Generalized Robertson Walker Research Articles

Some characterizations of Lorentzian manifolds

Generalized Robertson–Walker (GRW) spacetime is the generalization of the Robertson–Walker (RW) spacetime and a further generalization of GRW spacetime is the twisted spacetime. In this paper, we generalize the results of the paper [C. A. Mantica, Y. J. Suh and U. C. De, A note on generalized Robertson–Walker spacetimes, Int. J. Geom. Methods Mod. Phys. 13 (2016), Article Id: 1650079, 9 pp., doi: 101142/s0219887816500791 ]. We prove that a Ricci simple Lorentzian manifold with vanishing quasi-conformal curvature tensor is a RW spacetime. Further, we prove that a Ricci simple Lorentzian manifold with harmonic quasi-conformal curvature tensor is a GRW spacetime. As a consequence, we obtain several corollaries. Finally, we have cited some examples of the obtained results.

Cosmological perfect-fluids in f(R) gravity

We show that an [Formula: see text]-dimensional generalized Robertson–Walker (GRW) space-time with divergence-free conformal curvature tensor exhibits a perfect fluid stress–energy tensor for any [Formula: see text] gravity model. Furthermore, we prove that a conformally flat GRW space-time is still a perfect fluid in both [Formula: see text] and quadratic gravity where other curvature invariants are considered.

The stability of closed linear Weingarten spacelike hypersurfaces in foliated spacetimes

This paper discusses the stability concerning closed linear Weingarten spacelike hypersurfaces [Formula: see text] immersed in foliated spacetimes. Firstly, we show that a totally umbilic [Formula: see text] is strongly stable. Secondly, given a generalized Robertson–Walker spacetime [Formula: see text] whose warping function [Formula: see text] is under a suitable restriction, we prove that if [Formula: see text] is strongly stable, then it is totally umbilic.

$$\phi $$ ϕ -Parabolicity and the Uniqueness of Spacelike Hypersurfaces Immersed in a Spatially Weighted GRW Spacetime

In this paper, we extend a technique due to Romero et al. (Class Quantum Gravity 30:1–13, 2013; Int J Geom Methods Mod Phys 10:1360014, 2013; J Math Anal Appl 419:355–372, 2014) establishing sufficient conditions to guarantee the parabolicity of complete spacelike hypersurfaces immersed in a weighted generalized Robertson–Walker spacetime whose fiber has $$\phi $$ -parabolic universal Riemannian covering. As some applications of this criteria, we obtain uniqueness results concerning spacelike hypersurfaces immersed in spatially weighted generalized Robertson–Walker spacetimes. Furthermore, Calabi–Bernstein type results are also given.

Spacelike Hypersurfaces in Weighted Generalized Robertson-Walker Space-Times

Applying generalized maximum principle and weak maximum principle, we obtain several uniqueness results for spacelike hypersurfaces immersed in a weighted generalized Robertson-Walker (GRW) space-time under suitable geometric assumptions. Furthermore, we also study the special case when the ambient space is static and provide some results by using Bochner’s formula.

A New Calabi–Bernstein Type Result in Spatially Closed Generalized Robertson–Walker Spacetimes

The aim of this article is to study the uniqueness of a complete spacelike hypersurface $${\sum^{n}}$$ immersed with constant mean curvature H in a spatially closed generalized Robertson–Walker spacetime $${\overline{M}^{n+1} = -I {\times_{f}} {M^{n}}}$$ , whose Riemannian fiber $${M^n}$$ has positive curvature. Supposing that the warping function f is such that −log f is convex and $${H{f^{\prime}} \leq 0}$$ along $${\sum^{n}}$$ , we show that $${\sum^{n}}$$ must be isometric to a totally geodesic slice of $${\overline{M}^{n+1}}$$ . When $${\overline{M}^{n+1}}$$ is a Lorentzian product space, we obtain a new Calabi–Bernstein type result concerning the CMC spacelike hypersurface equation.

Uniform circular motion in general relativity: existence and extendibility of the trajectories

The concept of uniform circular motion in a general spacetime is introduced as a particular case of a planar motion. The initial value problem of the corresponding differential equation is analysed in detail. Geometrically, an observer that obeys a uniform circular motion is characterized as a Lorentzian helix. The completeness of inextensible trajectories is studied in generalized Robertson–Walker spacetimes and in a relevant family of pp-wave spacetimes. Under reasonable assumptions, the physical interpretation of such results is that a uniform circular observer lives forever, providing the absence of the singularities defined by these timelike curves.

On Einstein warped products with a quarter-symmetric connection

This paper characterizes the warping functions for a multiply generalized Robertson–Walker space-time to get an Einstein space [Formula: see text] with a quarter-symmetric connection for different dimensions of [Formula: see text] (i.e. (1). dim [Formula: see text] (2). dim [Formula: see text]) when all the fibers are Ricci flat. Then we have also computed the warping functions for a Ricci flat Einstein multiply warped product spaces M with a quarter-symmetric connection for different dimensions of [Formula: see text] (i.e. (1). dim [Formula: see text] (2). dim [Formula: see text] (3). dim [Formula: see text]) and all the fibers are Ricci flat. In the last section, we have given two examples of multiply generalized Robertson–Walker space-time with respect to quarter-symmetric connection.

Generalized Robertson–Walker spacetimes — A survey

Generalized Robertson–Walker spacetimes extend the notion of Robertson–Walker spacetimes, by allowing for spatial non-homogeneity. A survey is presented, with main focus on Chen's characterization in terms of a timelike concircular vector. Together with their most important properties, some new results are presented.

Marginally trapped submanifolds in generalized Robertson–Walker spacetimes

In this paper we consider codimension two marginally trapped submanifolds in the family of general Robertson–Walker spacetimes. In particular, we derive some rigidity results for this type of submanifolds which guarantee that, under appropriate hypothesis, the only ones are those contained in slices. We also derive some interesting non-existence results for weakly trapped submanifolds. In particular, we give applications to some cases of physical relevance such as the Einstein-de Sitter spacetime and certain open regions of de Sitter spacetime, including the so called steady state spacetime. Our results will be an application of the (finite) maximum principle for closed manifolds and, more generally, of the weak maximum principle for stochastically complete manifolds.

On uniqueness of the foliation by comoving observers restspaces of a Generalized Robertson–Walker spacetime

A characterization of the foliation by spacelike slices of an $(n+1)$-dimensional spatially closed Generalized Robertson-Walker spacetime is given by means of studying a natural mean curvature type equation on spacelike graphs. Under some natural assumptions, of physical or geometric nature, all the entire solutions of such an equation are obtained. In particular, the case of entire spacelike graphs in de Sitter spacetime is faced and completely solved by means of a new application of a known integral formula.

Rigidity of complete spacelike hypersurfaces in spatially weighted generalized Robertson–Walker spacetimes

Our purpose in this paper is to apply some maximum principles in order to study the rigidity of complete spacelike hypersurfaces immersed in a spatially weighted generalized Robertson–Walker (GRW) spacetime, which is supposed to obey the so called strong null convergence condition. Under natural constraints on the weight function and on the f-mean curvature, we establish sufficient conditions to guarantee that such a hypersurface must be a slice of the ambient spacetime. In this setting, we also obtain new Calabi–Bernstein type results concerning entire graphs in a spatially weighted GRW spacetime.

On the Weyl and Ricci tensors of Generalized Robertson-Walker space-times

We prove theorems about the Ricci and the Weyl tensors on Generalized Robertson-Walker space-times of dimension n ≥ 3. In particular, we show that the concircular vector introduced by Chen decomposes the Ricci tensor as a perfect fluid term plus a term linear in the contracted Weyl tensor. The Weyl tensor is harmonic if and only if it is annihilated by Chen’s vector, and any of the two conditions is necessary and sufficient for the Generalized Robertson-Walker (GRW) space-time to be a quasi-Einstein (perfect fluid) manifold. Finally, the general structure of the Riemann tensor for Robertson-Walker space-times is given, in terms of Chen’s vector. In n = 4, a GRW space-time with harmonic Weyl tensor is a Robertson-Walker space-time.

Uniqueness of Spacelike Hypersurfaces in a GRW Spacetime via Higher Order Mean Curvatures

We study the problem of uniqueness concerning complete spacelike hypersurfaces immersed in a generalized Robertson–Walker (GRW) spacetime, whose fiber obeys suitable curvature constraints. In this setting, we apply some maximum principles in order to guarantee that such a spacelike hypersurface must be a slice of the ambient space, provided that some of their higher order mean curvatures satisfies appropriated controls. Furthermore, we also establish nonparametric results concerning entire spacelike graphs in GRW spacetimes.

Generalized Robertson-Walker Space-Time Admitting Evolving Null Horizons Related to a Black Hole Event Horizon.

A new technique is used to study a family of time-dependent null horizons, called “Evolving Null Horizons” (ENHs), of generalized Robertson-Walker (GRW) space-time such that the metric satisfies a kinematic condition. This work is different from our early papers on the same issue where we used (1 + n)-splitting space-time but only some special subcases of GRW space-time have this formalism. Also, in contrast to previous work, we have proved that each member of ENHs is totally umbilical in . Finally, we show that there exists an ENH which is always a null horizon evolving into a black hole event horizon and suggest some open problems.

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.

A note on generalized Robertson–Walker space-times

A generalized Robertson–Walker (GRW) space-time is the generalization of the classical Robertson–Walker space-time. In the present paper, we show that a Ricci simple manifold with vanishing divergence of the conformal curvature tensor admits a proper concircular vector field and it is necessarily a GRW space-time. Further, we show that a stiff matter perfect fluid space-time or a mass-less scalar field with time-like gradient and with divergence-free Weyl tensor are GRW space-times.

Null hypersurfaces in generalized Robertson–Walker spacetimes

We study the geometry of null hypersurfaces M in generalized Robertson–Walker spacetimes. First we characterize such null hypersurfaces as graphs of generalized eikonal functions over the fiber and use this characterization to show that such hypersurfaces are parallel if and only if their fibers are also parallel. We further use this technique to construct several examples of null hypersurfaces in both de Sitter and anti de Sitter spaces. Then we characterize all the totally umbilical null hypersurfaces M in a Lorentzian space form (viewed as a quadric in a semi-Euclidean ambient space) as intersections of the space form with a hyperplane. Finally we study the totally umbilical spacelike hypersurfaces of null hypersurfaces in space forms and characterize them as planar sections of M .

Scalar curvature of spacelike hypersurfaces and certain class of cosmological models for accelerated expanding universes

We study the scalar curvature of spacelike hypersurfaces in the family of cosmological models known as generalized Robertson–Walker spacetimes, and give several rigidity results under appropriate mathematical and physical assumptions. On the other hand, we show that this family of spacetimes provides suitable models obeying the null convergence condition to explain accelerated expanding universes.

Complete spacelike hypersurfaces in generalized Robertson–Walker and the null convergence condition: Calabi–Bernstein problems

We study constant mean curvature spacelike hypersurfaces in generalized Robertson–Walker spacetimes $$\overline{M}= I \times _f F$$ which are spatially parabolic (i.e. its fiber F is a (non-compact) complete Riemannian parabolic manifold) and satisfy the null convergence condition. In particular, we provide several rigidity results under appropriate mathematical and physical assumptions. We pay special attention to the case where the GRW spacetime is Einstein. As an application, some Calabi–Bernstein type results are given.