Abstract
Local four-dimensional tensor decomposition formulae for generic vectors and 2-tensors in spacetime, in terms of scalar and antisymmetric covariant tensor potentials, are studied within the framework of tensor distributions. Earlier first-order decompositions are extended to include the case of four-dimensional symmetric 2-tensors and new second-order decompositions are introduced.
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Bampi, F. and Caviglia, G.: Third-order tensor potentials for the Riemann and Weyl tensors, Gen. Relativity Gravitation 15 (1983), 375.
Baumeister, R.: Clebsh representation and variational principles in the theory of relativistic dynamical systems, Utilitas Math. 16 (1979), 43.
Bruhat, Y.: The Cauchy problem, In: L. Witten (ed.), Gravitation, An Introduction to Current Research, Wiley, New York, 1962, p. 130.
Cantor, M.: Boundary value problems for asymptotically homogeneous elliptic second order operators, J. Differential Equations 34 (1979), 102.
Choquet-Bruhat, Y.: Hyperbolic partial differential equations on a manifold, In: C. M. DeWitt and J. A. Wheeler (eds), Battelle Rencontres, Benjamin, New York, 1968, p. 84.
Choquet-Bruhat, Y., DeWitt-Morette, C. and Dillard-Bleik, M.: Analysis, Manifolds and Physics. Part I: Basics, North-Holland, Amsterdam, 1977.
Crupi, G.: Considerazioni sul teorema di Clebsch e sul lemma di Finzi, Istituto Lombardo Rend. Sci. A 100 (1966), 951.
Deser, S.: Covariant decomposition of symmetric tensors and the gravitational Cauchy problem, Ann. Inst. H. Poincaré 7 (1967), 149.
Finzi, B.: Sul principio della minima azione e sulle equazioni elettromagnetiche che ne derivano, Rend. Sc. fis. mat. nat. Lincei 12 (1952), 378.
Finzi, B. and Pastori, M.: Calcolo tensoriale e applicazioni, Zanichelli, Bologna, 1961.
Gaffet, B.: On generalized vorticity-conservation laws, J. Fluid Mech. 156 (1985), 141.
Goncharov, V. and Pavlov, V.: Some remarks on the physycal foundation of the Hamiltonian description of fluid motions, European J. Mech. B Fluids 16 (1997), 509.
Illge, R.: On potentials for several classes of spinor and tensor fields in curved spacetimes, Gen. Relativity Gravitation 20 (1988), 551.
Lichnerowicz, A.: Propagateurs et commutateurs en relativité générale, Publications Mathématiques 10, Institut des Hautes Études Scientifiques, Paris, 1961.
Lichnerowicz, A.: Théorie des rayons en hydrodynamique et magnétohydrodinamique relativiste, Ann. Inst. H. Poincaré 7 (1967), 271.
Lichnerowicz, A.: Ondes de choc et hypothéses de compressibilité en magnétohydrodynamique relativiste, Comm. Math. Phys. 12 (1969), 145.
Lichnerowicz, A.: Magnetohydrodynamics: Waves and Shock Waves in Curved Space-Time, Math. Phys. Stud. 14, Kluwer Acad. Publ., Dordrecht, 1994.
Marchioro, C. and Pulvirenti, M.: Mathematical Theory of Incompressible Nonviscous Fluids, Springer-Verlag, New York, 1994.
Massa, E. and Pagani, E.: Is the Riemann tensor derivable from a tensor potential?, Gen. Relativity Gravitation 16 (1984), 805.
Maugin, G.: Sur la transformation de Clebsch et la magnétohydrodynamique relativiste, C.R. Acad. Sci. Paris Sér. A-B 274 (1972), A602.
Monroe, D. K.: Local transverse-traceless tensor operators in general relativity, J. Math. Phys. 9 (1981), 1994.
Persico, E.: Introduzione alla fisica matematica, Zanichelli, Bologna, 1952.
Rund, H.: Clebsch potentials and variational principles in the theory of dynamical systems, Arch. Ration. Mech. Anal. 65 (1977), 305.
Rund, H.: Clebsch representations and relativistic dynamical systems, Arch. Ration. Mech. Anal. 71 (1979), 199.
Rund, H.: Clebsch potentials in the theory of electromagnetic fields admitting electric and magnetic charge distributions, J. Math. Phys. 18 (1977), 84.
Scwarz, G.: Hodge Decomposition-A Method for Solving Boundary Value Problems, Lecture Notes in Math. 1607, Springer, Berlin, 1995.
Specovius-Neugebauer, M.: The Helmholtz decomposition of weighted Lr-spaces, Comm. Partial Differential Equations 15 (1990), 273.
Wenzelburger, J.: A kinematical model for continuous distributions of dislocations, J. Geom. Phys. 24 (1998), 334.
York, J.W., Jr.: Covariant decomposition of symmetric tensors in the theory of gravitation, Ann. Inst. H. Poincaré 21 (1974), 319.
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Gemelli, G. Second-Order Covariant Tensor Decomposition in Curved Spacetime. Mathematical Physics, Analysis and Geometry 3, 195–216 (2000). https://doi.org/10.1023/A:1011419231097
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DOI: https://doi.org/10.1023/A:1011419231097