Generalized Robertson Walker Spacetimes Research Articles

$K$-Ricci-Bourguignon Almost Solitons

We in this current article introduce and characterize a $K$-Ricci-Bourguignon almost solitons in perfect fluid spacetimes and generalized Robertson-Walker spacetimes. First, we demonstrate that if a perfect fluid spacetime admits a $K$-Ricci-Bourguignon almost soliton, then the integral curves produced by the velocity vector field are geodesics and the acceleration vector vanishes. Then we establish that if perfect fluid spacetimes permit a gradient $K$-Ricci-Bourguignon soliton with Killing velocity vector field, then either state equation of the perfect fluid spacetime is presented by $p=\frac{3-n}{n-1}\sigma$ , or the gradient $K$-Ricci-Bourguignon soliton is shrinking or expanding under some condition. Moreover, we illustrate that the spacetime represents a perfect fluid spacetime and the divergence of the Weyl tensor vanishes if a generalized Robertson-Walker spacetime admits a $K$-Ricci-Bourguignon almost soliton.

On some basic curvature invariants of screen homothetic lightlike hypersurfaces in a GRW spacetime

This study is focused on the investigation of lightlike hypersurfaces of a generalized Robertson-Walker (GRW) spacetime. Recently, Poyraz (2022) [51,52] established some basic inequalities involving various curvature invariants of screen homothetic lightlike hypersurfaces of GRW spacetimes, like k-scalar curvature and k-Ricci curvature. In this work, we consider other basic curvature invariants, namely the scalar curvature and δ-Casorati curvatures, and derive new inequalities for such hypersurfaces of a GRW spacetime. We also find the conditions for which the equality cases in these inequalities hold and give some applications in Lorentzian geometry.

Conharmonically flat and conharmonically symmetric warped product manifolds

This article presents characterizations of warped product manifolds based on the flatness and symmetry of the conharmonic curvature tensor. It is proved that when a warped product manifold is conharmonically flat, both the base and fiber manifolds exhibit constant sectional curvatures. In addition, the specific forms of the conharmonic curvature tensor are derived for both the base and fiber manifolds. It is demonstrated that in a conharmonically symmetric warped product manifold, the fiber manifold has a constant sectional curvature, while the base manifold is both Cartan-symmetric and conharmonically symmetric. In this scenario, the form of the conharmonic curvature tensor on the fiber manifold is determined. We characterize the generalized Robertson–Walker (GRW) space-time through the flatness and the symmetry of the conharmonic curvature tensor. It is shown that a conharmonically flat (symmetric) GRW space-time is a perfect fluid. Finally, a conharmonically flat standard static space-time is investigated.

Characterizations of η–ρ-Einstein solitons in spacetimes and f(ℛ)-gravity

A generalized Robertson–Walker spacetime is not, in general, a perfect fluid spacetime and the converse is not, in general, true. In this paper, we show that if a perfect fluid spacetime admits an [Formula: see text]–[Formula: see text]-Einstein soliton, then the integral curves generated by the velocity vector field [Formula: see text] are geodesics and the acceleration vector vanishes. Also, we show that if a perfect fluid spacetime with Killing velocity vector field admits a gradient [Formula: see text]–[Formula: see text]-Einstein soliton, then the spacetime represents either dark matter era or, the acceleration vector vanishes. Next, we show that if a generalized Robertson–Walker spacetime admits an [Formula: see text]–[Formula: see text]-Einstein soliton, then it becomes a perfect fluid spacetime. Finally, we prove that in a dark matter fluid with vanishing vorticity satisfying gradient [Formula: see text]–[Formula: see text]-Einstein soliton in [Formula: see text]-gravity with constant Ricci scalar, either the energy density and isotropic pressure are constant or, the potential function [Formula: see text] remains invariant under the velocity vector [Formula: see text]. Also, for two models [Formula: see text] and [Formula: see text] ([Formula: see text] = constant and [Formula: see text] is the Ricci scalar of the spacetime), various energy conditions in terms of the Ricci scalar are examined and state that the Universe is in an accelerating phase and satisfies the weak, null and dominant energy conditions, but violate the strong energy condition.

Characterizations of spacetimes admitting critical point equation and f(r)-gravity

In general, a perfect fluid spacetime is not a generalized Robertson–Walker spacetime and the converse is also not true. In this paper, it is shown that if a perfect fluid spacetime satisfies the critical point equation, then either the spacetime becomes a generalized Robertson–Walker spacetime and represents dark era or the vorticity of the fluid vanishes as well as the spacetime is expansion free. Besides, we prove that if a generalized Robertson–Walker spacetime with constant scalar curvature satisfies the critical point equation, then the spacetime becomes a perfect fluid spacetime. Next, the existence of critical point equation is established by a non-trivial example. Finally, we discuss the critical point equation in [Formula: see text]-gravity. For the model [Formula: see text] ([Formula: see text] = constant and [Formula: see text] is the scalar curvature of the spacetime), various energy conditions in terms of the scalar curvature are examined and state that the Universe is in an accelerating phase and satisfies the weak, null, dominant, and strong energy conditions.

Killing and 2-Killing Vector Fields on Doubly Warped Products

We provide a condition for a 2-Killing vector field on a compact Riemannian manifold to be Killing and apply the result to doubly warped product manifolds. We establish a connection between the property of a vector field on a doubly warped product manifold and its components on the factor manifolds to be Killing or 2-Killing. We also prove that a Killing vector field on the doubly warped product gives rise to a Ricci soliton factor manifold if and only if it is an Einstein manifold. If a component of a Killing vector field on the doubly warped product is of a gradient type, then, under certain conditions, the corresponding factor manifold is isometric to the Euclidean space. Moreover, we provide necessary and sufficient conditions for a doubly warped product to reduce to a direct product. As applications, we characterize the 2-Killing vector fields on the doubly warped spacetimes, particularly on the standard static spacetime and on the generalized Robertson–Walker spacetime.

Lorentzian manifolds: A characterization with a type of semi-symmetric non-metric connection

This paper characterizes the Lorentzian manifolds endowed with a semi-symmetric non-metric [Formula: see text]-connection (briefly, ssnmρc). First, the existence of semi-symmetric non-metric connection (ssnmc) on Lorentzian manifold is established, and it is shown that an [Formula: see text]-dimensional Lorentzian manifold equipped with an ssnmρc is a generalized Robertson–Walker spacetime. We also establish the condition for a Lorentzian manifold together with an ssnmρc to be a Robertson–Walker spacetime. In this series, the properties of Ricci semi-symmetric Lorentzian manifold endowed with an ssnmρc, almost Ricci solitons and gradient Ricci solitons are explored. Finally, a non-trivial example of Lorentzian manifold admits an ssnmρc, which is constructed to verify some of our results.

On the geometry and dynamics of relativistic particles in generalized Robertson–Walker spacetimes

In this work, we consider geometrical models describing spinning particles in the context of generalized Robertson–Walker spacetimes. We study spacelike and timelike trajectories for massive and massless particles governed by two different Lagrangians: one depending on the torsion and the other on the curvature of the trajectory. For both, we are able to relate the curvature and torsion of such trajectories with the curvature of the spatial fiber of the cosmological model. Moreover, several conserved quantities are described as well as their possible relations with physical parameters of the particle.

Characterizations of GRW spacetimes admitting Ricci–Yamabe solitons

In this paper, we investigate Ricci–Yamabe solitons (RYSs) and gradient Ricci–Yamabe solitons (gradient RYSs) in generalized Robertson–Walker (GRW) spacetimes. At first, we prove that if a GRW spacetime admits a RYS, then it becomes a perfect fluid spacetime (PFS) and the divergence of the Weyl tensor vanishes. Also, a GRW spacetime admitting a RYS is of Petrov type [Formula: see text], [Formula: see text] or [Formula: see text] and in case of four dimension, the spacetime turns into a Robertson–Walker spacetime. Next, we show that if a GRW spacetime of constant scalar curvature admits a gradient RYS, then it becomes a PFS and the divergence of the Weyl tensor vanishes.

Curvature properties of spacelike hypersurfaces in a RW spacetime

Curvature invariants of both intrinsic and extrinsic nature play a significant role in elucidating the geometry of a spacetime. In particular, these invariants are useful in detecting event horizon of black holes. Notable examples of spacetimes are provided by the generalized Robertson-Walker (GRW) models. An (m+1)-dimensional GRW spacetime is a Lorentzian warped product I×fN endowed with the warped metric gˆ=−dt2+f2(t)g, having as base an open real interval I equipped with the metric −dt2 and as fiber any Riemannian space (N,g) of dimension m, where f is a smooth positive-valued function on I. In this article, we focus our study on the GRW spacetimes having the fiber a Riemannian space with metric g=gk of constant sectional curvature k and denoted by L1m+1(k,f), i.e. RW spacetimes. Using an approach originally developed in Decu et al. (2008) [46], we obtain in this work the lower bound of the (generalized) normalized Casorati curvatures for a spacelike hypersurface H in the GRW spacetime L1m+1(k,f) in terms of the (normalized) scalar curvature of H, the constant sectional curvature k of the fiber, the normal hyperbolic angle of the hypersurface and the warping function f. We also derive the conditions under which this lower bound is reached. We finally apply the results to some basic cosmological models, namely Lorentz-Minkowski, de Sitter, Einstein-de Sitter, anti de Sitter and steady state spacetimes.

ON THE GEOMETRY OF SPACELIKE MEAN CURVATURE FLOW SOLITONS IMMERSED IN A GRW SPACETIME

Abstract We investigate geometric aspects of complete spacelike mean curvature flow solitons of codimension one in a generalized Robertson–Walker (GRW) spacetime $-I\times _{f}M^n$ , with base $I\subset \mathbb R$ , Riemannian fiber $M^n$ and warping function $f\in C^\infty (I)$ . For this, we apply suitable maximum principles to guarantee that such a mean curvature flow soliton is a slice of the ambient space and to obtain nonexistence results concerning these solitons. In particular, we deal with entire graphs constructed over the Riemannian fiber $M^n$ , which are spacelike mean curvature flow solitons, and we also explore the geometry of a conformal vector field to establish topological and further rigidity results for compact (without boundary) mean curvature flow solitons in a GRW spacetime. Moreover, we study the stability of spacelike mean curvature flow solitons with respect to an appropriate stability operator. Standard examples of spacelike mean curvature flow solitons in GRW spacetimes are exhibited, and applications related to these examples are given.

Almost conformal Ricci solitons on generalized Robertson–Walker space-times

Almost conformal Ricci solitons on generalized Robertson–Walker space-times

Prescribed mean curvature flow of non-compact space-like Cauchy hypersurfaces

In this paper we consider the prescribed mean curvature flow of a non-compact space-like Cauchy hypersurface of bounded geometry in a generalized Robertson–Walker space-time. We prove that the flow preserves the space-likeness condition and exists for infinite time. We also prove convergence in the setting of manifolds with boundary. Our discussion generalizes previous work by Ecker, Huisken, Gerhardt and others with respect to a crucial aspects: we consider any non-compact Cauchy hypersurface under the assumption of bounded geometry. Moreover, we specialize the aforementioned works by considering globally hyperbolic Lorentzian space-times equipped with a specific class of warped product metrics.

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.