pith. machine review for the scientific record. sign in

arxiv: 2605.03172 · v1 · submitted 2026-05-04 · 🌀 gr-qc · math-ph· math.DG· math.MG· math.MP

Recognition: unknown

Stability of Synthetic Timelike Ricci Bounds under C⁰-Limits and Applications to Impulsive Gravitational Waves

Andrea Mondino, Clemens S\"amann, Vanessa Ryborz

Authors on Pith no claims yet

Pith reviewed 2026-05-08 17:23 UTC · model grok-4.3

classification 🌀 gr-qc math-phmath.DGmath.MGmath.MP
keywords synthetic curvature-dimension conditiontimelike Ricci curvatureC0-limitsimpulsive gravitational wavesLorentzian optimal transportstrong energy conditionlow-regularity spacetimes
0
0 comments X

The pith

Synthetic timelike Ricci curvature bounds are stable under locally uniform convergence of smooth Lorentzian metrics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the synthetic curvature-dimension condition TCD^e_p(K,N) passes to the limit when a sequence of smooth Lorentzian metrics converges locally uniformly, assuming uniform global hyperbolicity. This stability implies that limits of vacuum spacetimes satisfy the strong energy condition in the synthetic sense. The result is applied to show that impulsive gravitational waves, which have only Lipschitz metrics, satisfy these timelike Ricci bounds. This approach uses optimal transport to control the curvature in the limit without a priori bounds.

Core claim

The synthetic timelike curvature-dimension condition TCD^e_p(K,N) is stable under locally uniform convergence of smooth Lorentzian metrics when the sequence is uniformly globally hyperbolic. Consequently, smooth locally uniform limits of vacuum spacetimes satisfy the strong energy condition. This stability is leveraged to establish that large classes of impulsive gravitational waves satisfy synthetic timelike Ricci curvature lower bounds, with upper bounds also holding in the Minkowski background.

What carries the argument

The synthetic curvature-dimension condition TCD^e_p(K,N) via Lorentzian optimal transport, serving as a formulation of the Hawking-Penrose strong energy condition.

If this is right

  • Uniform limits of vacuum spacetimes satisfy the strong energy condition synthetically.
  • Impulsive gravitational waves satisfy synthetic lower bounds on timelike Ricci curvature.
  • In Minkowski spacetime, impulsive waves also satisfy synthetic upper Ricci bounds.
  • The Eschenburg-Galloway-Newman splitting theorem does not extend to infinitesimally Minkowskian TCD^e_p(0,N) Lorentzian length spaces.
  • No direct Lorentzian analogue of the Cheeger-Colding almost splitting theorem exists under almost non-negative timelike Ricci curvature.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The stability allows geometric analysis of singularities in general relativity where classical curvature tensors are not defined.
  • The contrast with Riemannian geometry indicates that Lorentzian lower curvature bounds may lack certain rigidity properties.
  • Similar stability techniques could apply to other weak solutions of Einstein's equations from nonlinear wave interactions.

Load-bearing premise

A uniform global hyperbolicity condition must hold for the sequence of smooth metrics.

What would settle it

A sequence of smooth metrics converging locally uniformly to a limit that violates the TCD^e_p(K,N) condition while satisfying uniform global hyperbolicity would falsify the stability claim.

read the original abstract

We investigate the stability of timelike Ricci curvature lower bounds under low-regularity limits of Lorentzian metrics. Specifically, we prove that the synthetic curvature-dimension condition $TCD^e_p(K,N)$, which provides an optimal transport formulation of the Hawking-Penrose strong energy condition, is stable under locally uniform convergence of smooth Lorentzian metrics, provided a uniform global hyperbolicity assumption holds. As a consequence, smooth locally uniform limits of vacuum spacetimes satisfy the strong energy condition, even though curvature is not controlled a priori. As a main application, we study impulsive gravitational waves - spacetimes with Lipschitz continuous metrics - and show that large classes of such waves satisfy synthetic timelike Ricci curvature lower bounds. In the case of Minkowski background, we further establish synthetic upper Ricci curvature bounds. Our approach relies on constructing suitable smooth approximations with lower bounds on the timelike Ricci, and analyzing the limiting behavior via Lorentzian optimal transport. These results yield new geometric insights into low-regularity solutions of the Einstein equations and, in particular, provide a counterexample to the extension of the Eschenburg-Galloway-Newman Lorentzian splitting theorem to infinitesimally Minkowskian $TCD^e_p(0,N)$ Lorentzian length spaces. Moreover, our construction shows that a direct Lorentzian analogue of the Cheeger-Colding almost splitting theorem - under assumptions of almost non-negative timelike Ricci curvature and the existence of an almost maximizing line - cannot hold. This highlights a fundamental difference between the geometry of Riemannian and Lorentzian lower Ricci curvature bounds. We also apply the aforementioned stability theorem to weak solutions of the Einstein equations arising from the nonlinear interaction of impulsive gravitational waves.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proves stability of the synthetic timelike curvature-dimension condition TCD^e_p(K,N) under locally uniform (C^0) convergence of smooth Lorentzian metrics, assuming uniform global hyperbolicity of the sequence. As a consequence, smooth locally uniform limits of vacuum spacetimes satisfy the strong energy condition. The main applications are to impulsive gravitational waves with Lipschitz metrics, where large classes are shown to satisfy synthetic timelike Ricci lower bounds (and, for Minkowski backgrounds, upper bounds); the work also yields counterexamples to extensions of the Eschenburg-Galloway-Newman splitting theorem and to a direct Lorentzian analogue of the Cheeger-Colding almost-splitting theorem.

Significance. If the stability result holds, the paper supplies a useful tool for controlling synthetic timelike Ricci bounds in low-regularity Lorentzian geometry and for weak solutions of the Einstein equations. The applications to impulsive waves and the explicit counterexamples to splitting theorems are of genuine interest, as they illustrate concrete differences between the Lorentzian and Riemannian synthetic settings.

major comments (2)
  1. [Stability theorem (abstract and § on main result)] The stability theorem for TCD^e_p(K,N) is stated only under an additional uniform global hyperbolicity assumption on the sequence. Locally uniform convergence does not automatically preserve uniform control on the time-separation function or on causal diamonds, so the assumption is external to the convergence hypothesis and must be verified separately for each application, including the smooth approximations constructed for impulsive waves.
  2. [Applications to impulsive gravitational waves] In the construction of smooth approximations to the Lipschitz metrics of impulsive gravitational waves, it is not clear how the uniform global hyperbolicity (and the lower timelike Ricci bounds) are simultaneously maintained while passing to the C^0 limit; this verification is load-bearing for the claim that such spacetimes satisfy TCD^e_p(K,N).
minor comments (1)
  1. [Notation and definitions] Clarify the precise dependence of the constants K and N on the approximating sequence and on the background metric in the Minkowski case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each major point below and will revise the paper to improve clarity where appropriate.

read point-by-point responses

  1. Referee: [Stability theorem (abstract and § on main result)] The stability theorem for TCD^e_p(K,N) is stated only under an additional uniform global hyperbolicity assumption on the sequence. Locally uniform convergence does not automatically preserve uniform control on the time-separation function or on causal diamonds, so the assumption is external to the convergence hypothesis and must be verified separately for each application, including the smooth approximations constructed for impulsive waves.

    Authors: We agree that uniform global hyperbolicity is an additional hypothesis not implied by C^0 convergence alone, since the latter does not automatically yield uniform control on the time-separation function or causal diamonds. This is the reason the assumption is stated explicitly in the theorem and abstract. In the applications to impulsive gravitational waves the approximating sequence is constructed to satisfy the uniform global hyperbolicity condition; we will add a short clarifying paragraph after the statement of the stability theorem explaining why the assumption is necessary and how it is verified in each application. revision: partial

  2. Referee: [Applications to impulsive gravitational waves] In the construction of smooth approximations to the Lipschitz metrics of impulsive gravitational waves, it is not clear how the uniform global hyperbolicity (and the lower timelike Ricci bounds) are simultaneously maintained while passing to the C^0 limit; this verification is load-bearing for the claim that such spacetimes satisfy TCD^e_p(K,N).

    Authors: We thank the referee for highlighting this point. In the construction (detailed in Section 5), the mollifiers are chosen to preserve both the uniform global hyperbolicity of the background and the lower timelike Ricci bounds along the approximating sequence; the passage to the C^0 limit then invokes the stability theorem. We acknowledge that the simultaneous preservation could be made more explicit and will expand the relevant paragraphs in the revised version to include a step-by-step verification of these properties. revision: yes

Circularity Check

0 steps flagged

Minor self-citation for TCD definition; central stability proof via optimal transport is independent

full rationale

The derivation constructs smooth approximations to the given C^0 metrics that satisfy TCD^e_p(K,N) by assumption, then passes to the limit using Lorentzian optimal transport to obtain stability of the synthetic bound. The uniform global hyperbolicity hypothesis is stated explicitly as an additional requirement and is not derived from the convergence itself. No equation or step reduces the claimed stability or the strong-energy-condition consequence to a fitted parameter or to a self-citation chain that itself lacks independent verification. The application to impulsive waves follows the same approximation-plus-limit strategy without circular renaming or self-definition. This yields a score of 2 for routine foundational citations while the core argument remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the definition of the synthetic condition TCD^e_p(K,N) via Lorentzian optimal transport, standard properties of globally hyperbolic Lorentzian manifolds, and the existence of suitable smooth approximations with controlled timelike Ricci curvature.

axioms (2)
  • domain assumption Uniform global hyperbolicity of the approximating metrics
    Invoked to pass to the limit while preserving the synthetic bound.
  • domain assumption Existence of smooth approximations with lower bounds on timelike Ricci curvature
    Used to construct the sequence whose limit satisfies TCD^e_p.

pith-pipeline@v0.9.0 · 5629 in / 1375 out tokens · 34141 ms · 2026-05-08T17:23:19.697213+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

71 extracted references · 9 canonical work pages · 1 internal anchor

  1. [1]

    K., and McCann, R

    Tobias Beran, Mathias Braun, Matteo Calisti, Nicola Gigli, Robert J. McCann, Argam Ohanyan, Felix Rott, and Clemens S¨ amann. A nonlinear d’Alembert comparison theorem and causal differential calculus on metric measure spacetimes. Preprint, arXiv:2408.15968, 2024

    work page arXiv 2024

  2. [2]

    Braginskii and Leonid P

    Vladimir B. Braginskii and Leonid P. Grishchuk. Kinematic resonance and the memory effect in free mass gravitational antennas.Zh. Eksp. Teor. Fiz, 89:744, 1985

    1985

  3. [3]

    R´ enyi’s entropy on Lorentzian spaces

    Mathias Braun. R´ enyi’s entropy on Lorentzian spaces. Timelike curvature-dimension conditions.J. Math. Pures Appl. (9), 177:46–128, 2023

    2023

  4. [4]

    New perspectives on the d’Alembertian from general relativity

    Mathias Braun. New perspectives on the d’Alembertian from general relativity. An invitation.Indag. Math., 2025

    2025

  5. [5]

    Comparison theory for Lipschitz spacetimes

    Mathias Braun and Marta S´ alamo Candal. Comparison theory for Lipschitz spacetimes. Preprint, arXiv:2603.24195, 2026

    work page arXiv 2026

  6. [6]

    McCann, Argam Ohanyan, and Clemens S¨ amann

    Mathias Braun, Nicola Gigli, Robert J. McCann, Argam Ohanyan, and Clemens S¨ amann. A Lorentzian splitting theorem for continuously differentiable metrics and weights. Preprint, arXiv:2507.06836, 2025

    work page arXiv 2025

  7. [7]

    Mathias Braun and Robert J. McCann. Causal convergence conditions through variable timelike Ricci curvature bounds. Preprint, arXiv:2312.17158, to appear inMem. Europ. Math. Soc., 2023

    work page arXiv 2023

  8. [8]

    Gregory A. Burnett. The high-frequency limit in general relativity.J. Math. Phys., 30(1):90–96, 1989

    1989

  9. [9]

    Busemann

    H. Busemann. Timelike spaces.Dissertationes Math. Rozprawy Mat., 53:52, 1967. ISSN: 0012-3862

    1967

  10. [10]

    Bykov, E

    A. Bykov, E. Minguzzi, and S. Suhr. Lorentzian metric spaces and GH-convergence: the unbounded case.Lett. Math. Phys., 115(3):Paper No. 63, 66, 2025

    2025

  11. [11]

    A review of Lorentzian synthetic theory of timelike Ricci curva- ture bounds.Gen

    Fabio Cavalletti and Andrea Mondino. A review of Lorentzian synthetic theory of timelike Ricci curva- ture bounds.Gen. Relativity Gravitation, 54(11):Paper No. 137, 39, 2022

    2022

  12. [12]

    Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications.Camb

    Fabio Cavalletti and Andrea Mondino. Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications.Camb. J. Math., 12(2):417–534, 2024

    2024

  13. [13]

    Jeff Cheeger and Tobias H. Colding. Lower bounds on Ricci curvature and the almost rigidity of warped products.Ann. Math. (2), 144(1):189–237, 1996

    1996

  14. [14]

    The splitting theorem for manifolds of nonnegative Ricci curvature

    Jeff Cheeger and Detlef Gromoll. The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differ. Geom., 6:119–128, 1971

    1971

  15. [15]

    Nonlinear nature of gravitation and gravitational-wave experiments.Phys

    Demetrios Christodoulou. Nonlinear nature of gravitation and gravitational-wave experiments.Phys. Rev. Lett., 67(12):1486–1489, 1991

    1991

  16. [16]

    Chru´ sciel

    Piotr T. Chru´ sciel. Elements of causality theory.preprint arXiv:1110.6706, 2011

    work page arXiv 2011

  17. [17]

    Chru´ sciel and James D

    Piotr T. Chru´ sciel and James D. E. Grant. On Lorentzian causality with continuous metrics.Classical Quantum Gravity, 29(14):145001, 32, 2012

    2012

  18. [18]

    DiPerna and Andrew J

    Ronald J. DiPerna and Andrew J. Majda. Oscillations and concentrations in weak solutions of the incompressible fluid equations.Commun. Math. Phys., 108:667–689, 1987

    1987

  19. [19]

    The splitting theorem for space-times with strong energy condition.J

    Jost-Hinrich Eschenburg. The splitting theorem for space-times with strong energy condition.J. Differ. Geom., 27(3):477–491, 1988

    1988

  20. [20]

    Flores and Miguel S´ anchez

    Jos´ e L. Flores and Miguel S´ anchez. On the geometry of pp-wave type spacetimes.Analytical and numerical approaches to mathematical relativity, pages 79–98, 2006

    2006

  21. [21]

    Galloway

    Gregory J. Galloway. The Lorentzian splitting theorem without the completeness assumption.J. Differ. Geom., 29(2):373–387, 1989. 88

    1989

  22. [22]

    Strings and other distributional sources in general relativity.Phys

    Robert Geroch and Jennie Traschen. Strings and other distributional sources in general relativity.Phys. Rev. D, 36:1017–1031, Aug 1987

    1987

  23. [23]

    Nicola Gigli.The splitting theorem in non-smooth context, volume 1609 ofMem. Am. Math. Soc. Providence, RI: American Mathematical Society (AMS), 2026

    2026

  24. [24]

    Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows.Proc

    Nicola Gigli, Andrea Mondino, and Giuseppe Savar´ e. Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows.Proc. Lond. Math. Soc. (3), 111(5):1071–1129, 2015

    2015

  25. [25]

    Maximizers in Lipschitz spacetimes are either timelike or null.Classical Quantum Gravity, 35(8):087001, 6, 2018

    Melanie Graf and Eric Ling. Maximizers in Lipschitz spacetimes are either timelike or null.Classical Quantum Gravity, 35(8):087001, 6, 2018

    2018

  26. [26]

    Griffiths and Jiˇ r´ ı Podolsk´ y.Exact space-times in Einstein’s general relativity

    Jerry B. Griffiths and Jiˇ r´ ı Podolsk´ y.Exact space-times in Einstein’s general relativity. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 2009

    2009

  27. [27]

    Oscillations in wave map systems and homogenization of the Einstein equations in symmetry.Arch

    Andr´ e Guerra and Rita Teixeira da Costa. Oscillations in wave map systems and homogenization of the Einstein equations in symmetry.Arch. Ration. Mech. Anal., 249(1):48, 2025. Id/No 9

    2025

  28. [28]

    A triangle comparison theorem for Lorentz manifolds.Indiana University Mathematics Journal, 31(3):289–308, 1982

    Steven G Harris. A triangle comparison theorem for Lorentz manifolds.Indiana University Mathematics Journal, 31(3):289–308, 1982

    1982

  29. [29]

    Burnett’s conjecture in generalized wave coordinates

    C´ ecile Huneau and Jonathan Luk. Burnett’s conjecture in generalized wave coordinates. Preprint, arXiv:2403.03470, 2024

    work page arXiv 2024

  30. [30]

    High-frequency solutions to the Einstein equations.Classical Quan- tum Gravity, 41(14):48, 2024

    C´ ecile Huneau and Jonathan Luk. High-frequency solutions to the Einstein equations.Classical Quan- tum Gravity, 41(14):48, 2024. Id/No 143002

    2024

  31. [31]

    Trilinear compensated compactness and Burnett’s conjecture in general relativity.Ann

    C´ ecile Huneau and Jonathan Luk. Trilinear compensated compactness and Burnett’s conjecture in general relativity.Ann. Sci. ´Ec. Norm. Sup´ er. (4), 57(2):385–472, 2024

    2024

  32. [32]

    Isaacson

    Richard A. Isaacson. Gravitational radiation in the limit of high frequency. i. the linear approximation and geometrical optics.Phys. Rev., 166:1263–1271, Feb 1968

    1968

  33. [33]

    Isaacson

    Richard A. Isaacson. Gravitational radiation in the limit of high frequency. ii. nonlinear terms and the effective stress tensor.Phys. Rev., 166:1272–1280, Feb 1968

    1968

  34. [34]

    Rough solutions of the Einstein-vacuum equations.Ann

    Sergiu Klainerman and Igor Rodnianski. Rough solutions of the Einstein-vacuum equations.Ann. of Math. (2), 161(3):1143–1193, 2005

    2005

  35. [35]

    E. H. Kronheimer and R. Penrose. On the structure of causal spaces.Proc. Cambridge Philos. Soc., 63:481–501, 1967

    1967

  36. [36]

    Ricci curvature bounds and rigidity for non-smooth Riemannian and semi-Riemannian metrics.Manuscr

    Michael Kunzinger, Argam Ohanyan, and Alessio Vardabasso. Ricci curvature bounds and rigidity for non-smooth Riemannian and semi-Riemannian metrics.Manuscr. Math., 176(4):39, 2025. Id/No 53

    2025

  37. [37]

    Lorentzian length spaces.Ann

    Michael Kunzinger and Clemens S¨ amann. Lorentzian length spaces.Ann. Global Anal. Geom., 54(3):399–447, 2018

    2018

  38. [38]

    Michael Kunzinger, Roland Steinbauer, and James A. Vickers. The Penrose singularity theorem in regularityC 1,1.Classical Quantum Gravity, 32(15):12, 2015. Id/No 155010

    2015

  39. [39]

    Lorentz meets Lipschitz.Adv

    Christian Lange, Alexander Lytchak, and Clemens S¨ amann. Lorentz meets Lipschitz.Adv. Theor. Math. Phys., 25(8):2141–2170, 2021

    2021

  40. [40]

    The regularity of geodesics in impulsivepp- waves.Gen

    Alexander Lecke, Roland Steinbauer, and Robert ˇSvarc. The regularity of geodesics in impulsivepp- waves.Gen. Relativity Gravitation, 46(1):Art. 1648, 8, 2014

    2014

  41. [41]

    LeFloch and Cristinel Mardare

    Philippe G. LeFloch and Cristinel Mardare. Definition and stability of Lorentzian manifolds with distributional curvature.Port. Math. (N.S.), 64(4):535–573, 2007. 89

    2007

  42. [42]

    Ricci curvature for metric-measure spaces via optimal transport.Ann

    John Lott and Cedric Villani. Ricci curvature for metric-measure spaces via optimal transport.Ann. Math. (2), 169(3):903–991, 2009

    2009

  43. [43]

    Local propagation of impulsive gravitational waves.Comm

    Jonathan Luk and Igor Rodnianski. Local propagation of impulsive gravitational waves.Comm. Pure Appl. Math., 68(4):511–624, 2015

    2015

  44. [44]

    Nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations.Camb

    Jonathan Luk and Igor Rodnianski. Nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations.Camb. J. Math., 5(4):435–570, 2017

    2017

  45. [45]

    Robert J. McCann. Displacement convexity of Boltzmann’s entropy characterizes the strong energy condition from general relativity.Camb. J. Math., 8(3):609–681, 2020

    2020

  46. [46]

    Robert J. McCann. A synthetic null energy condition.Comm. Math. Phys., 405:Paper No. 38, 2024

    2024

  47. [47]

    Robert J. McCann. Trading linearity for ellipticity: a nonsmooth approach to Einstein’s theory of gravity and the Lorentzian splitting theorems. Preprint, arXiv:2501.00702, 2025

    work page arXiv 2025

  48. [48]

    Limit curve theorems in Lorentzian geometry.J

    Ettore Minguzzi. Limit curve theorems in Lorentzian geometry.J. Math. Phys., 49(9):092501, 18, 2008

    2008

  49. [49]

    Further observations on the definition of global hyperbolicity under low regularity

    Ettore Minguzzi. Further observations on the definition of global hyperbolicity under low regularity. Classical and Quantum Gravity, 40(18):185001, 2023

    2023

  50. [50]

    Lorentzian metric spaces and their Gromov-Hausdorff convergence

    Ettore Minguzzi and Stefan Suhr. Lorentzian metric spaces and their Gromov-Hausdorff convergence. Lett. Math. Phys., 114:Paper No. 73, 63, 2024

    2024

  51. [51]

    On the equivalence of distributional and synthetic Ricci curvature lower bounds.J

    Andrea Mondino and Vanessa Ryborz. On the equivalence of distributional and synthetic Ricci curvature lower bounds.J. Funct. Anal., 289(8):82, 2025. Id/No 111035

    2025

  52. [52]

    [25]Mondino, A., and Suhr, S.An optimal transport formulation of the Einstein equations of general relativity.J

    Andrea Mondino and Clemens S¨ amann. Lorentzian Gromov-Hausdorff convergence and pre- compactness. Preprint, arXiv:2504.10380, 2025

    work page arXiv 2025

  53. [53]

    An optimal transport formulation of the Einstein equations of general relativity.J

    Andrea Mondino and Stefan Suhr. An optimal transport formulation of the Einstein equations of general relativity.J. Eur. Math. Soc. (JEMS), 25(3):933–994, 2023

    2023

  54. [54]

    Compacite par compensation.Ann

    Francois Murat. Compacite par compensation.Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., 5:489– 507, 1978

    1978

  55. [55]

    Richard P. A. C. Newman. A proof of the splitting conjecture of S.-T. Yau.J. Differ. Geom., 31(1):163– 184, 1990

    1990

  56. [56]

    Structure of space-time

    Roger Penrose. Structure of space-time. InBattelle Rencontres. New York: Benjamin, 1968

    1968

  57. [57]

    The geometry of impulsive gravitational waves

    Roger Penrose. The geometry of impulsive gravitational waves. InGeneral relativity (papers in honour of J. L. Synge), pages 101–115. Clarendon Press, Oxford, 1972. ISBN: 9780198511267

    1972

  58. [58]

    The global existence, uniqueness andC 1-regularity of geodesics in nonexpanding impulsive gravitational waves.Classical and Quantum Gravity, 32(2):025003, 2014

    Jiˇ r´ ı Podolsk` y, Clemens S¨ amann, Roland Steinbauer, and RobertˇSvarc. The global existence, uniqueness andC 1-regularity of geodesics in nonexpanding impulsive gravitational waves.Classical and Quantum Gravity, 32(2):025003, 2014

    2014

  59. [59]

    Cut-and-paste for impulsive gravitational waves with Λ: the geometric picture.Phys

    Jiˇ r´ ı Podolsk´ y, Clemens S¨ amann, Roland Steinbauer, and RobertˇSvarc. Cut-and-paste for impulsive gravitational waves with Λ: the geometric picture.Phys. Rev. D, 100(2):024040, 8, 2019

    2019

  60. [60]

    Penrose junction conditions with Λ: geometric insights into low- regularity metrics for impulsive gravitational waves.Gen

    Jiˇ r´ ı Podolsk´ y and Roland Steinbauer. Penrose junction conditions with Λ: geometric insights into low- regularity metrics for impulsive gravitational waves.Gen. Relativity Gravitation, 54(9):Paper No. 96, 24, 2022

    2022

  61. [61]

    Infinitesimal Minkowskianity for manifolds with continuous Lorentzian metrics

    Vanessa Ryborz. Infinitesimal Minkowskianity for manifolds with continuous Lorentzian metrics.arXiv preprint arXiv:2604.22420, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  62. [62]

    Global hyperbolicity for spacetimes with continuous metrics.Ann

    Clemens S¨ amann. Global hyperbolicity for spacetimes with continuous metrics.Ann. Henri Poincar´ e, 17(6):1429–1455, 2016. 90

    2016

  63. [63]

    A brief introduction to non-regular spacetime geometry.Internationale Mathematis- che Nachrichten, 256, 2024

    Clemens S¨ amann. A brief introduction to non-regular spacetime geometry.Internationale Mathematis- che Nachrichten, 256, 2024

    2024

  64. [64]

    Cut-and-paste for impul- sive gravitational waves with Λ: the mathematical analysis.Lett

    Clemens S¨ amann, Benedict Schinnerl, Roland Steinbauer, and Robert ˇSvarc. Cut-and-paste for impul- sive gravitational waves with Λ: the mathematical analysis.Lett. Math. Phys., 114(2):Paper No. 58, 31, 2024

    2024

  65. [65]

    Geodesics in nonexpanding impulsive gravitational waves with Λ, part I.Classical Quantum Gravity, 33(11):115002, 33, 2016

    Clemens S¨ amann, Roland Steinbauer, Alexander Lecke, and Jiˇ r´ ı Podolsk´ y. Geodesics in nonexpanding impulsive gravitational waves with Λ, part I.Classical Quantum Gravity, 33(11):115002, 33, 2016

    2016

  66. [66]

    Smith and Daniel Tataru

    Hart F. Smith and Daniel Tataru. Sharp local well-posedness results for the nonlinear wave equation. Ann. Math. (2), 162(1):291–366, 2005

    2005

  67. [67]

    Comment on ‘Memory effect for impulsive gravitational waves’.Classical Quantum Gravity, 36(9):098001, 14, 2019

    Roland Steinbauer. Comment on ‘Memory effect for impulsive gravitational waves’.Classical Quantum Gravity, 36(9):098001, 14, 2019

    2019

  68. [68]

    On the geometry of metric measure spaces

    Karl-Theodor Sturm. On the geometry of metric measure spaces. I.Acta Math., 196:65–131, 2006

    2006

  69. [69]

    On the geometry of metric measure spaces

    Karl-Theodor Sturm. On the geometry of metric measure spaces. II.Acta Math., 196:133–177, 2006

    2006

  70. [70]

    Remarks about synthetic upper Ricci bounds for metric measure spaces.Tˆ ohoku Math

    Karl-Theodor Sturm. Remarks about synthetic upper Ricci bounds for metric measure spaces.Tˆ ohoku Math. J. (2), 73(4):539–564, 2021

    2021

  71. [71]

    Compensated compactness and applications to partial differential equations

    Luc Tartar. Compensated compactness and applications to partial differential equations. Nonlinear analysis and mechanics: Heriot-Watt Symp., Vol. 4, Edinburgh 1979, Res. Notes Math. 39, 136-212, 1979. 91

    1979