Submanifolds in space-time with unphysical extra dimensions, cosmology and warped brane world models

Submanifolds in spacetime with an auxiliary unphysical extra dimension and four-dimensional open and closed FRW universes

The explicit coordinate transformations which show the equivalence between the FRW metrics of four-dimensional open and closed universes and the metrics induced on appropriate submanifolds in a five-dimensional pseudo-Euclidean space-time are presented.

Geodesic motion in the neighbourhood of submanifolds embedded in warped product spaces

We study the classical geodesic motions of nonzero rest mass test particles and photons in (3+1+n)- dimensional warped product spaces. An important feature of these spaces is that they allow a natural decoupling between the motions in the (3+1)-dimensional spacetime and those in the extra n dimensions. Using this decoupling and employing phase space analysis we investigate the conditions for confinement of particles and photons to the (3+1)- spacetime submanifold. In addition to providing information regarding the motion of photons, we also show that these motions are not constrained by the va…

Focusing of branes in warped backgrounds

Branes are embedded surfaces in a given background (bulk) spacetime. Assuming a warped bulk, we investigate, in analogy with the case for geodesics, the notion of {\em focusing} of families of such embedded, extremal 3--branes in a five dimensional background . The essential tool behind our analysis, is the well-known generalised Raychaudhuri equations for surface congruences. In particular, we find explicit solutions of these equations, which seem to show that families of 3--branes can focus along lower dimensional submanifolds depending on where the initial expansions are specified. We concl…

Multiply-warped product metrics and reduction of Einstein equations

It is shown that for every multidimensional metric in the multiply warped product form $\bar{M} = K\times_{f_1} M_1\times_{f_2}M_2$ with warp functions $f_1$, $f_2$, associated to the submanifolds $M_1$, $M_2$ of dimensions $n_1$, $n_2$ respectively, one can find the corresponding Einstein equations $\bar{G}_{AB}=-\barΛ\bar{g}_{AB}$, with cosmological constant $\barΛ$, which are reducible to the Einstein equations $G_{αβ} = -Λ_1 g_{αβ}$ and $G_{ij} =-Λ_2 h_{ij}$ on the submanifolds $M_1$, $M_2$, with cosmological constants ${Λ_1}$ and ${Λ_2}$, respectively, where $\barΛ$, ${Λ_1}$ and ${Λ_2}$ a…

Space and Space-Time geodesics in Schwarzschild geometry

Geodesic orbit equations in the Schwarzschild geometry of general relativity reduce to ordinary conic sections of Newtonian mechanics and gravity for material particles in the non-relativistic limit. On the contrary, geodesic orbit equations for a proper spatial submanifold of Schwarzschild metric at any given coordinate-time correspond to an unphysical gravitational repulsion in the non-relativistic limit. This demonstrates at a basic level the centrality and critical role of relativistic time and its intimate pseudo-Riemannian connection with space. Correspondingly, a commonly popularized de…

四维有效嵌套翘曲几何中的模动力学及若干宇宙学启示

Moduli Dynamics in Effective Nested Warped Geometry in Four Dimensions and Some Cosmological Implications

本文研究嵌套翘曲膜世界中模场的四维有效动力学，发现dS膜上第一个模方向存在亚稳态极小值，并论证沿该方向的慢滚暴胀最可行，而沿更高模方向的慢滚因需要非物理的半径层级而不受观测支持。