Warped Product Spacetimes Research Articles

Doubly warped product manifolds with the structure of hyperbolic Ricci solitons

In this paper we study hyperbolic Ricci solitons on doubly warped product manifolds. The necessary conditions are obtained for a hyperbolic Ricci soliton with the structure of a doubly warped product to be an Einstein manifold when we consider the potential field as a Killing or a conformal vector field. Also, we examined the behavior of hyperbolic Ricci solitons on doubly warped product space-times.

Null distance and Gromov–Hausdorff convergence of warped product spacetimes

Null distance and Gromov–Hausdorff convergence of warped product spacetimes

Sequential warped product manifolds with a semi-symmetric metric connection

In the present paper, we study a new generalization of warped product manifolds, called sequential warped product manifolds, with respect to a semi-symmetric metric connection. We obtain relations for covariant derivatives, Riemannian curvature, Ricci curvature and scalar curvature of the sequential warped product manifolds with respect to the semi-symmetric connection, respectively, and demonstrate the relationship between them and curvatures with respect to the Levi-Civita connection. Also, we consider sequential warped product space-time models, namely sequential generalized Robertson- Walker space-times and sequential standard static space-times, with semi-symmetric metric connections and obtain conditions for such space-times to be Einstein.

On the dynamics of relativistic particles with torsion in warped product spacetimes

In this work, we study spacelike trajectories of a relativistic particle model whose Lagrangian is given by the total torsion of its worldline. Once the field equations are established, we investigate the trajectories in two physically relevant families of four-dimensional spacetimes: generalized Robertson–Walker and standard static spacetimes. For the former we obtain both characterization as well as non-existence results for Frenet curves of different orders, which are the curves capable of modeling trajectories. For standard static spacetimes, we obtain a full characterization of the critical curves and present an example in which we can explicitly compute non-trivial trajectories.

Ricci solitons on singly warped product manifolds and applications

The purpose of this article is to study implications of a Ricci soliton warped product manifold to its base and fiber manifolds. First, it is proved that if a warped product manifold is Ricci soliton then its factors are Ricci soliton. Then we study Ricci soliton on warped product manifolds admitting either a conformal vector field or a concurrent vector field. Finally, we study Ricci soliton on some warped product space-times.

Hyperbolicity and causality of Einstein-Gauss-Bonnet gravity in warped product spacetimes

In Einstein-Gauss-Bonnet gravity, for a group of warped product spacetimes, we get a generalized master equation for the perturbation of tensor type. We show that the "effective metric" or "acoustic metric" for the tensor perturbation equation can be defined even without a static condition. Since this master equation does not depend on the mode expansion, the hyperbolicity and causality of the tensor perturbation equation can be investigated for every mode of the perturbation. Based on the master equation, we study the hyperbolicity and causality for all relavent vacuum solutions of this theory. For each solution, we give the exact hyperbolic condition of the tensor perturbation equations. Our approach can also applied to dynamical spacetimes, and Vaidya spacetime have been investigated as an example.

Properties of the Null Distance and Spacetime Convergence

AbstractThe null distance for Lorentzian manifolds was recently introduced by Sormani and Vega. Under mild assumptions on the time function of the spacetime, the null distance gives rise to an intrinsic, conformally invariant metric that induces the manifold topology. We show when warped products of low regularity and globally hyperbolic spacetimes endowed with the null distance are (local) integral current spaces. This metric and integral current structure sets the stage for investigating convergence analogous to Riemannian geometry. Our main theorem is a general convergence result for warped product spacetimes relating uniform, Gromov–Hausdorff, and Sormani–Wenger intrinsic flat convergence of the corresponding null distances. In addition, we show that nonuniform convergence of warping functions in general leads to distinct limiting behavior, such as limits that disagree.

Gray’s decomposition on doubly warped product manifolds and applications

A. Gray presented an interesting O(n) invariant decomposition of the covariant derivative of the Ricci tensor. Manifolds whose Ricci tensor satisfies the defining property of each orthogonal class are called Einstein-like manifolds. In the present paper, we answered the following question: Under what condition(s), does a factor manifold Mi,i = 1,2 of a doubly warped product manifold M =f2 M1 x f1 M2 lie in the same Einstein- like class of M? By imposing sufficient and necessary conditions on the warping functions, an inheritance property of each class is proved. As an application, Einstein-like doubly warped product space-times of type A,B or P are considered.

Sequential warped products: Curvature and conformal vector fields

In this note, we introduce a new type of warped products called as sequential warped products to cover a wider variety of exact solutions to Einstein?s field equation. First, we study the geometry of sequential warped products and obtain covariant derivatives, curvature tensor, Ricci curvature and scalar curvature formulas. Then some important consequences of these formulas are also stated. We provide characterizations of geodesics and two different types of conformal vector fields, namely, Killing vector fields and concircular vector fields on sequential warped product manifolds. Finally, we consider the geometry of two classes of sequential warped product space-time models which are sequential generalized Robertson-Walker space-times and sequential standard static space-times.

Five-Dimensional Warped Product Space-Time with Time-Dependent Warping and a Scalar Field in the Bulk

We have considered gravity in a five-dimensional warped product space-time, with a time-dependent warp factor and a time-dependent extra dimension. The five-dimensional field equations are derived for a spatially flat FRW brane and the energy conditions and the nature of bulk geometry have been examined. It is found that the expansion of the four-dimensional universe depends on its location along the extra dimension and will be different at different locations in the bulk spacetime. At low energies, the trapping of fields within the brane implies a specific correlation between the warp factor and the extra-dimensional scale factor. Generally, the bulk is not conformally flat. At high energies, the bulk is assumed to be sourced by a scalar field with self-interaction. The analysis shows that the potential of the scalar field source of gravity at a given position along the fifth dimension, is related to the Hubble parameter on the brane at that position in the bulk.

Classification of Einstein equations with cosmological constant in warped product space-time

We classify all warped product space-times in three categories as (i) generalized twisted product structures, (ii) base conformal warped product structures and (iii) generalized static space-times and then we obtain the Einstein equations with the corresponding cosmological constant by which we can determine uniquely the warp functions in these warped product space-times.

Warped product space-times

Many classical results in relativity theory concerning spherically symmetric space-times have easy generalizations to warped product space-times, with a two-dimensional Lorentzian base and arbitrary dimensional Riemannian fibers. We first give a systematic presentation of the main geometric constructions, with emphasis on the Kodama vector field and the Hawking energy; the construction is signature independent. This leads to proofs of general Birkhoff-type theorems for warped product manifolds; our theorems in particular apply to situations where the warped product manifold is not necessarily Einstein, and thus can be applied to solutions with matter content in general relativity. Next we specialize to the Lorentzian case and study the propagation of null expansions under the assumption of the dominant energy condition. We prove several non-existence results relating to the Yamabe class of the fibers, in the spirit of the black-hole topology theorem of Hawking–Galloway–Schoen. Finally we discuss the effect of the warped product ansatz on matter models. In particular we construct several cosmological solutions to the Einstein–Euler equations whose spatial geometry is generally not isotropic.

The W2-curvature tensor on warped product manifolds and applications

The purpose of this paper is to study the [Formula: see text]-curvature tensor on (singly) warped product manifolds as well as on generalized Robertson–Walker and standard static space-times. Some different expressions of the [Formula: see text]-curvature tensor on a warped product manifold in terms of its relation with [Formula: see text]-curvature tensor on the base and fiber manifolds are obtained. Furthermore, we investigate [Formula: see text]-curvature flat warped product manifolds. Many interesting results describing the geometry of the base and fiber manifolds of a [Formula: see text]-curvature flat warped product manifold are derived. Finally, we study the [Formula: see text]-curvature tensor on generalized Robertson–Walker and standard static space-times; we explore the geometry of the fiber of these warped product space-time models that are [Formula: see text]-curvature flat.

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. Introducing the concept of an energy function along the phase path in terms of the extra-dimensional coordinate, we have examined how the energy function depends on the warp factor. The energy function serves as a measure of the amount of perturbation of geodesic paths along the extra dimension in the region close to the brane. Then we studied the geodesic motion under a conventional metric perturbation in the form of homothetic motion and conformal motion and examined the nature of critical points for a Mashhoon-Wesson-type metric, for timelike and null geodesics when the cross-diagonal term of the Einstein tensor vanishes. Finally we investigated the motion for null and timelike geodesics under the condition when the cross-diagonal element of the Einstein tensor is non-vanishing and examined the effects of perturbation on the critical points of the dynamical system.

Conformal Vector Fields on Doubly Warped Product Manifolds and Applications

This article aimed to study and explore conformal vector fields on doubly warped product manifolds as well as on doubly warped spacetime. Then we derive sufficient conditions for matter and Ricci collineations on doubly warped product manifolds. A special attention is paid to concurrent vector fields. Finally, Ricci solitons on doubly warped product spacetime admitting conformal vector fields are considered.

2-Killing vector fields on warped product manifolds

This paper provides a study of 2-Killing vector fields on warped product manifolds as well as characterization of this structure on standard static and generalized Robertson–Walker space-times. Some conditions for a 2-Killing vector field on a warped product manifold to be parallel are obtained. Moreover, some results on the curvature of a warped product manifolds in terms of 2-Killing vector fields are derived. Finally, we apply our results to describe 2-Killing vector fields of some well-known warped product space-time models.

Gravity in five-dimensional warped product spacetimes with time-dependent warp factor

We have considered gravity in a 5-dimensional warped product spacetime with a time-dependent warp factor and time-dependent extra dimension. The braneworld has been described by a spatially flat FRW-type metric. The five-dimensional field equations have been constructed and solved, and the status of the energy conditions have been examined. In the high energy regime, the bulk is assumed to be sourced by a scalar field. The effective cosmological constant of the 4-dimensional universe is a variable quantity monitored by the time-dependent warp factor, and leads to a geometric interpretation of dynamical dark energy.

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 effective cosmological constant of the four-dimensional universe is a variable quantity monitored by the time-dependent warp factor. The universe is initially decelerated, but subsequently makes a transition to an accelerated phase at later times.

Brane cosmology and motion of test particles in five-dimensional warped product spacetimes

In the braneworld scenario ordinary standard model matter and on-gravitational fields are confined by some trapping mechanism to the 4-dimensional universe constituting the D3-branes which are embedded in a (4 + n)-dimensional manifold referred to as the bulk (n being the number of extra dimensions). The notion of particle confinement is necessary for theories with non-compact extra dimensions, otherwise, the particles would escape from our 4-dimensional world along unseen directions. In this paper, we have considered a five-dimensional warped product space-time having an exponential warping function which depends both on time as well as on the extra coordinates and a non-compact fifth dimension. Further, we have assumed that the lapse function may either be a constant or a function of both time and of the extra coordinates, and have studied the nature of the geodesics of test particles and analyzed the conditions of stability in this geometrical framework. We have also discussed the possible cosmology of the corresponding (3 + 1)-dimensional hypersurfaces.

Inverse approach to Einstein’s equations for fluids with vanishing anisotropic stress tensor

We expand previous work on an inverse approach to Einstein Field Equations where we include fluids with energy flux and consider the vanishing of the anisotropic stress tensor. We consider the approach using warped product spacetimes of class $B_1$. Although restricted, these spacetimes include many exact solutions of interest to compact object studies and to cosmological models studies. The question explored here is as follows: given a spacetime metric, what fluid flow (timelike congruence), if any, could generate the spacetime via Einstein's equations. We calculate the flow from the condition of a vanishing anisotropic stress tensor and give results in terms of the metric functions in the three canonical types of coordinates. A condition for perfect fluid sources is also provided. The framework developed is algorithmic and suited for the study and validation of exact solutions using computer algebra systems. The framework can be applied to solutions in comoving and non-comoving frames of reference, and examples in different types of coordinates are worked out.