|
Generalized pseudo-Riemannian geometry
Author(s):
Michael
Kunzinger;
Roland
Steinbauer
Journal:
Trans. Amer. Math. Soc.
354
(2002),
4179-4199.
MSC (2000):
Primary 46F30;
Secondary 46T30, 46F10, 83C05
Posted:
June 3, 2002
Retrieve article in:
PDF DVI PostScript
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
Generalized tensor analysis in the sense of Colombeau's construction is employed to introduce a nonlinear distributional pseudo-Riemannian geometry. In particular, after deriving several characterizations of invertibility in the algebra of generalized functions, we define the notions of generalized pseudo-Riemannian metric, generalized connection and generalized curvature tensor. We prove a ``Fundamental Lemma of (pseudo-) Riemannian geometry'' in this setting and define the notion of geodesics of a generalized metric. Finally, we present applications of the resulting theory to general relativity.
References:
-
- 1.
- Aragona, J., Juriaans, S. O. Some structural properties of the topological ring of Colombeau's generalized numbers. Comm. Algebra, 29 (5), 2201-2230, 2001.
- 2.
- Balasin, H. Distributional energy-momentum tensor of the extended Kerr geometry. Class. Quant. Grav., pages 3353-3362, 1997. MR 99f:83009
- 3.
- Balasin, H. Distributional aspects of general relativity: The example of the energy-momentum tensor of the extended Kerr geometry. In Grosser, M., Hörmann, G., Kunzinger, M., Oberguggenberger, M., editor, Nonlinear Theory of Generalized Functions, volume 401 of CRC Research Notes, pages 231-239, Boca Raton, 1999. CRC Press. MR 2000g:83008
- 4.
- Bhatia, R. Perturbation Bounds for Matrix Eigenvalues, volume 162 of Pitman Research Notes in Mathematics. Longman, Harlow, U.K., 1987. MR 88k:15020
- 5.
- Bourbaki, N. Algebra I, Chapters 1-3. Elements of Mathematics. Hermann, Paris, and Addison-Wesley, Massachusetts, 1974. MR 50:6689
- 6.
- Clarke, C. J. S., Vickers, J. A., Wilson, J. P. Generalised functions and distributional curvature of cosmic strings. Class. Quant. Grav., 13, 1996. MR 97g:83085
- 7.
- Colombeau, J. F. New Generalized Functions and Multiplication of Distributions. North Holland, Amsterdam, 1984. MR 86c:46042
- 8.
- Colombeau, J. F. Elementary Introduction to New Generalized Functions. North Holland, Amsterdam, 1985. MR 87f:46064
- 9.
- Dapic, N., Kunzinger, M., Pilipovic, S. Symmetry group analysis of weak solutions. Proc. London Math. Soc., 84(3):686-710, 2002. CMP 2002:09
- 10.
- De Roever, J. W., Damsma, M. Colombeau algebras on a
 -manifold. Indag. Mathem., N.S., 2(3), 1991. MR 93e:46046 - 11.
- Geroch, R., Traschen, J. Strings and other distributional sources in general relativity. Phys. Rev. D, 36(4):1017-1031, 1987. MR 89d:83048
- 12.
- Grosser, M., Farkas, E., Kunzinger, M., Steinbauer, R. On the foundations of nonlinear generalized functions I, II. Mem. Amer. Math. Soc. 153, 2001.
- 13.
- Grosser, M., Kunzinger, M., Steinbauer, R., Vickers, J. A global theory of algebras of generalized functions. Adv. Math., 166(1):50-72, 2002. CMP 2002:09
- 14.
- Heinzle, J. M., Steinbauer, R. Remarks on the distributional Schwarzschild geometry. J. Math. Phys., 43(3):1493-1508, 2002.
- 15.
- Israel, W. Singular hypersurfaces and thin shells in general relativity. Nuovo Cim., 44B(1):1-14, 1966.
- 16.
- Kamleh, W. Signature changing space-times and the new generalised functions. Preprint, gr-qc/0004057, 2000.
- 17.
- Klainerman, S., Nicoló, F. On local and global aspects of the Cauchy problem in general relativity. Class. Quant. Grav., 16:R73-R157, 1999. MR 2000h:83006
- 18.
- Klainerman, S., Rodnianski, I. Rough solutions of the Einstein-vacuum equations. Preprint, math.AP/0109173, 2001.
- 19.
- Kriegl, A., Michor, P. W. The Convenient Setting of Global Analysis, volume 53 of Math. Surveys Monogr. Amer. Math. Soc., Providence, RI, 1997. MR 98i:58015
- 20.
- Kunzinger, M. Generalized functions valued in a smooth manifold. Monatsh. Math., to appear (available electronically at http://arxiv.org/abs/math.FA/ 0107051), 2002.
- 21.
- Kunzinger, M., Oberguggenberger, M. Group analysis of differential equations and generalized functions. SIAM J. Math. Anal., 31(6):1192-1213, 2000. MR 2001c:35010
- 22.
- Kunzinger, M., Steinbauer, R. A rigorous solution concept for geodesic and geodesic deviation equations in impulsive gravitational waves. J. Math. Phys., 40:1479-1489, 1999. MR 2001a:83018
- 23.
- Kunzinger, M., Steinbauer, R. A note on the Penrose junction conditions. Class. Quant. Grav., 16:1255-1264, 1999. MR 2000c:83023
- 24.
- Kunzinger, M., Steinbauer, R. Foundations of a nonlinear distributional geometry. Acta Appl. Math., 71, 179-206, 2002.
- 25.
- Mallios, A. Geometry of vector sheaves. Vols. I, II. Mathematics and its Applications, 439. Kluwer Academic Publishers, Dordrecht, 1998. MR 99g:58002; MR 99g:58003
- 26.
- Mallios, A., Rosinger, E. E. Abstract differential geometry, differential algebras of generalized functions, and de Rham cohomology. Acta Appl. Math., 55(3):231-250, 1999. MR 2000e:58003
- 27.
- Mallios, A., Rosinger, E. E. Space-time foam dense singularities and de Rham cohomology. Acta Appl. Math., 67: 59-89, 2001. MR 2002e:58004
- 28.
- Ligeza, J., Tvrdy, M. On systems of linear algebraic equations in the Colombeau algebra. Math. Bohem., 124(1):1-14, 1999. MR 2000c:46079
- 29.
- Mansouri, R., Nozari, K. A new distributional approach to signature change. Gen. Relativity Gravitation, 32(2):235-269, 2000. MR 2001f:83136
- 30.
- Marsden, J. E. Generalized Hamiltonian mechanics. Arch. Rat. Mech. Anal., 28(4):323-361, 1968. MR 37:534
- 31.
- Oberguggenberger, M. Multiplication of Distributions and Applications to Partial Differential Equations, volume 259 of Pitman Research Notes in Mathematics. Longman, Harlow, 1992. MR 94d:46044
- 32.
- Oberguggenberger, M., Kunzinger, M. Characterization of Colombeau generalized functions by their point values. Math. Nachr., 203:147-157, 1999. MR 2000d:46053
- 33.
- O'Neill, B. Semi-Riemannian Geometry (with Applications to Relativity). Academic Press, New York, 1983. MR 85f:53002
- 34.
- Parker, P. Distributional geometry. J. Math. Phys., 20(7):1423-1426, 1979. MR 81f:83030
- 35.
- Penrose, R., Rindler, W. Spinors and space-time I. Cambridge University Press, 1984. MR 86h:83002
- 36.
- Rendall, A. Local and Global Existence Theorems for the Einstein Equations. Living Rev. Relativity, 3(1). [Online Article]: cited on 2002-01-20, http://www. livingreviews.org/Articles/Volume3/2000-1rendall/. MR 2000i:58053
- 37.
- Simanca, S. R. Pseudo-differential Operators, volume 236 of Pitman Research in Notes in Mathematics. Longman, Harlow, 1990. MR 91i:35222
- 38.
- Steinbauer, R. The ultrarelativistic Reissner-Nordstrøm field in the Colombeau algebra. J. Math. Phys., 38:1614-1622, 1997. MR 98f:83017
- 39.
- Steinbauer, R. Geodesics and geodesic deviation for impulsive gravitational waves. J. Math. Phys., 39:2201-2212, 1998. MR 99a:83027
- 40.
- Vickers, J. A. Nonlinear generalized functions in general relativity. In Grosser, M., Hörmann, G., Kunzinger, M., Oberguggenberger, M., editor, Nonlinear Theory of Generalized Functions, volume 401 of CRC Research Notes, pages 275-290, Boca Raton, 1999. CRC Press. MR 2000g:83056
- 41.
- Vickers, J., Wilson, J. A nonlinear theory of tensor distributions. ESI-Preprint (available electronically at http://www.esi.ac.at/ESI-Preprints.html), 566, 1998.
- 42.
- Vickers, J. A., Wilson, J. P. Invariance of the distributional curvature of the cone under smooth diffeomorphisms. Class. Quantum. Grav., 16:579-588, 1999. MR 2000a:83096
- 43.
- Wilson, J. P. Distributional curvature of time dependent cosmic strings. Class. Quantum Grav., 14:3337-3351, 1997. MR 99e:83018
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(2000):
46F30,
46T30, 46F10, 83C05
Retrieve articles in all Journals with MSC
(2000):
46F30,
46T30, 46F10, 83C05
Additional Information:
Michael
Kunzinger
Affiliation:
Department of Mathematics, University of Vienna, Strudlhofg. 4, A-1090 Wien, Austria
Email:
Michael.Kunzinger@univie.ac.at
Roland
Steinbauer
Affiliation:
Department of Mathematics, University of Vienna, Strudlhofg. 4, A-1090 Wien, Austria
Email:
roland.steinbauer@univie.ac.at
DOI:
10.1090/S0002-9947-02-03058-1
PII:
S 0002-9947(02)03058-1
Keywords:
Algebras of generalized functions,
Colombeau algebras,
generalized tensor fields,
generalized metric,
(generalized) pseudo-Riemannian geometry,
general relativity.
Received by editor(s):
August 9, 2001
Received by editor(s) in revised form:
January 31, 2002
Posted:
June 3, 2002
Additional Notes:
This work was in part supported by research grant P12023-MAT of the Austrian Science Fund
Copyright of article:
Copyright
2002,
American Mathematical Society
|
Comments: webmaster@ams.org