Robertson-Walker Spacetimes Research Articles

Multiple positive radial solutions for Dirichlet problem of the prescribed mean curvature spacelike equation in a Friedmann–Lemaître–Robertson–Walker spacetime

In this paper, we consider the radially symmetric spacelike solutions of a nonlinear Dirichlet problem for the prescribed mean curvature spacelike equation in a Friedmann–Lemaître–Robertson–Walker spacetime. By using a conformal change of variable, this problem can be translated an equivalent problem in the Minkowski spacetime. By using the lower and upper solution method, fixed point, a priori bounds and topological degree method, we obtain the existence, nonexistence and multiplicity of radially symmetric spacelike solutions.

On gradient normalized Ricci-harmonic solitons in sequential warped products

<p>Our investigation involved sequentially warped product manifolds that contained gradient-normalized Ricci-harmonic solitons. We presented the primary connections for a gradient-normalized Ricci-harmonic soliton on sequential warped product manifolds. In practical applications, our research investigated gradient-normalized Ricci-harmonic solitons for sequential generalized Robertson-Walker spacetimes and sequential standard static space-times. Our finding generalized all results proven in <sup>[<xref ref-type="bibr" rid="b26">26</xref>]</sup>.</p>

Geometric phases of nonlinear elastic [formula omitted]-rotors via Cartan’s moving frames

We study the geometric phases of nonlinear elastic N-rotors with continuous rotational symmetry. In the Hamiltonian framework, the geometric structure of the phase space is a principal fiber bundle, i.e., a base, or shape manifold B, and fibers F along the symmetry direction attached to it. The symplectic structure of the Hamiltonian dynamics determines the connection and curvature forms of the shape manifold. Using Cartan’s structural equations with zero torsion we find an intrinsic (pseudo) Riemannian metric for the shape manifold. Without lose of generality, we show that one has the freedom to define the rotation sign of the total angular momentum A of the elastic rotors as either positive or negative, e.g., counterclockwise or clockwise, respectively, or viceversa. This endows the base manifold B with two distinct metrics both compatible with the geometric phase. In particular, the metric is pseudo-Riemannian if A<0, and the shape manifold is a 2D Robertson-Walker spacetime with positive curvature. For A>0, the shape manifold is the hyperbolic plane H2 with negative curvature. We then generalize our results to free elastic N-rotors. We show that the associated shape manifold B is reducible to the product manifold of N−1 hyperbolic planes H2 (A>0), or 2D Robertson-Walker spacetimes (A<0) depending on the convection used to define the rotation sign of the total angular momentum. We then consider elastic N-rotors subject to time-dependent self-equilibrated moments. The N-dimensional shape manifold of the extended autonomous system has a structure similar to that of the (N−1)-dimensional shape manifold of free elastic rotors. The Riemannian structure of the shape manifold provides an intrinsic measure of the closeness of one shape to another in terms of curvature, or induced geometric phase.

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.

Conformally flat generalized Ricci recurrent spacetimes and curvature-squared gravity

First, we illustrate that a conformally flat generalized Ricci recurrent spacetime is a perfect fluid spacetime. As a consequence, we prove that such a spacetime represents a dust matter fluid, stiff matter and dark energy era under certain restrictions on the Ricci scalar. Then we establish that such a spacetime is a Robertson–Walker spacetime. Also, we note that in a conformally flat generalized Ricci recurrent spacetime, the fluid is shear-free, vorticity-free and its flow vector is hypersurface orthogonal. We investigate a conformally flat generalized Ricci recurrent spacetime as a solution of curvature-squared gravity theory. In this study, 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, dominant, and strong energy conditions.

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.

Geometry of Twisted Products and Applications on Static Perfect Fluid Spacetimes

In this paper, first we study the harmonicity of the functions and forms on the twisted products, and then we determine its sectional curvature. We explore some characteristics of static perfect fluid and static vacuum spacetimes on twisted product manifolds by proving the existence and obstructions on Ricci curvature. Finally, we study the problem of the existence static perfect fluid spacetime associated with the twisted generalized Robertson-Walker and standard static spacetime metrics.

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.

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.

Impact of m-projective curvature tensor in f (, G)-gravity and f (,  m )-gravity

This article deals with the characterization of a spacetime and modified gravity, such as -gravity and -gravity endowed with the m-projective curvature tensor. At first, we show that a m-projectively flat perfect fluid spacetime satisfying Einstein’s field equations represents a dark energy era. Also, we prove that such a spacetime obeying the strong energy condition is locally isometric to Minkowski spacetime. Moreover, it is established that a m-projectively flat perfect fluid spacetime is either a de-Sitter spacetime or a Minkowski spacetime. In addition, we show that a perfect fluid spacetime admitting harmonic m-projective curvature tensor is a Robertson-Walker spacetime. Next, we study m-projectively flat perfect fluid spacetime as a solution of -gravity and -gravity theories and explain their physical meaning. Several energy conditions in terms of Ricci scalar are investigated with the models and . For these models, the weak, null and dominant energy conditions are satisfied, while the strong energy condition is violated, which is good agreement with the recent observational studies and this reveals that the current Universe is in accelerating phase.

New insights from GW170817 in the dynamical system analysis of Einstein Gauss–Bonnet gravity

In this work, we explore the dynamical system phase space of Einstein–Gauss–Bonnet theory in the cosmological minisuperspace in the light of GW170817. This approach binds the main features of the theory through a system of autonomous differential equations, in the context of a flat Friedmann–Lemaître–Robertson–Walker spacetime. We analyze the critical points that feature in this system to assess their stability criteria. The phase space of this form of scalar-tensor gravity is very rich due to the fourth-order contributions of the Gauss–Bonnet invariant together with the second order contribution of the scalar field together with their coupling dynamics. We find additional critical points as compared with previous works in the literature which may be important for understanding the larger evolution of standard background cosmology within this class of gravitational models.

Jerk Parameter and Modified Theory

The accelerated expansion of the universe during recent times is well known in cosmology, whereas during early times, there was decelerated expansion. The ΛCDM model is consistent with most observations, but there are some issues with it. In addition, the transition from early deceleration to late-time acceleration cannot be explained by general relativity. Hence, it is worthwhile to examine modified gravity theories to explain this transition and to get a better understanding of dark energy. In this work, dark energy in modified f(R,T) gravity is investigated, where R is the Ricci scalar and T is the trace of the energy momentum tensor. Normally, the simplest form of f(R,T) is used, viz., f(R)=R+λT. In this work, the more complicated form f(R,T)=R+RT is investigated in Friedmann–Lemaître–Robertson–Walker spacetime. This form has not been well studied. Since the jerk parameter in general relativity is constant and j=1, in order to have as small a departure from general relativity as possible, the jerk parameter j=1 is also assumed here. This enables the complete solution for the scale factor to be found. One of these forms is used for a complete analysis and is compared with the usually studied form f(R,T)=R+RT. The solution can also be broken down into a power-law form at early times (deceleration) and an exponential form at late times (acceleration), which makes the analysis simpler. Surprisingly, each of these forms is also a solution to the differential equation j=1 (though they are not solutions to the general solution). The energy conditions are also studied, and plots are provided. It is shown that viable models can be obtained without the need for the introduction of a cosmological constant, which reduces to the ΛCDM at late times.

Raychaudhuri equation and bouncing cosmology

This work deals with an exhaustive study of bouncing cosmology in the background of homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker spacetime. The geometry of the bouncing point has been studied extensively and used as a tool to classify the models from the point of view of cosmology. Raychaudhuri equation (RE) has been furnished in these models to classify the bouncing point as regular point or singular point. Behavior of time-like geodesic congruence in the neighborhood of the bouncing point has been discussed using the Focusing Theorem which follows as a consequence of the RE. An analogy of the RE with the evolution equation for a linear harmonic oscillator has been made and an oscillatory bouncing model has been discussed in this context.

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.

Quantum field corrections to the equation of state of freely streaming matter in the Friedman–Lemaître–Robertson–Walker space-time

We calculate the energy density and pressure of a scalar field after its decoupling from a thermal bath in the spatially flat Friedman–Lemaître–Robertson–Walker space-time, within the framework of quantum statistical mechanics. By using the density operator determined by the condition of local thermodynamic equilibrium, we calculate the mean value of the stress-energy tensor of a real scalar field by subtracting the vacuum expectation value at the time of the decoupling. The obtained expressions of energy density and pressure involve corrections with respect to the classical free-streaming solution of the relativistic Boltzmann equation, which may become relevant even at long times.

Decay of solutions of the wave equation in cosmological spacetimes—a numerical analysis

We numerically evolve spherically symmetric solutions to the linear wave equation on some expanding Friedmann–Lemaître–Robertson–Walker spacetimes and study the respective asymptotics for large times. We find a quantitative relation between the expansion rate of the underlying background Universe and the decay rate of linear waves, also in the context of spatially-hyperbolic spacetimes, for which rigorous proofs of decay rates are not generally known. A prominent role in the decay mechanism is shown to be played by tails, i.e. scattered waves propagating in the interior of the lightcone.

Comparison of energy conditions in general relativity and modified theories of gravity

To make a comparison of energy conditions in the theory of general relativity and in the modified theories, we have considered [Formula: see text], [Formula: see text] and [Formula: see text] theories (where [Formula: see text] and [Formula: see text] are the Ricci scalar and trace of the energy–momentum tensor, respectively, while [Formula: see text]) to test the validity of all the four energy conditions. These energy conditions had been derived to check the viability of cosmological as well as astrophysical models. In this paper, we consider the standard Friedmann–Robertson–Walker spacetime representing the homogeneous and isotropic universe, and investigate the available literature on testing energy conditions as well as calculate these conditions ourselves. In order to provide the comparative results, we test these conditions analytically as well as graphically and present how energy bounds in modified theories depend on the values of the involved parameters and its validity within certain limits. We discuss here, that which of the modified theories comply with certain energy conditions, and hence provide a supporting argument on the existence of modified theories.

Noether symmetry approach in scalar-torsion f(T,phi ) gravity

The Noether Symmetry approach is applied to study an extended teleparallel f(T,phi ) gravity that contains the torsion scalar T and the scalar field phi in the context of an Friedmann–Lemaître–Robertson–Walker space-time. We investigate the Noether symmetry approach in f(T,phi ) gravity formalism with the specific form of f(T,phi ) and analyze how to demonstrate a nontrivial Noether vector. The Noether symmetry method is a helpful resource for generating models and finding out the exact solution of the Lagrangian. In this article, we go through how the Noether symmetry approach enables us to define the form of the function f(T,phi ) and obtain exact cosmological solutions. We also find the analytical cosmological solutions to the field equations, that is consistent with the Noether symmetry. Our results demonstrate that the obtained solutions enable an accelerated expansion of the Universe. We have also obtained the present value of the Hubble parameter, deceleration parameter, and effective equation of state parameter, which is fit in the range of current cosmological observations.