, G. A. (2018) · Inventiones mathematicae · doi:10.1007/s00222-017-0780-y · Cited by 25 · see citing works
The Initial Value Problem in General Relativity
Isenberg, J. (2014) · Springer handbooks · doi:10.1007/978-3-642-41992-8_16 · Cited by 24 · see citing works
Newtonian Limits of Isolated Cosmological Systems on Long Time Scales
Liu, C., & Oliynyk, T. A. (2018) · Annales Henri Poincaré · doi:10.1007/s00023-018-0686-2 · Cited by 23 · see citing works
Rough sound waves in 3D compressible Euler flow with vorticity
Disconzi, M. M., Luo, C., Mazzone, G., et al. (2022) · Selecta Mathematica · doi:10.1007/s00029-021-00733-3 · Cited by 23 · see citing works
On the well-posedness of relativistic viscous fluids
Disconzi, M. M. (2014) · Nonlinearity · doi:10.1088/0951-7715/27/8/1915 · Cited by 23 · see citing works
Ancient Dynamics in Bianchi Models: Approach to Periodic Cycles
Liebscher, S., Härterich, J., Webster, K., et al. (2011) · Communications in Mathematical Physics · doi:10.1007/s00220-011-1248-3 · Cited by 23 · see citing works
Do We Have any Viable Solution to the Measurement Problem?
Adlam, E. (2023) · Foundations of Physics · doi:10.1007/s10701-023-00686-x · Cited by 22 · see citing works
“A time-evolution formulation of the Einstein equations does exist [66,67], but the original global formulation remains central to research in the field and there seems no obvious reason to think that the time-evolution formulation must be more fundamental.”
BackgroundFuture Stability of the FLRW Spacetime for a Large Class of Perfect Fluids
Liu, C., & Wei, C. (2021) · Annales Henri Poincaré · doi:10.1007/s00023-020-00987-1 · Cited by 21 · see citing works
Time Functions on Lorentzian Length Spaces
Burtscher, A., & García-Heveling, L. (2024) · Annales Henri Poincaré · doi:10.1007/s00023-024-01461-y · Cited by 20 · see citing works
“Cauchy surfaces represent the natural sets to pose initial conditions for the Einstein equations (for a selfcontained exposition see [56]).”
The Lichnerowicz equation on compact manifolds with boundary
Holst, M., & Tsogtgerel, G. (2013) · Classical and Quantum Gravity · doi:10.1088/0264-9381/30/20/205011 · Cited by 20 · see citing works
On a viable first-order formulation of relativistic viscous fluids and its applications to cosmology
Disconzi, M. M., Kephart, T. W., & Scherrer, R. J. (2017) · International Journal of Modern Physics D · doi:10.1142/s0218271817501462 · Cited by 20 · see citing works
“, [63, 99]), the relevant conclusion of proposition A.”
Background4 ON THE WELL-POSEDNESS OF RELATIVISTIC VISCOUS FLUIDSPreprint
Disconzi, M. M. (2016) · Cited by 20 · see citing works
3 THE INITIAL VALUE PROBLEM IN GENERAL RELATIVITYPreprint
Isenberg, J. A. (2016) · Cited by 20 · see citing works
Smooth Gowdy-symmetric generalized Taub–NUT solutions
Beyer, F., & Hennig, J. (2012) · Classical and Quantum Gravity · doi:10.1088/0264-9381/29/24/245017 · Cited by 19 · see citing works
“More details and references can be found in [45, 43].”
BackgroundInfluentialLaplace’s demon tries on Aristotle’s cloak: on two approaches to determinism
Placek, T. (2016) · Synthese · doi:10.1007/s11229-016-1158-1 · Cited by 17 · see citing works
“Ringström (2009)) is that the non-globally hyperbolic developments of initial data sets are rare, in some measure-theoretical sense, with a measure defined on the space of relevant solutions to EFE.”
InfluentialOn proving future stability of cosmological solutions with accelerated expansion
Ringström, H. (2015) · Surveys in Differential Geometry · doi:10.4310/sdg.2015.v20.n1.a10 · Cited by 17 · see citing works
Self-citationRelativistic spacetimes and definitions of determinism
Doboszewski, J. (2019) · European Journal for Philosophy of Science · doi:10.1007/s13194-019-0248-6 · Cited by 17 · see citing works
“First, the theorem is so far proven only for vacuum spacetimes and simplified matter models (Ringström 2009).”
BackgroundAspects of $$C^0$$ causal theory
Ling, E. (2020) · General Relativity and Gravitation · doi:10.1007/s10714-020-02708-9 · Cited by 18 · see citing works
“1], but also pointed out in [17] is another argument using the Hopf–Rinow and the Whitney embedding theorems (the latter of course still requires a partition of unity argument).”
MethodFinster, F., & Röken, C. (2015) · University of Regensburg Publication Server (University of Regensburg) · Cited by 17 · see citing works
Recent developments in mathematical aspects of relativistic fluids
Disconzi, M. M. (2024) · Living Reviews in Relativity · doi:10.1007/s41114-024-00052-x · Cited by 16 · see citing works
Choquet–Bruhat, Y. (2014) · doi:10.1093/oso/9780199666454.003.0007 · Cited by 16 · see citing works
Beyer, F., Escobar, L., & Frauendiener, J. (2016) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.93.043009 · Cited by 16 · see citing works
A comment on the construction of the maximal globally hyperbolic Cauchy development
Wong, W. W.-Y., Polytechnique, E., & Lausanne, F. D. (2014) · Cited by 16 · see citing works
Conformal Methods in General Relativity
Kroon, J. A. V. (2022) · Cambridge University Press eBooks · doi:10.1017/9781009291309 · Cited by 14 · see citing works
An exact smooth Gowdy-symmetric generalized Taub–NUT solution
Beyer, F., & Hennig, J. (2014) · Classical and Quantum Gravity · doi:10.1088/0264-9381/31/9/095010 · Cited by 15 · see citing works
“Or would quite general solutions in general relativity always suffer from such defects? The former alternative is proposed in Penrose’s famous strong cosmic censorship conjecture [33, 26, 12, 36, 37], according to which the maximal”
BackgroundGlobally hyperbolic spacetimes: slicings, boundaries and counterexamples
Sánchez, M. (2022) · General Relativity and Gravitation · doi:10.1007/s10714-022-03002-6 · Cited by 15 · see citing works
Future stability of the 1 + 3 Milne model for the Einstein–Klein–Gordon system
Wang, J. (2019) · Classical and Quantum Gravity · doi:10.1088/1361-6382/ab4dd3 · Cited by 15 · see citing works
Solutions to the Einstein-scalar field constraint equations with a small TT-tensor
Gicquaud, R., & Nguyen, C. (2016) · Calculus of Variations and Partial Differential Equations · doi:10.1007/s00526-016-0963-1 · Cited by 13 · see citing works
Polish spaces of causal curves
Miller, T. (2017) · Journal of Geometry and Physics · doi:10.1016/j.geomphys.2017.02.006 · Cited by 13 · see citing works
Inverse problems for first-order hyperbolic equations with time-dependent coefficients
Floridia, G., & Takase, H. (2021) · Journal of Differential Equations · doi:10.1016/j.jde.2021.10.007 · Cited by 13 · see citing works
Double null data and the characteristic problem in general relativity
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Boundary conditions for the gravitational field
Winicour, J. (2012) · Classical and Quantum Gravity · doi:10.1088/0264-9381/29/11/113001 · Cited by 12 · see citing works
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Semilinear wave model for critical collapse
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On maximal globally hyperbolic vacuum space-times
Chruściel, P. T. (2013) · Journal of Fixed Point Theory and Applications · doi:10.1007/s11784-013-0141-4 · Cited by 11 · see citing works
Dirac Equation with External Potential and Initial Data on Cauchy SurfacesPreprint
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Dirac equation with external potential and initial data on Cauchy surfaces
Deckert, D.-A., & Merkl, F. (2014) · Journal of Mathematical Physics · doi:10.1063/1.4902376 · Cited by 10 · see citing works
Strong Cosmic Censorship in Orthogonal Bianchi Class B Perfect Fluids and Vacuum Models
Radermacher, K. (2019) · Annales Henri Poincaré · doi:10.1007/s00023-018-00756-1 · Cited by 10 · see citing works
A Unified Approach to the Klein–Gordon Equation on Bianchi Backgrounds
Ringström, H. (2019) · Communications in Mathematical Physics · doi:10.1007/s00220-019-03325-7 · Cited by 10 · see citing works
Self-citationBelnap, N., Müller, T., & Placek, T. (2021) · doi:10.1093/oso/9780190884314.001.0001 · Cited by 11 · see citing works
Kinematical Foundations of Loop Quantum Cosmology
Fleischhack, C. (2016) · Springer eBooks · doi:10.1007/978-3-319-26902-3_11 · Cited by 10 · see citing works
“It is a very remarkable result that our initial-value problem can now be solved at least locally [34, 50, 48] iff these constraints are met.”
BackgroundCosmological singularity theorems for f(R) gravity theories
Alani, I., & Santillán, O. (2016) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2016/05/023 · Cited by 8 · see citing works
A Class of Dust-Like Self-Similar Solutions of the Massless Einstein–Vlasov System
Rendall, A. D., & Velázquez, J. J. L. (2011) · Annales Henri Poincaré · doi:10.1007/s00023-011-0094-3 · Cited by 8 · see citing works
The Einstein−Λ flow on product manifolds
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Asymptotics of solutions of a hyperbolic formulation of the constraint equations
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“In its most common formulation, see [1, 21, 36], one performs a 3 + 1-split and thereby decomposes the spacetime and all corresponding geometric tensor fields into spatial and timelike components.”
BackgroundKamiński, W. (2023) · Communications in Mathematical Physics · doi:10.1007/s00220-023-04703-y · Cited by 8 · see citing works
“It means a series of important properties (see [18]): 1.”
BackgroundBeig, R. (2023) · Classical and Quantum Gravity · doi:10.1088/1361-6382/acc307 · Cited by 8 · see citing works
“Ringström [15]), but a detailed presentation seems to be lacking.”
BackgroundA class of dust-like self-similar solutions of the massless Einstein-Vlasov system
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Conformal Cyclic Cosmology, gravitational entropy and quantum information
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“Typically, one assumes a stronger property of global hyperbolicity, which guarantees the well-posedness of the Cauchy problem [1].”
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Hyperbolic Evolution Equations, Lorentzian Holonomy, and Riemannian Generalised Killing Spinors
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Uniqueness and Non-Uniqueness Results for Spacetime Extensions
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Dynamics of solutions of the Einstein equations with twisted Gowdy symmetry
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From holography to the geometry of the spacetime
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Lorentzian Geometry: Holonomy, Spinors, and Cauchy Problems
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A class of cosmological models with spatially constant sign-changing curvature
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“In the 3-steps CBGL construction, once an initial data set (g̃, K̃, ψ̃, π̃) is given, continuity for Step 3 is ensured by the notion of Cauchy stability (see Ringström, [36]).”
MethodInfluentialGlobal existence of Dirac-wave maps with curvature term on expanding spacetimes
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Hadamard States for Linearized Gravity on Spacetimes with Compact Cauchy Surfaces
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Explorations of the infinite regions of spacetime
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Construction of global solutions to the linearized field equations for causal variational principles
Finster, F., & Kraus, M. (2023) · Methods and Applications of Analysis · doi:10.4310/maa.2023.v30.n2.a2 · Cited by 5 · see citing works
- Authors
- Ringström, H.
- Affiliations
- KTH Royal Institute of Technology (SE)
- Corresponding author
- Ringström, Hans
- ORCID
- Ringström, Hans
- Venue
- ESI Lectures in Mathematics and Physics
- Publisher
- EMS Press
- Published
- 2009-06-08
- ISSN
- 2943-4920 (ISSN-L)
- 2943-4939
- DOI
- 10.4171/053
- OpenAlex
- W4300641598
- Cited by
- 262
- References
- 0
- Citations by year
- 2026: 7 · 2025: 13 · 2024: 18 · 2023: 26 · 2022: 10 · 2021: 27 · 2020: 12 · 2019: 12 · 2018: 9 · 2017: 13 · 2016: 15 · 2015: 7 · 2014: 26 · 2013: 52 · 2012: 5
- FWCI
- 4.60
- Type
- book
- Language
- en
- Open access
- closed
- Locations
- ESI lectures in mathematics and physics (publishedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Advanced Differential Geometry Research
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Mathematical Physics Problems
- Keywords
- Cauchy problem, general relativity, Einstein field equations, initial value problems, well-posedness, spacetime geometry
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