A regularisation approach to causality theory for \(C^{1,1}\)-Lorentzian metrics. (English) Zbl 1308.83026
Summary: We show that many standard results of Lorentzian causality theory remain valid if the regularity of the metric is reduced to \(C^{1,1}\). Our approach is based on regularisations of the metric adapted to the causal structure.
MSC:
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 83C10 | Equations of motion in general relativity and gravitational theory |
| 53Z05 | Applications of differential geometry to physics |
| 22E10 | General properties and structure of complex Lie groups |
| 54F65 | Topological characterizations of particular spaces |
References:
| [1] | Choquet-Bruhat, Y., Geroch, R.: Global aspects of the Cauchy problem in general relativity. Commun. Math. Phys. 14, 329-335 (1969) · Zbl 0182.59901 · doi:10.1007/BF01645389 |
| [2] | Klainerman, S., Rodnianski, I.: Rough solution for the Einstein vacuum equations. Ann. Math. 161(2), 1143-1193 (2005) · Zbl 1089.83006 · doi:10.4007/annals.2005.161.1143 |
| [3] | Maxwell, D.: Rough solutions of the Einstein constraint equations. J. Reine Angew. Math. 590, 1-29 (2006) · Zbl 1088.83004 · doi:10.1515/CRELLE.2006.001 |
| [4] | Klainerman, S., Rodnianski, I., Szeftel, J.: Overview of the proof of the bounded \[L^2\] L2 curvature conjecture, arXiv:1204.1772 [gr-qc], 2012 · Zbl 1330.53089 |
| [5] | Lichnerowicz, A.: Théories relativistes de la gravitation et de l’électromgnétisme. Masson, Paris (1955) · Zbl 0065.20704 |
| [6] | Griffiths, J., Podolský, J.: Exact Space-Times in Einstein’s General Relativity. Cambridge University Press, Cambridge (2009) · Zbl 1184.83003 · doi:10.1017/CBO9780511635397 |
| [7] | Penrose, R.: Techniques of Differential Topology in Relativity (Conf. Board of the Mathematical Sciences Regional Conf. Series in Applied Mathematics vol 7). SIAM, Philadelphia, PA (1972) · Zbl 0042.15605 |
| [8] | O’Neill, B.: Semi-Riemannian Geometry. With Applications to Relativity. Pure and Applied Mathematics 103. Academic Press, New York (1983) · Zbl 0531.53051 |
| [9] | Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian Geometry (Monographs and Textbooks in Pure and Applied Mathematics vol 202) 2nd edn. New York, Dekker (1996) · Zbl 1246.83025 |
| [10] | Kriele, M.: Spacetime. Foundations of General Relativity and Differential Geometry. Lecture Notes in Physics 59. Springer, Berlin (1999) · Zbl 0933.83001 |
| [11] | Minguzzi, E., Sanchez, M.: The causal hierarchy of spacetimes in recent developments in pseudo-Riemannian geometry. ESI Lect. Math. Phys., Eur. Math. Soc. Publ. House, Zürich (2008) · Zbl 0966.83022 |
| [12] | Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973) · Zbl 0265.53054 · doi:10.1017/CBO9780511524646 |
| [13] | Senovilla, J.M.M.: Singularity theorems and their consequences. Gen. Relativ. Gravit. 30(5), 701-848 (1998) · Zbl 0924.53045 · doi:10.1023/A:1018801101244 |
| [14] | García-Parrado, A., Senovilla, J.M.M.: Causal structures and causal boundaries. Class. Quantum Gravity 22, R1-R84 (2005) · Zbl 1073.83001 · doi:10.1088/0264-9381/22/9/R01 |
| [15] | Chruściel, P.T.: Elements of causality theory, arXiv:1110.6706 · Zbl 1073.83001 |
| [16] | Clarke, C.J.S.: The analysis of spacetime singularities. Cambridge Lect. Notes Phys. 1. Cambridge University Press (1993) · Zbl 0835.53093 |
| [17] | Sorkin, R.D., Woolgar, E.: A causal order for spacetimes with \[C^0\] C0 Lorentzian metrics: Proof of compactness of the space of causal curves class. Quantum Gravity 13, 1971-1994 (1996) · Zbl 0966.83022 · doi:10.1088/0264-9381/13/7/023 |
| [18] | Fathi, A., Siconolfi, A.: On smooth time functions. Math. Proc. Camb. Philos. Soc. 155, 1-37 (2011) · Zbl 1246.53095 |
| [19] | Chruściel, P.T., Grant, J.D.E.: On Lorentzian causality with continuous metrics. Class. Quantum Gravity 29(14), 145001 (2012) · Zbl 1246.83025 · doi:10.1088/0264-9381/29/14/145001 |
| [20] | Hartman, P.: On geodesic coordinates. Am. J. Math. 73, 949-954 (1951) · Zbl 0044.17306 · doi:10.2307/2372125 |
| [21] | Hartman, P., Wintner, A.: On the problems of geodesics in the small. Am. J. Math. 73, 132-148 (1951) · Zbl 0042.15605 · doi:10.2307/2372166 |
| [22] | Whitehead, J.H.C.: Convex regions in the geometry of paths. Q. J. Math., Oxf. Ser. 3, 33-42, (1932). Addendum, ibid. 4, 226 (1933) · Zbl 0004.13102 |
| [23] | Kunzinger, M., Steinbauer, R., Stojković, M.: The exponential map of a \[C^{1,1}\] C1,1-metric. Differ. Geom. Appl. 34, 14-24 (2014) · Zbl 1291.53016 · doi:10.1016/j.difgeo.2014.03.005 |
| [24] | Chen, B.-L., LeFloch, P.: Injectivity radius of Lorentzian manifolds. Commun. Math. Phys. 278, 679-713 (2008) · Zbl 1156.53040 · doi:10.1007/s00220-008-0412-x |
| [25] | Minguzzi, E.: Convex neighborhoods for Lipschitz connections and sprays, arXiv:1308.6675 · Zbl 1330.53021 |
| [26] | Hirsch, M.W.: Differential Topology. Springer, Berlin (1976) · Zbl 0356.57001 · doi:10.1007/978-1-4684-9449-5 |
| [27] | Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.: Geometric Theory of Generalized Functions. Kluwer, Dordrecht (2001) · Zbl 0998.46015 |
| [28] | Hörmann, G., Kunzinger, M., Steinbauer, R.: Wave equations on non-smooth space-times. Evolution equations of hyperbolic and Schrödinger type, pp. 163-186, Progr. Math., 301, Birkhäuser/Springer, Berlin (2012) · Zbl 1262.58018 |
| [29] | Amann, H.: Ordinary Differential Equations. De Gruyter, Berlin (1990) · Zbl 0708.34002 · doi:10.1515/9783110853698 |
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