Generalized Robertson Walker Spacetimes Research Articles

On complete trapped submanifolds in globally hyperbolic spacetimes

The aim of this manuscript is to obtain rigidity and non-existence results for parabolic spacelike submanifolds with causal mean curvature vector field in orthogonally splitted spacetimes, and in particular, in globally hyperbolic spacetimes. We also obtain results regarding the geometry of submanifolds by ensuring, under some mild hypothesis, the non-existence of local minima or maxima of certain distinguished function. Furthermore, in this last case the submanifold does not need to be parabolic or even complete. As an application in General Relativity, we obtain several nice results regarding (non-necessarily closed) trapped surfaces in a huge family of spacetimes. In fact, we show how our technique allows us to recover some relevant previous results for trapped surfaces in both, standard static spacetimes and generalized Robertson–Walker spacetimes.

On sequential warped product manifolds admitting gradient Ricci-harmonic solitons

We consider sequential warped product manifolds admitting gradient Ricci-harmonic solitons. We give the main relations for a gradient Ricci-harmonic soliton on sequential warped product manifolds. As physical applications, we consider gradient Ricci-harmonic solitons on sequential generalized Robertson-Walker space-times and sequential standard static space-times.

On Calabi–Bernstein’s Problem for Uniqueness of Complete Spacelike Hypersurfaces in Weighted Generalized Robertson–Walker Spacetimes

In this paper, we study Calabi–Bernstein’s problem for the uniqueness of complete constant weighted mean curvature spacelike hypersurfaces Σn in weighted generalized Robertson–Walker spacetimes −I×ρMfn. Under appropriate geometric assumptions, we confirm that Σn is a spacelike slice.

Relativistic spacetimes admitting almost Schouten solitons

In this paper, we investigate almost Schouten solitons and almost gradient Schouten solitons in spacetimes of general relativity. At first, it is proven that if a generalized Robertson–Walker spacetime permits an almost Schouten soliton, then it becomes a perfect fluid spacetime as well as the spacetime represents a dark matter era. Besides this, we investigate almost gradient Schouten solitons in generalized Robertson–Walker spacetimes. Moreover, a spacetime obeying almost Schouten solitons whose potential vector field is a non-homothetic conformal vector field is of Petrov type [Formula: see text] or [Formula: see text].

Some physical properties of generalized quasi-Einstein spacetimes under Gray’s decomposition

In this study, we analyze generalized quasi-Einstein spacetimes endowed with Gray’s decomposition, as well as generalized Robertson–Walker spacetimes. It is shown that the Ricci tensor of a generalized quasi-Einstein spacetime assumes the form of a perfect fluid in all Gray’s subspaces under certain restrictions. Finally, it is established that a generalized quasi-Einstein generalized Robertson–Walker spacetime is a perfect fluid spacetime.

Characterizations of -flat curvature tensor on spacetimes and -gravity

The prime goal of this article is to characterize spacetimes endowed with -flat curvature tensor. It is demonstrated that a 4-dimensional -flat perfect fluid spacetime is either a de-Sitter spacetime or locally isometric to Minkowski spacetime under certain energy condition. Moreover, it is established that a Ricci simple -flat spacetime becomes a Robertson-Walker spacetime. In addition, we shown that a Ricci simple spacetime with harmonic -curvature tensor is a generalized Robertson-Walker spacetime. Also, we explain several conclusions based on -flat generalized Robertson-Walker spacetimes. Finally, we investigate -flat spacetimes that satisfy the -gravity and -gravity.

Hyperbolic Ricci soliton on warped product manifolds

In this paper, we investigate hyperbolic Ricci soliton as the special solution of hyperbolic geometric flow on warped product manifolds. Then, especially, we study these manifolds admitting either a conformal vector field or a concurrent vector field. Also, the question that:? whether or not a hyperbolic soliton reduces to an Einstein manifold?? is considered and answered. Finally, we obtain some necessary conditions for generalized Robertson-Walker space-time to be a hyperbolic Ricci soliton.

Einstein warped product spaces on Lie groups

We consider a compact Lie group with bi-invariant metric, coming from the Killing form. In this paper, we study Einstein warped product space, $M = M_1 \times_{f_1} M_2$ for the cases, $(i)$ $M_1$ is a Lie group $(ii)$ $M_2$ is a Lie group and $(iii)$ both $M_1$ and $M_2$ are Lie groups. Moreover, we obtain the conditions for an Einstein warped product of Lie groups to become a simple product manifold. Then, we characterize the warping function for generalized Robertson-Walker spacetime, $(M = I \times_{f_1} G_2, - dt^2 + f_1^2 g_2)$ whose fiber $G_2$, being semi-simple compact Lie group of $\dim G_2>2$, having bi-invariant metric, coming from the Killing form.

A new approach on Einstein equation in super spacetime

In this paper, we introduce a new kind of super warped product spaces [Formula: see text], [Formula: see text], and [Formula: see text], where [Formula: see text] is a supermanifold of dimension [Formula: see text], [Formula: see text] is a superdomain with [Formula: see text] and [Formula: see text], subject to the warp functions [Formula: see text], [Formula: see text], and [Formula: see text], respectively. In each super warped product space, [Formula: see text], [Formula: see text], and [Formula: see text], it is shown that Einstein equations [Formula: see text], with cosmological term [Formula: see text] are reducible to the Einstein equations [Formula: see text] on the super space [Formula: see text] with cosmological term [Formula: see text], where [Formula: see text] and [Formula: see text] are functions of f(t), [Formula: see text], and [Formula: see text], as well as (m, n). This dependence points to the origin of cosmological terms which turn out to be within the warped structure of the super spacetime. By using the generalized Robertson–Walker spacetime, as a super spacetime, and demanding for constancy of [Formula: see text], we can determine the warp functions and [Formula: see text] which result in finding the solutions for Einstein equations [Formula: see text] and [Formula: see text]. We have discussed the cosmological solutions, for each kind of super warped product space, in the special case of [Formula: see text].

Characterizations of mixed quasi-Einstein spacetimes under Gray’s decomposition

In this study, we analyze mixed quasi-Einstein spacetimes endowed with Gray’s decomposition, as well as generalized Robertson–Walker spacetimes. For mixed quasi-Einstein spacetimes, the shape of the Ricci tensor in all [Formula: see text]-invariant subspaces can be identified using Gray’s decomposition of the gradient of the Ricci tensor. In case one, such a spacetime is found to be static; in three cases, the Ricci tensor is found to be in the form of a  perfect fluid; and in the other three situations, the spacetime becomes a quasi-Einstein spacetime under certain restrictions on the associated vector fields. Finally, it is established that a mixed quasi-Einstein generalized  Robertson–Walker spacetime is a perfect fluid spacetime.

Complete Stationary Spacelike Surfaces in an n-Dimensional Generalized Robertson–Walker Spacetime

Several uniqueness results for non-compact complete stationary spacelike surfaces in an \(n(\ge 3)\)-dimensional Generalized Robertson Walker spacetime are obtained. In order to do that, we assume a natural inequality involving the Gauss curvature of the surface, the restrictions of the warping function and the sectional curvature of the fiber to the surface. This inequality gives the parabolicity of the surface. Using this property, a distinguished non-negative superharmonic function on the surface is shown to be constant, which implies that the stationary spacelike surface must be totally geodesic. Moreover, non-trivial examples of stationary spacelike surfaces in the four dimensional Lorentz–Minkowski spacetime are exposed to show that each of our assumptions is needed.

Mean curvature flow of graphs in generalized Robertson–Walker spacetimes with perpendicular Neumann boundary condition

We prove the longtime existence for the mean curvature flow problem with a perpendicular Neumann boundary condition in a generalized Robertson–Walker (GRW) spacetime that obeys the null convergence condition. In addition, we prove that the metric of such a solution is conformal to the one of the leaf of the GRW in asymptotic time. Furthermore, if the initial hypersurface is mean convex, then the evolving hypersurfaces remain mean convex during the flow.

Causal completions as Lorentzian pre-length spaces

In this work we revisit the notion of the (future) causal completion of a globally hyperbolic spacetime and endow it with the structure of a Lorentzian pre-length space. We further carry out this construction for a certain class of generalized Robertson-Walker spacetimes.

Generalized quasi-Einstein metrics and applications on generalized Robertson–Walker spacetimes

In this paper, we study generalized quasi-Einstein manifolds (Mn, g, V, λ) satisfying certain geometric conditions on its potential vector field V whenever it is harmonic, conformal, and parallel. First, we construct some integral formulas and obtain some triviality results. Then, we find some necessary conditions to construct a quasi-Einstein structure on (Mn, g, V, λ). Moreover, we prove that for any generalized Ricci soliton (M̄=I×fM,ḡ,ξ̄,λ), where ḡ is a generalized Robertson–Walker spacetime metric and the potential field ξ̄=h∂t+ξ is conformal, (M̄,ḡ) can be considered as the model of perfect fluids in general relativity. Moreover, the fiber (M, g) also satisfies the quasi-Einstein metric condition. Therefore, the state equation of (M̄=I×fM,ḡ) is presented. We also construct some explicit examples of generalized quasi-Einstein metrics by using a four-dimensional Walker metric.

Gradient solitons on twisted product manifolds and their applications in general relativity

In this paper, first, we find the necessary and sufficient condition for a Riemannian manifold to be the locally warped product. Then we investigate the existence of different types of gradient solitons, such as gradient (almost) Yamabe soliton, conformal soliton and gradient Ricci soliton on the twisted product manifolds. We also study the concircular flatness condition on a twisted product and examine the Einstein-type relations on its base and fiber manifold. Moreover, we introduce the notions of twisted generalized Robertson–Walker spacetime and twisted standard static spacetime. We get an ordinary differential equation (ODE) that determines the twisting function of the former and the exact form of the twisting function for the latter one.

Perfect Fluid Spacetimes and Gradient Solitons

In this article, we investigate perfect fluid spacetimes equipped with concircular vector field. At first, in a perfect fluid spacetime admitting concircular vector field, we prove that the velocity vector field annihilates the conformal curvature tensor. In addition, in dimension 4, we show that a perfect fluid spacetime is a generalized Robertson–Walker spacetime with Einstein fibre. It is proved that if a perfect fluid spacetime furnished with concircular vector field admits a second order symmetric parallel tensor P, then either the equation of state of the perfect fluid spacetime is characterized by p=frac{3-n}{n-1} sigma , or the tensor P is a constant multiple of the metric tensor. Finally, The perfect fluid spacetimes with concircular vector field whose Lorentzian metrics are Ricci soliton, gradient Ricci soliton, gradient Yamabe solitons, and gradient m -quasi Einstein solitons, are characterized.

Characterizations of Lorentzian manifolds

The focus of this paper is to characterize the Lorentzian manifolds equipped with a semi-symmetric non-metric ρ-connection [briefly, (M,∇̃)]. The conditions for a Lorentzian manifold to be a generalized Robertson–Walker spacetime are established and vice versa. We prove that an n-dimensional compact (M,∇̃) is geodesically complete. We also study the properties of almost Ricci solitons and gradient almost Ricci solitons on Lorentzian manifolds and Yang pure space, respectively. Finally, we study the properties of semisymmetric (M,∇̃), and it is proven that (M,∇̃) is semisymmetric if and only if it is a Robertson–Walker spacetime.

A Note on Generalized Quasi-Einstein and (λ, n + m)-Einstein Manifolds with Harmonic Conformal Tensor

Sufficient conditions for a Lorentzian generalized quasi-Einstein manifold M,g,f,μ to be a generalized Robertson–Walker spacetime with Einstein fibers are derived. The Ricci tensor in this case gains the perfect fluid form. Likewise, it is proven that a λ,n+m-Einstein manifold M,g,w having harmonic Weyl tensor, ∇jw∇mwCjklm=0 and ∇lw∇lw<0 reduces to a perfect fluid generalized Robertson–Walker spacetime with Einstein fibers. Finally, M,g,w reduces to a perfect fluid manifold if φ=−m∇lnw is a φRic-vector field on M and to an Einstein manifold if ψ=∇w is a ψRic-vector field on M. Some consequences of these results are considered.

Torqued vector fields on generalized Ricci solitons and Lorentzian twisted products

In this study, some relations between the generalized Ricci solitons with generalized quasi-Einstein manifolds and twisted products are established. Then, an explicit example of generalized quasi-Einstein spacetime endowed with the spatially homogeneous and anisotropic Bianchi type-V metric is constructed. Also, we get certain identities about the Riemannian and the Ricci tensors on a Lorentzian twisted product [Formula: see text] admitting a timelike torqued vector field. We proved that such spacetime admitting a timelike torqued vector field having a closed torqued form is a generalized Robertson–Walker spacetime. Therefore, from Mantica and Molinari’s classifications, this spacetime becomes a model of perfect fluids. As a physical application, it is shown that the magnetic parts of the Weyl tensor of such spacetime vanish and so its possible Petrov types are [Formula: see text] or [Formula: see text].

Imperfect Fluid Generalized Robertson Walker Spacetime Admitting Ricci-Yamabe Metric

In the present paper, we investigate the nature of Ricci-Yamabe soliton on an imperfect fluid generalized Robertson-Walker spacetime with a torse-forming vector fieldξ. Furthermore, if the potential vector fieldξof the Ricci-Yamabe soliton is of the gradient type, the Laplace-Poisson equation is derived. Also, we explore the harmonic aspects ofη-Ricci-Yamabe soliton on an imperfect fluidGRWspacetime with a harmonic potential functionψ. Finally, we examine necessary and sufficient conditions for a1-formη, which is theg-dual of the vector fieldξon imperfect fluidGRWspacetime to be a solution of the Schrödinger-Ricci equation.