Weyl tensor

Abstract: We carefully investigate the gravitational equations of the brane world, in which all the matter forces except gravity are confined on the 3-brane in a 5-dimensional spacetime with ${Z}_{2}$ symmetry. We derive the effective gravitational equations on the brane, which reduce to the conventional Einstein equations in the low energy limit. From our general argument we conclude that the first Randall-Sundrum-type theory predicts that the brane with a negative tension is an antigravity world and hence should be excluded from the physical point of view. Their second-type theory where the brane has a positive tension provides the correct signature of gravity. In this latter case, if the bulk spacetime is exactly anti--de Sitter spacetime, generically the matter on the brane is required to be spatially homogeneous because of the Bianchi identities. By allowing deviations from anti--de Sitter spacetime in the bulk, the situation will be relaxed and the Bianchi identities give just the relation between the Weyl tensor and the energy momentum tensor. In the present brane world scenario, the effective Einstein equations cease to be valid during an era when the cosmological constant on the brane is not well defined, such as in the case of the matter dominated by the potential energy of the scalar field.

Abstract: I: Calculus of Variations. Minkowskian Spaces.- 1. Problems in the calculus of variations in parametric form.- 2. The tangent space. The indicatrix.- 3. The metric tensor and the osculating indicatrix.- 4. The dual tangent space. The figuratrix.- 5. The Hamiltonian function.- 6. The trigonometric functions and orthogonality.- 7. Definitions of angle.- 8. Area and Volume.- II: Geodesics: Covariant Differentiation.- 1. The differential equations satisfied by the geodesics.- 2. The explicit expression for the second derivatives in the differential equations of the geodesies.- 3. The differential of a vector.- 4. Partial differentiation of vectors.- 5. Elementary properties of ?-differentiation.- III: The "Euclidean Connection" of E. Cartan.- 1. The fundamental postulates of Cartan.- 2. Properties of the covariant derivative.- 3. The general geometry of paths: the connection of Berwald.- 4. Further connections arising from the general geometry of paths.- 5. The osculating Riemannian space.- 6. Normal coordinates.- IV: The Theory of Curvature.- 1. The commutation formulae.- 1 . Commutation formulae resulting from ?-differentiation.- 2 . The three curvature tensors of Cartan.- 3 . Alternative derivation of the curvature tensors by means of exterior forms.- 2. Identities satisfied by the curvature tensors.- 3. The Bianchi identities.- 4. Geodesic deviation Ill.- 5. The first and second variations of the length integral.- 6. The curvature tensors arising from Berwald's connection.- 7. Spaces of constant curvature.- 8. The projective curvature tensors.- 1 . The generalised Weyl tensor.- 2 . The projective connection.- 3 . Projectively flat spaces spaces with rectilinear geodesies.- V: The Theory of Subspaces.- 1. The theory of curves.- 2. The projection factors.- 3. The induced connection parameters. .- 4. Fundamental aspects of the theory of subspaces based on the euclidean connection.- 1 . The normal curvature and associated tensors.- 2 . The D-symbolism.- 3 . The generalised equations of Gauss, Codazzi and Kuhne.- 5. The Lie derivative and its application to the theory of subspaces.- 6. Surfaces imbedded in an F3.- 7. Fundamental aspects of the theory of subspaces from the point of view of the locally Minkowskian metric.- 1 . Normal curvature.- 2 . The two second fundamental forms.- 3 . Principal directions.- 4 . The equations of Gauss and Codazzi.- 5 . Subspaces of arbitrary dimension.- 8. The differential geometry of the indicatrix and the geometrical significance of the tensor Sijhk.- 9. Comparison between the induced and the intrinsic connection parameters.- VI: Miscellaneous Topics.- 1. Groups of motions.- 2. Conformai geometry.- 3. The equivalence problem.- 4. The theory of non-linear connections.- 5. The local imbedding theories.- 6. Two-dimensional Finsler spaces.- 1 . Formal Aspects.- 2 . Certain projective changes applied to F2. Spaces with rectilinear geodesics.- 3 . Two-dimensional Finsler spaces whose principal scalar is a function of position only. Landsberg spaces.- Appendix: Bibliographical references to related topics.- Symbols.