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We develop a new approach on the (1+3) threading of spacetime (M, g) with respect to a congruence of curves defined by an arbitrary timelike vector field. The study is based on spatial tensor fields and on the Riemannian spatial connection ∇ , which behave as 3D geometric objects. We obtain new formulas for local components of the Ricci tensor field of (M, g) with respect to the threading frame field, in terms of the Ricci tensor field of ∇ and of kinematic quantities. Also, new expressions for time covariant derivatives of kinematic quantities are stated. In particular, a new form of Raychaudhuri's equation enables us to prove Lemma 6.3, which completes a well-known lemma used in the proof of the Penrose-Hawking singularity theorems. Finally, we apply the new (1 + 3) formalism to the study of the dynamics of a Kerr-Newman black hole.
Let \documentclass[12pt]{minimal}\begin{document}$(\bar{M}, \bar{g})$\end{document}(M¯,g¯) be a 5D warped space defined by the 4D spacetime (M, g) and the warped function A. By using the extrinsic curvature of the horizontal distribution, we obtain the classification of all spaces \documentclass[12pt]{minimal}\begin{document}$(\bar{M}, \bar{g})$\end{document}(M¯,g¯) satisfying Einstein equations \documentclass[12pt]{minimal}\begin{document}$\bar{G} = -\bar{\lambda }\bar{g}$\end{document}G¯=−λ¯g¯. This enables us to describe all the exact solutions for the warped metric \documentclass[12pt]{minimal}\begin{document}$\bar{g}$\end{document}g¯ by means of 4D exact solutions.
Slant curves are introduced in three-dimensional warped products with Euclidean factors. These curves are characterised by the scalar product between the normal at the curve and the vertical vector field, and an important feature is that the case of constant Frenet curvatures implies a proper mean curvature vector field. A Lancret invariant is obtained and the Legendre curves are analysed as a particular case. An example of a slant curve is given for the exponential warping function; our example illustrates a proper (that is, not reducible to the two-dimensional) case of the Lancret theorem of three-dimensional hyperbolic geometry. We point out an eventuality relationship with the geometry of relativistic models.
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