The geodesic motion near hypersurfaces in the warped products spacetime

曲积空间中子流形邻域的测地运动

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

研究了（3+1+n）维翘曲乘积空间中非零静止质量的试验粒子和光子的经典测地线运动。这些空间的一个重要特征是，它们允许（3+1）维时空中的运动与额外n维时空中的运动之间的自然解耦。利用这种解耦和相空间分析，我们调查的条件约束的粒子和光子的（3+1）-时空子流形。除了提供有关光子运动的信息，我们还表明，这些运动不受外在曲率的值。在伪黎曼流形的情况下，我们得到了测地线禁闭的一般条件，并建立了这种禁闭的稳定性条件。这些结果也推广了作者最近关于余维为1的超曲面的嵌入的结果。

A Fundamental Theorem for Hypersurfaces in Semi-Riemannian Warped Products

We give necessary and sufficient conditions for a semi-Riemannian manifold of arbitrary signature to be locally isometrically immersed into certain warped products. Then, we describe a way to use the structure equations of such immersions to construct foliations of marginally trapped surfaces in a four-dimensional Lorentzian spacetimes. We point out that, sometimes, Gauss and Codazzi equations are not sufficient to ensure the existence of a local isometric immersion of a semi-Riemannian manifold as a hypersurface of another manifold. We finally give two low-dimensional examples to illustrate o…

Geodesics, Mass and the Uncertainty Principle in a Warped de Sitter Space-time

We present the explicit solution to the geodesic equations in a warped de Sitter space-time proposed by Randall-Sundrum. We find that a test particle moves in the bulk and is not restricted on a 3-brane (to be taken as our universe). On the 3-brane, the test particle moves with uniform velocity, giving the appearance that it is not subject to a force. But computing the particle's energy using the energy-momentum tensor yields a time-dependent energy that suggests a time-dependent mass. Thus, the extra force, which is the effect of the warped extra dimension on the particle's motion on the 3-br…

Five-dimensional warped product space-time with time-dependent warp factor and cosmology of the four-dimensional universe

In this paper, we have studied a 5-dimensional warped product space-time with a time-dependent warp factor. This warp factor plays an important role in localizing matter to the 4-dimensional hypersurface constituting the observed universe and leads to a geometric interpretation of dynamical dark energy. The five-dimensional field equations are constructed and its solutions are obtained. The nature of modifications produced by this warp factor in the bulk geometry is discussed. The hypersurface is described by a flat FRW-type metric in the ordinary spatial dimension. It is found that the effect…

Siklos时空的全测地超曲面与平行超曲面

Totally Geodesic and Parallel Hypersurfaces of Siklos Spacetimes

本文研究Siklos时空，系统分类并描述其全测地与平行超曲面，并刻画了一大类极小超曲面。

Finsler时空中的全测地零超曲面与表面重力的恒定性

Totally geodesic null hypersurfaces and constancy of surface gravity in Finsler spacetimes

本文定义并研究了Finsler时空中的全测地零超曲面，证明了在零收敛条件和特定引力方程下，超曲面上Ricci 1-形式消失，从而将Lorentz情形中紧致全测地零超曲面的主要结果推广到Lorentz-Finsler框架，包括表面重力恒定性和拓扑分类。

Particle Motion and Perturbed Dynamical System in Warped Product Spacetimes

In this paper we have used the dynamical systems analysis to study the dynamics of a five-dimensional universe in the form of a warped product spacetime with a spacelike dynamic extra dimension. We have decomposed the geodesic equations to get the motion along the extra dimension and have studied the associated dynamical system when the cross-diagonal element of the Einstein tensor vanishes, and also when it is non-vanishing. In the first case, introducing the concept of an energy function along the phase path in terms of the extra-dimensional coordinate, we have examined how the energy functi…

Wormholes with a warped extra dimension?

We investigate the role of a specifically warped extra dimension in constructing examples of higher dimensional spacetimes representing Lorentzian wormholes. The warping chosen is largely inspired by the well-known non-static Witten bubble of nothing, though our spacetimes are static and geometrically different. Vacuum solutions in $D\geq 5$ dimensions and others (non-asymptotically flat) with `perfectly normal' matter stress energy are interpreted as possible Lorentzian wormholes. Asymptotically flat wormholes in $D\geq 5$ with `exotic matter' and within this class of spacetimes also appear t…

锥结构中的类光超曲面与时间最小测地线

Lightlike hypersurfaces and time-minimizing geodesics in cone structures

本文将洛伦兹几何中的类光超曲面概念推广到无度规的锥结构，证明其具有唯一锥测地线叶状结构，并借此在全局双曲时空中获得锥测地线的时间最小化性质。

真空时空上的共形测地线

Conformal geodesics on vacuum space-times

讨论了一般时空、真空时空和曲积时空上的共形测地线的性质，并导出了共形偏差方程组。这些结果被用来说明如何在Schwarzschild-Kruskal时空上构造全局共形高斯坐标，该坐标光滑地且无退化地延伸到未来和过去的零无穷远。

动力学叶理上5D翘曲乘积空间的典型暴胀

Quintessential inflation from 5D warped product spaces on a dynamical foliation

假设在翘曲乘积空间类上存在一个5D纯动力学标量场，我们研究了在4D超曲面上模拟暴胀和典型场景的可能性，方法是在第五坐标上实现动力学叶理而不是常数叶理.我们得到了在四维动力学超曲面上，具有几何诱导标量势和诱导准真空状态方程的诱导混沌暴胀是可能的。虽然在一个恒定的叶理上，宇宙可以被认为是今天占主导地位的物质，但在一个4D动力学超曲面家族中，宇宙可以经历一段加速膨胀的时期，减速参数接近-1。这种影响的动力学叶理结果可以忽略不计，在暴胀时代允许一个混乱的情况下，并成为相当大的时代，允许一个典型的情况。

Confinement of test particles in warped spacetimes

We investigate test particle trajectories in warped spacetimes with a thick brane warp factor, a cosmological on--brane line element and a time dependent extra dimension. The geodesic equations are reduced to a first order autonomous dynamical system. Using analytical methods, we arrive at some useful general conclusions regarding possible trajectories. Oscillatory motion, suggesting confinement about the location of the thick brane, arises for a growing warp factor. On the other hand, we find runaway trajectories (exponential-like) for a decaying warp factor. Variations of the extra dimension…

The asymptotic behaviour of the Hawking energy along null asymptotically flat hypersurfaces

In this work we obtain the limit of the Hawking energy of a large class of foliations along general null hypersurfaces $Ω$ satisfying a weak notion of asymptotic flatness. The foliations are not required to be either geodesic or approaching large spheres at infinity. The limit is obtained in terms of a reference background geodesic foliation approaching large spheres and a positive function, constant along the null generators on $Ω$, which describes the relation between the two foliations at infinity. The integrand in the limit expression has interesting covariance and invariance properties wi…

翘曲乘积空间上5维纯动力学标量场形式的诱导暴胀

Induced inflation from a 5D purely kinetic scalar field formalism on warped product spaces

考虑到一个可分离的和纯动力学的5D标量场在一个扭曲的产品度规的背景下，我们提出了一个新的和更普遍的方法，诱导4D标量势的4D常数叶理的5D时空。我们得到一个真正的四维标量场的感应势，而不是一个有效的四维标量场的势。在这个形式中，我们可以恢复通常的4D暴胀形式与几何诱导的暴胀势。此外，暴胀子模的量子限制是自然地获得至少一类翘曲因子的模型。除了4D暴胀物理，这种形式的结果是独立的4D超曲面的选择。

The role of pseudo-hypersurfaces in non-holonomic motion

The geometry of hypersurfaces is generalized to pseudo-hypersurfaces, which are defined by Pfaff equations. The general methods are then applied to modeling the kinematics of motion constrained by a single linear, non-holonomic constraint. They are then applied to the example of a charge moving in an electromagnetic field, and the Lorentz equation of motion is shown to represent a geodesic that is constrained to lie in a pseudo-hypersurface that is defined by the potential 1-form.

Singularity theorems for warped products and the stability of spatial extra dimensions

New singularity theorems are derived for generic warped-product spacetimes of any dimension. The main purpose is to analyze the stability of (compact or large) extra dimensions against dynamical perturbations. To that end, the base of the warped product is assumed to be our visible 4-dimensional world, while the extra dimensions define the fibers, hence we consider "extra-dimensional evolution". Explicit conditions on the warping function that lead to geodesic incompleteness are given. These conditions can be appropriately rewritten, given a warping function, as restrictions on the intrinsic g…