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Spacetimes with distributional semi-Riemannian metrics and their curvature. (English) Zbl 1435.53018

Summary: We develop a comprehensive geometric framework for defining spaces \(\mathcal{G} (M, E)\) of nonlinear generalized sections of vector bundles \(E \to M\) containing spaces of distributional sections \(\mathcal{D}^\prime (M, E)\). Our theory incorporates classical differential geometric operations (like tensor products, covariant derivatives and Lie derivatives), is localizable and fully compatible with smooth and distributional tensor calculus. As an application to the treatment of singular metrics, we calculate the curvature of the conical metric used to describe cosmic strings.

MSC:

53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53Z05 Applications of differential geometry to physics
46F99 Distributions, generalized functions, distribution spaces
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[1] Abraham, R.; Marsden, J. E.; Ratiu, T., (Manifolds, Tensor Analysis, and Applications. Manifolds, Tensor Analysis, and Applications, Applied Mathematical Sciences, vol. 75 (1988), Springer-Verlag: Springer-Verlag New York) · Zbl 0875.58002
[2] Aragona, J.; Biagioni, H. A., Intrinsic definition of the Colombeau algebra of generalized functions, Anal. Math., 17, 2, 75-132 (1991) · Zbl 0765.46019
[3] Clarke, C.; Vickers, J.; Wilson, J., Generalized functions and distributional curvature of cosmic strings, Classical Quantum Gravity, 13, 9, 2485-2498 (1996) · Zbl 0859.53074
[4] Colombeau, J. F., (New Generalized Functions and Multiplication of Distributions. New Generalized Functions and Multiplication of Distributions, North-Holland Mathematics Studies, vol. 84 (1984), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam) · Zbl 0532.46019
[5] Colombeau, J. F., (Elementary Introduction to New Generalized Functions. Elementary Introduction to New Generalized Functions, North-Holland Mathematics Studies, vol. 113 (1985), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam) · Zbl 0584.46024
[6] Colombeau, J. F.; Meril, A., Generalized functions and multiplication of distributions on \(\mathcal{C}^\infty\) manifolds, J. Math. Anal. Appl., 186, 2, 357-364 (1994) · Zbl 0819.46026
[7] Dahia, F.; Romero, C., Conical space-times: a distribution theory approach, Modern Phys. Lett. A, 14, 1879-1893 (1999), arXiv:gr-qc/9801109
[8] Geroch, R.; Traschen, J., Strings and other distributional sources in general relativity, Phys. Rev. D (3), 36, 4, 1017-1031 (1987)
[9] Gottfried, K., (Topological Vector Spaces II. Topological Vector Spaces II, Grundlehren der Mathematischen Wissenschaften, vol. 237 (1979), Springer-Verlag: Springer-Verlag New York) · Zbl 0417.46001
[10] Greub, W.; Halperin, S.; Vanstone, R., (Connections, Curvature, and Cohomology. Vol. I: De Rham Cohomology of Manifolds and Vector Bundles. Connections, Curvature, and Cohomology. Vol. I: De Rham Cohomology of Manifolds and Vector Bundles, Pure and Applied Mathematics, vol. 47 (1972), Academic Press: Academic Press New York) · Zbl 0322.58001
[11] Grosser, M.; Farkas, E.; Kunzinger, M.; Steinbauer, R., On the foundations of nonlinear generalized functions I, Mem. Am. Math. Soc., 153, 729 (2001) · Zbl 0985.46026
[12] Grosser, M.; Kunzinger, M.; Oberguggenberger, M.; Steinbauer, R., (Geometric Theory of Generalized Functions with Applications to General Relativity. Geometric Theory of Generalized Functions with Applications to General Relativity, Mathematics and its Applications, vol. 537 (2001), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht) · Zbl 0998.46015
[13] Grosser, M.; Kunzinger, M.; Steinbauer, R.; Vickers, J. A., A global theory of algebras of generalized functions, Adv. Math., 166, 1, 50-72 (2002) · Zbl 0995.46054
[14] Grosser, M.; Kunzinger, M.; Steinbauer, R.; Vickers, J. A., A global theory of algebras of generalized functions. II. Tensor distributions, New York J. Math., 18, 139-199 (2012) · Zbl 1252.46030
[15] Grosser, M.; Nigsch, E. A., Full and special Colombeau algebras, Proc. Edinb. Math. Soc., 61, 4, 961-994 (2018) · Zbl 1430.46034
[16] Hörmann, G., Conical spacetimes and global hyperbolicity, Novi Sad J. Math., 45, 1, 215-229 (2015) · Zbl 1474.83004
[17] Horváth, J., Topological Vector Spaces and Distributions, Vol. 1 (1966), Addison-Wesley: Addison-Wesley Reading, Mass. · Zbl 0143.15101
[18] Jarchow, H., Locally Convex Spaces (1981), B. G. Teubner: B. G. Teubner Stuttgart · Zbl 0466.46001
[19] Jelínek, J., An intrinsic definition of the Colombeau generalized functions, Commentat. Math. Univ. Carol., 40, 1, 71-95 (1999) · Zbl 1060.46513
[20] Kahn, D. W., Introduction to Global Analysis (2007), Dover Publications: Dover Publications Mineola, N.Y. · Zbl 1118.58001
[21] Kolář, I.; Michor, P. W.; Slovák, J., Natural Operations in Differential Geometry (1993), Springer-Verlag: Springer-Verlag Berlin · Zbl 0782.53013
[22] Kriegl, A.; Michor, P. W., (The Convenient Setting of Global Analysis. The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, vol. 53 (1997), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0889.58001
[23] Lang, S., (Algebra. Algebra, Graduate Texts in Mathematics, vol. 211 (2002), Springer: Springer New York) · Zbl 0984.00001
[24] Lee, J. M., (Introduction to Smooth Manifolds. Introduction to Smooth Manifolds, Graduate Texts in Mathematics, vol. 218 (2013), Springer: Springer New York) · Zbl 1258.53002
[25] Lee, D. A.; LeFloch, P. G., The positive mass theorem for manifolds with distributional curvature, Comm. Math. Phys., 339, 99-120 (2015), arXiv:1408.4431 · Zbl 1330.53062
[26] Lott, J., Ricci measure for some singular Riemannian metrics, Math. Ann., 365, 1, 449-471 (2016) · Zbl 1343.53031
[27] Nigsch, E. A., The functional analytic foundation of Colombeau algebras, J. Math. Anal. Appl., 421, 1, 415-435 (2015), arXiv:1305.1460 · Zbl 1314.46055
[28] Nigsch, E. A., Nonlinear generalized sections of vector bundles, J. Math. Anal. Appl., 440, 183-219 (2016), arXiv:1409.2962 · Zbl 1380.46059
[29] Nigsch, E. A., On regularization of vector distributions on manifolds, Forum Math., 28, 6, 1131-1141 (2016), arXiv:1504.02237 · Zbl 1364.46064
[30] Oberguggenberger, M., (Multiplication of Distributions and Applications to Partial Differential Equations. Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Research Notes in Mathematics, volume. 259 (1992), Longman: Longman Harlow, U.K.) · Zbl 0818.46036
[31] O’Neill, B., Semi-Riemannian Geometry (1983), Academic Press: Academic Press New York · Zbl 0531.53051
[32] Reina, B.; Senovilla, J. M.M.; Vera, R., Junction conditions in quadratic gravity: thin shells and double layers, Classical Quantum Gravity, 33, 10, 41 (2016), ISSN: 0264-9381; 1361-6382/e · Zbl 1338.83159
[33] Schaefer, H. H., Topological Vector Spaces (1971), Springer-Verlag: Springer-Verlag New York · Zbl 0212.14001
[34] Schwartz, L., Sur l’impossibilité de la multiplication des distributions, C. R. Acad. Sci., 239, 847-848 (1954) · Zbl 0056.10602
[35] Schwartz, L., Espaces de fonctions différentiables à valeurs vectorielles, J. Anal. Math., 4, 1, 88-148 (1955) · Zbl 0066.09601
[36] Schwartz, L., Théorie des distributions (1966), Hermann: Hermann Paris · Zbl 0149.09501
[37] Steinbauer, R.; Vickers, J. A., The use of generalized functions and distributions in general relativity, Classical Quantum Gravity, 23, 10, r91-r114 (2006) · Zbl 1096.83001
[38] Steinbauer, R.; Vickers, J. A., On the Geroch-Traschen class of metrics, Classical Quantum Gravity, 26, 6 (2009) · Zbl 1162.83320
[39] Tonita, A., Distributional sources for black hole initial data, Classical Quantum Gravity, 29, 1, Article 015001 pp. (2012), arXiv:1107.4691 · Zbl 1235.83019
[40] Vickers, J. A., Distributional geometry in general relativity, J. Geom. Phys., 62, 3, 692-705 (2012) · Zbl 1242.53098
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