Abstract
A pair (M, Γ) is defined as a Riemannian manifold M of normal hyperbolic type carrying a distinguished time-like congruence Γ. The spatial tensor algebraD associated with the pair (M, Γ) is discussed. A general definition of the concept of spatial tensor analysis over (M, Γ) is then proposed. Basically, this includes a spatial covariant differentiation\(\tilde \nabla \) and a time-derivative\(\tilde \nabla _T \), both acting onD and commuting with the process of raising and lowering the tensor indices. The torsion tensor fields of the pair\(\left( {\tilde \nabla ,\tilde \nabla _T } \right)\) are discussed, as well as the corresponding structural equations. The existence of a distinguished spatial tensor analysis over (M, Γ) is finally established, and the resulting mathematical structure is examined in detail.
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References
Bondi, H. (1960).Cosmology, (University Press, Cambridge).
Adler, A., Bazin, M. and Schiffer, M. (1965).Introduction to General Relativity, (McGraw Hill, New York).
Robertson, H.P. (1935).Astrophys. J.,82, 284.
Massa, E. (1971).Commun. Math. Phys.,20, 279.
Massa, E. (1971).Commun. Math. Phys.,22, 321.
Massa, E. (1966).Nuovo Cim.,B42, 178.
Møller, C. (1952).The Theory of Relativity (Clarendon Press, Oxford).
Cattaneo, C. (1958).Nuovo Cim.,10, 318.
Cattaneo, C. (1959).Ann. Mat.,XLVIII, 361.
Cattaneo, C. (1959).C.R. Acad. Sci.,248, 179.
Cattaneo, C. (1959).Nuovo Cim.,11, 733.
Cattaneo, C. (1961).Introduzione alla teoria Einsteniana della gravitazione, (Veschi, Roma).
Cattaneo, C. (1963).C.R. Acad. Sci.,256, 3974.
Cattaneo Gasparini, I. (1961).C.R. Acad. Sci.,252, 3722.
Cattaneo Gasparini, I. (1963).Rend. Mat.,252, 3722.
Cattaneo Gasparini, I. (1963).C.R. Acad. Sci.,256, 2089.
Ferrarese, G. (1963).Rend. Mat.,22, 147.
Helgason, S. (1962).Differnetial Geometry and Symmetric Spaces, (Academic Press, New York).
Eisenhart, L.P. (1956).Riemannian Geometry, (University Press, Princeton).
Flanders, H. (1963).Differential Forms with Applications to the Physical Sciences, (Academic Press, New York).
Sternberg, S. (1964).Lectures on Differential Geometry, (Prentice Hall, Englewood Cliffs, N.J.).
Israel, W. (1970). Differential Forms in General Relativity,Commun. Dublin Inst. Adv. Stud., Ser. A, No. 19.
Trautman, A. (1972).On the Structure of the Einstein-Cartan Equations, (Report for Convegno di Relatività held in Rome in February 1972), (to be published on the proceedings of the Convegno).
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This work was assisted by funds from the C.N.R. under the aegis of the activity of the National Group for Mathematical Physics.
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Massa, E. Space tensors in general relativity I: Spatial tensor algebra and analysis. Gen Relat Gravit 5, 555–572 (1974). https://doi.org/10.1007/BF02451398
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DOI: https://doi.org/10.1007/BF02451398