We investigate the dynamics of a spherically symmetric vaccum on a Randall and Sundrum 3-brane world. Under certain natural conditions, the effective Einstein equations on the brane form a closed system for spherically symmetric dark radiation. We determine exact dynamical and inhomogeneous solutions, which are shown to depend on the brane cosmological constant, on the dark radiation tidal charge and on its initial energy configuration. We identify the conditions defining these solutions as singular or as globally regular. Finally, we discuss the confinement of gravity to the vicinity of the brane and show that a phase transition to a regime where gravity is not bound to the brane may occur at short distances during the collapse of positive dark energy density on a realistic de Sitter brane.
In the Randall-Sundrum scenario we investigate the dynamics of a spherically symmetric 3-brane world when matter fields are present in the bulk. To analyze the 5-dimensional Einstein equations we employ a global conformal transformation whose factor characterizes the Z 2 symmetric warp. We find a new set of exact dynamical collapse solutions which localize gravity in the vicinity of the brane for a stress-energy tensor of conformal weight -4 and a warp factor that depends only on the coordinate of the fifth dimension. Geometries which describe the dynamics of inhomogeneous dust and generalized dark radiation on the brane are shown to belong to this set. The conditions for singular or globally regular behavior and the static marginally bound limits are discussed for these examples. Also explicitly demonstrated is complete consistency with the effective point of view of a 4-dimensional observer who is confined to the brane and makes the same assumptions about the bulk degrees of freedom.
Scientific research involves mathematical modelling in the context of an interactive balance between theory, experiment and computation. However, computational methods and tools are still far from being appropriately integrated in the high school and university curricula in science and mathematics. In this paper, it is discussed the relevance of mathematical modelling and illustrated how a computer modelling tool (Modellus, a free tool available on the Internet and developed at FCTUNL) can be used to embed modelling in high school and undergraduate courses. Modellus allows students to create and explore mathematical models using functions, differential and iterative equations, and visualize the behaviour of mathematical objects.
We consider in the Randall-Sundrum scenario the dynamics of a spherically symmetric 3-brane world when matter fields exist in the bulk. We determine exact 5-dimensional solutions which localize gravity near the brane and describe the dynamics of homogeneous polytropic matter on the brane. We show that these geometries are associated with a well defined conformal class of bulk matter fields. We analyze the effective polytropic dynamics on the brane identifying conditions which define it as singular or as globally regular.
In the Randall-Sundrum scenario, we analyse the dynamics of an AdS 5 braneworld when conformal matter fields propagate in five dimensions. We show that conformal fields of weight −4 are associated with stable geometries which describe the dynamics of inhomogeneous dust, generalized dark radiation and homogeneous polytropic dark energy on a spherically symmetric 3-brane embedded in the compact AdS 5 orbifold. We discuss aspects of the radion stability conditions and of the localization of gravity in the vicinity of the
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