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Cauchy-compact flat spacetimes with extreme BTZ

Abstract

Cauchy-compact flat spacetimes with extreme BTZ are Lorentzian analogue of complete hyperbolic surfaces of finite volume. Indeed, the latter are 2-manifolds locally modeled on the hyperbolic plane, with group of isometries \(\mathrm {PSL}_2(\mathbb {R})\), admitting finitely many cuspidal ends while the regular part of the former are 3-manifolds locally models on 3 dimensionnal Minkowski space, with group of isometries \(\mathrm {PSL}_2(\mathbb {R})\ltimes \mathbb {R}^3\), admitting finitely many ends whose neighborhoods are foliated by cusps. We prove a Theorem akin to the classical parametrization result for finite volume complete hyperbolic surfaces: the tangent bundle of the Teichmüller space of a punctured surface parametrizes globally hyperbolic Cauchy-maximal and Cauchy-compact locally Minkowski manifolds with extreme BTZ. Previous results of Mess, Bonsante and Barbot provide already a satisfactory parametrization of regular parts of such manifolds, the particularity of the present work lies in the consideration of manifolds with a singular geometrical structure with singularities modeled on extreme BTZ. We present a BTZ-extension procedure akin to the procedure compactifying finite volume complete hyperbolic surface by adding cusp points at infinity.

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Notes

  1. 1.

    Corollary 2.22 below shows it’s never the case for Cauchy-compact \({\mathbb {E}}^{1,2}_{0}\)-manifold whose fundamental group is anabelian if it has at least one BTZ-line. Indeed, for a Cauchy-complete Cauchy-maximal \({\mathbb {E}}^{1,2}_{}\)-manifold M with anabelian fundamental group, the image of the developing map is a convex of Minkowski space bounded by infinitely many lightlike planes, in particular, it admits a spacelike support plane. Hence, it may contain either a complete future geodesic ray or a complete past geodesic ray but never both.

References

  1. 1.

    Aghanim, N., Akrami, Y., Arroja, F., Ashdown, M., Aumont, J., Baccigalupi, C., Ballardini, M., Banday, A.J., Barreiro, R.B., et al.: Planck 2018 results. i. overview and the cosmological legacy of planck. Astronomy & Astrophysics (2019). https://doi.org/10.1051/0004-6361/201833880

  2. 2.

    Andersson, L., Barbot, T., Benedetti, R., Bonsante, F., Goldman, W.M., Labourie, F., Scannell, K.P., Schlenker, J.M.: Notes on a paper of Mess. Geometriae Dedicata 126(1), 47–70 (2007). https://hal.archives-ouvertes.fr/hal-00642328. 26 pages

  3. 3.

    Barbot, T.: Globally hyperbolic flat spacetimes. Journal of Geometry and Physics 53(2), 123–165 (2005) https://doi.org/10.1016/j.geomphys.2004.05.002. https://hal.archives-ouvertes.fr/hal-00012988

  4. 4.

    Barbot, T., Bonsante, F., Schlenker, J.M.: Collisions of particles in locally AdS spacetimes I. Local description and global examples. Comm. Math. Phys. 308(1), 147–200 (2011). https://doi.org/10.1007/s00220-011-1318-6

  5. 5.

    Bañados, M., Teitelboim, C., Zanelli, J.: Black hole in three-dimensional spacetime. Physical Review Letters 69(13), 1849–1851 (1992). https://doi.org/10.1103/physrevlett.69.1849

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Benedetti, R., Bonsante, F.: Canonical Wick rotations in 3-dimensional gravity. Mem. Amer. Math. Soc. 198(926), viii+164 (2009). https://doi.org/10.1090/memo/0926

  7. 7.

    Bernal, A.N., Sánchez, M.: On smooth Cauchy hypersurfaces and Gerochs splitting theorem. Comm. Math. Phys. 243(3), 461–470 (2003). https://doi.org/10.1007/s00220-003-0982-6

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Bernard, P., Suhr, S.: Lyapounov functions of closed cone fields: from Conley theory to time functions. Comm. Math. Phys. 359(2), 467–498 (2018). https://doi.org/10.1007/s00220-018-3127-7

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Bonahon, F., Wong, H.: Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations. Invent. Math. 204(1), 195–243 (2016). https://doi.org/10.1007/s00222-015-0611-y

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Bonahon, F., Wong, H.: Representations of the Kauffman bracket skein algebra II: Punctured surfaces. Algebr. Geom. Topol. 17(6), 3399–3434 (2017). https://doi.org/10.2140/agt.2017.17.3399

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Bonahon, F., Wong, H.: Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality. Quantum Topol. 10(2), 325–398 (2019). https://doi.org/10.4171/QT/125

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Bonsante, F.: Flat spacetimes with compact hyperbolic Cauchy surfaces. J. Differential Geom. 69(3), 441–521 (2005). http://projecteuclid.org/euclid.jdg/1122493997

  13. 13.

    Bonsante, F., Meusburger, C., Schlenker, J.M.: Recovering the geometry of a flat spacetime from background radiation. Ann. Henri Poincaré 15(9), 1733–1799 (2014). https://doi.org/10.1007/s00023-013-0300-6

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Bonsante, F., Seppi, A.: On Codazzi tensors on a hyperbolic surface and flat Lorentzian geometry. Int. Math. Res. Not. IMRN (2), 343–417 (2016). https://doi.org/10.1093/imrn/rnv144

  15. 15.

    Brunswic, L.: Surfaces de cauchy polyédrales des espaces temps-plats singuliers. Ph.D. thesis, Université d’Avignon et des Pays de Vaucluse (2017)

  16. 16.

    Brunswic, L.: On branched coverings of singular \((g,x)\)-manifolds (2020). arXiv:2010.10610

  17. 17.

    Brunswic, L., Buchert, T.: Gauss-bonnet-chern approach to the averaged universe. Classical and Quantum Gravity 37(21), 215022 (2020). https://doi.org/10.1088/1361-6382/abae45

    MathSciNet  Article  Google Scholar 

  18. 18.

    Buchert, T.: Dark energy from structure: a status report. General Relativity and Gravitation 40(2–3), 467–527 (2007). https://doi.org/10.1007/s10714-007-0554-8

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Carlip, S.: Quantum Gravity in 2+1 Dimensions. Cambridge Monographs on Mathematical Physics. Cambridge University Press (1998). https://doi.org/10.1017/CBO9780511564192

  20. 20.

    Choquet-Bruhat, Y., Geroch, R.: Global aspects of the Cauchy problem in general relativity. Comm. Math. Phys. 14, 329–335 (1969)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Ehresmann, C.: Sur les espaces localement homogènes. In: Œuvres complètes et commentées. I-1,2. Topologie algébrique et géométrie différentielle, Cahiers Topologie Géom. Différentielle, pp. 87–103. Geom. Topol. Publ., Coventry (1983). https://doi.org/10.2140/gtm.1998.1.511

  22. 22.

    Fathi, A., Siconolfi, A.: On smooth time functions. Math. Proc. Cambridge Philos. Soc. 152(2), 303–339 (2012). https://doi.org/10.1017/S0305004111000661

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Fock, V., Goncharov, A.: Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. Inst. Hautes Études Sci. 103, 1–211 (2006). https://doi.org/10.1007/s10240-006-0039-4

    Article  MATH  Google Scholar 

  24. 24.

    Fourés-Bruhat, Y.: Théoréme d’existence pour certains systémes d’équations aux dérivées partielles non linéaires. Acta Math. 88, 141–225 (1952). https://doi.org/10.1007/BF02392131

  25. 25.

    Fox, R.H.: Covering spaces with singularities. In: A symposium in honor of S. Lefschetz, pp. 243–257. Princeton University Press, Princeton, N.J. (1957)

  26. 26.

    Gauld, D.: Non-metrisable manifolds. Springer, Singapore (2014). https://doi.org/10.1007/978-981-287-257-9

  27. 27.

    Geroch, R.: Domain of dependence. J. Math. Phys. 11, 437–449 (1970)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Goldman, W.M.: The symplectic nature of fundamental groups of surfaces. Adv. Math. 54(2), 200–225 (1984). https://doi.org/10.1016/0001-8708(84)90040-9

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Goldman, W.M.: Geometric structures on manifolds and varieties of representations. In: Geometry of group representations (Boulder, CO, 1987), Contemp. Math., vol. 74, pp. 169–198. Amer. Math. Soc., Providence, RI (1988). https://doi.org/10.1090/conm/074/957518

  30. 30.

    Guillemin, V., Pollack, A.: Differential topology. AMS Chelsea Publishing, Providence, RI (2010). https://doi.org/10.1090/chel/370. Reprint of the 1974 original

  31. 31.

    Hulin, D., Troyanov, M.: Prescribing curvature on open surfaces. Math. Ann. 293(2), 277–315 (1992). https://doi.org/10.1007/BF01444716

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Katok, S.: Fuchsian groups. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL (1992)

    Google Scholar 

  33. 33.

    Mac Lane, S.: Categories for the working mathematician, Graduate Texts in Mathematics, vol. 5, 2nd edn. Springer, New York (1998)

  34. 34.

    Mess, G.: Lorentz spacetimes of constant curvature. Geom. Dedicata 126, 3–45 (2007). https://doi.org/10.1007/s10711-007-9155-7

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    Meusburger, C., Scarinci, C.: Generalized shear coordinates on the moduli spaces of three-dimensional spacetimes. J. Differential Geom. 103(3), 425–474 (2016). http://projecteuclid.org/euclid.jdg/1468517501

  36. 36.

    Minguzzi, E., Sánchez, M.: The causal hierarchy of spacetimes. In: Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys. pp. 299–358. Eur. Math. Soc., Zürich (2008). https://doi.org/10.4171/051-1/9. arXiv:gr-qc/0609119v3

  37. 37.

    Moon, H.B., Wong, H.: The roger-yang skein algebra and the decorated teichmuller space (2019)

  38. 38.

    O’Neill, B.: Semi-Riemannian geometry. Academic Press, New York (1983)

  39. 39.

    Penrose, R.: Techniques of differential topology in relativity. Society for Industrial and Applied Mathematics, Philadelphia, Pa. (1972). Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 7

  40. 40.

    Richards, I.: On the classification of noncompact surfaces. Trans. Amer. Math. Soc. 106, 259–269 (1963). https://doi.org/10.2307/1993768

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Ringström, H.: The Cauchy problem in general relativity. ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Zürich (2009). https://doi.org/10.4171/053

  42. 42.

    Sbierski, J.: On the existence of a maximal cauchy development for the einstein equations: a dezornification. Annales Henri Poincaré (2015)

  43. 43.

    Seifert, H.J.: Smoothing and extending cosmic time functions. Gen. Relat. Gravit. 8(10), 815–831 (1977). https://doi.org/10.1007/BF00759586

    MathSciNet  Article  MATH  Google Scholar 

  44. 44.

    Thurston, W.P.: Shapes of polyhedra and triangulations of the sphere. In: The Epstein birthday schrift, Geom. Topol. Monogr., vol. 1, pp. 511–549. Geom. Topol. Publ., Coventry (1998). https://doi.org/10.2140/gtm.1998.1.511

  45. 45.

    Troyanov, M.: Prescribing curvature on compact surfaces with conical singularities. Trans. Amer. Math. Soc. 324(2), 793–821 (1991). https://doi.org/10.2307/2001742

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

This work has been part of a PhD thesis supervised by Thierry Barbot at Université d’Avignon et des Pays de Vaucluse and is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement ERC advanced Grant 740021–ARTHUS, PI: Thomas Buchert).

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Brunswic, L. Cauchy-compact flat spacetimes with extreme BTZ. Geom Dedicata 214, 571–608 (2021). https://doi.org/10.1007/s10711-021-00629-8

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Keywords

  • Geometry
  • Geometry topology
  • Minkowski space
  • Singular geometrical structure
  • Lorentzian manifold

Mathematics Subject Classification

  • 57K35; 51H20; 83A05; 83C57; 57K20; 51M10