Walker Spacetime Research Articles

Second-order dust perturbations of the non-flat FLRW model with the positive cosmological constant

In this paper, a specific solution to the second-order cosmological perturbation theory is given. Perturbations are performed around any Friedmann–Lemaître–Robertson–Walker spacetime filled with dust and with a positive cosmological constant. In particular, with a possibly non-vanishing spatial curvature. The adopted symmetry condition allows us to simplify the equations, leaving us with a great deal of freedom to choose the density distribution. In the result, we get a relatively simple metric of an inhomogeneous cosmological model, which will give a perfect tool for studying the influence of the local inhomogeneities onto the cosmological observables.

Stable cosmologies with collisionless charged matter

It is shown that Milne models (a subclass of Friedmann–Lematre–Robertson–Walker (FLRW) spacetimes with negative spatial curvature) are nonlinearly stable in the set of solutions to the Einstein–Vlasov–Maxwell system, describing universes with ensembles of collisionless self-gravitating, charged particles. The system contains various slowly decaying borderline terms in the mutually coupled equations describing the propagation of particles and Maxwell fields. The effects of those terms are controlled using a suitable hierarchy based on the energy density of the matter fields.

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].

Observational limits on the Nash theory of gravity with matter contents

We generalize the original Nash theory of gravity by adding the matter fields by specifying a proper form of the field equations on more general footings for space with matter contents. We first derive the equations of motion in the flat Friedmann–Lemaitre–Robertson–Walker (FLRW) spacetime and examine the behaviors of the solutions by invoking specific forms of the Hubble parameter. We also classify the physical behaviors of the solutions by employing the stability analysis. We check the consistency of the model by considering particular cosmological parameters. We use observational data from Supernovae Ia (SNeIa) and baryonic acoustic oscillations (BAO) to constrain the parameters of the model. Interestingly, we discover the best fit of our model with [Formula: see text] and [Formula: see text] for [Formula: see text] and [Formula: see text] assuming the Hubble parameter [Formula: see text].

Emergence of smooth distance and apparent magnitude in a lumpy Universe

The standard interpretation of observations such as the peak apparent magnitude of Type Ia supernova made from one location in a lumpy Universe is based on the idealised Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. All possible corrections to this model due to inhomogeneities are usually neglected. Here, we use the result from the recent concise derivation of the area distance in an inhomogeneous Universe to study the monopole and Hubble residual of the apparent magnitude of Type Ia supernovae. We find that at low redshifts, the background FLRW spacetime model of the apparent magnitude receives corrections due to relative velocity perturbation in the observed redshift. We show how this velocity perturbation could contribute to a variance in the Hubble residual and how it could impact the calibration of the absolute magnitude of the Type Ia supernova in the Hubble flow. We also show that it could resolve the tension in the determination of the Hubble rate from the baryon acoustic oscillation and local measurements.

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.

Critical exponent of Fujita type for semilinear wave equations in Friedmann–Lemaître–Robertson–Walker spacetime

We consider the nonlinear massless wave equation belonging to some family of the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. We prove the global in time small data solutions for supercritical powers in the case of decelerating expansion universe.

The power spectrum of cosmological number densities

This paper studies the cosmological power spectrum (PS) of the differential and integral galaxy volume number densities, respectively, \(\gamma _i\) and \(\gamma _i^{*}\), constructed with the cosmological distances \(d_i\) \((i={\scriptstyle A},{\scriptstyle G},{\scriptstyle L},{\scriptstyle Z})\), where \(d_{\scriptscriptstyle A}\) is the angular diameter distance, \(d_{\scriptscriptstyle G}\) is the galaxy area distance, \(d_{\scriptscriptstyle L}\) is the luminosity distance and \(d_z\) is the redshift distance. Theoretical and observational quantities were obtained in the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime with a non-vanishing cosmological constant. The radial correlation \(\Xi _i\), a quantity defined in the context of these densities, is also discussed in the wave number domain. All observational quantities were computed using luminosity function (LF) data obtained from the FORS Deep Field (FDF) galaxy survey. The theoretical and observational power spectra of \(\gamma _i\), \(\gamma _i^{*}\), \(\Xi _i\) and the ratio \(\gamma _i / \gamma _i^*\) were calculated by performing Fourier transforms on values of these densities which were previously derived by Iribarrem (Astron. Astrophys. 539:A112, 2012) from the observational values \([\gamma ]_{{\textstyle {\scriptscriptstyle \mathrm {obs}}}}\) and \([\gamma ^*]_{{\textstyle {\scriptscriptstyle \mathrm {obs}}}}\) obtained by using the galactic absolute magnitudes and Schechter’s parameters of the galaxy LF presented in Gabasch (Astron. Astrophys. 421:41–58, 2004; Astron. Astrophys. 448:101, 2006). These parameters were evaluated from a I-band selected dataset of the FDF in the redshift range \(0.5 \le z \le 5.0\) for its blue bands and \(0.75 \le z \le 3.0\) for its red ones. The results show similar behavior of the power spectra obtained from \(\gamma \) and \(\gamma ^{*}\) using \(d_{\scriptscriptstyle L}\), \(d_z\) and \(d_{\scriptscriptstyle G}\) as distance measures. The PS of the densities defined with \(d_{\scriptscriptstyle A}\) have a different and inconclusive behavior because this cosmological distance reaches a maximum at \(z\approx 1.6\) in the adopted cosmological model. For the other distances, our results suggest that the PS of \([\gamma _i]_{{\textstyle {\scriptscriptstyle \mathrm {obs}}}}\), \([\gamma ^*_i]_{{\textstyle {\scriptscriptstyle \mathrm {obs}}}}\) and \([\gamma _i / \gamma ^{*}_i]_{{\textstyle {\scriptscriptstyle \mathrm {obs}}}}\) has a general behavior approximately similar to the power spectra obtained with the galaxy two-point correlation function and, by being sample size independent, they may be considered as alternative analytical tools to study the galaxy distribution.

Energy-Momentum Complex in Higher Order Curvature-Based Local Gravity

An unambiguous definition of gravitational energy remains one of the unresolved issues of physics today. This problem is related to the non-localization of gravitational energy density. In General Relativity, there have been many proposals for defining the gravitational energy density, notably those proposed by Einstein, Tolman, Landau and Lifshitz, Papapetrou, Møller, and Weinberg. In this review, we firstly explored the energy–momentum complex in an nth order gravitational Lagrangian L=Lgμν,gμν,i1,gμν,i1i2,gμν,i1i2i3,⋯,gμν,i1i2i3⋯in and then in a gravitational Lagrangian as Lg=(R¯+a0R2+∑k=1pakR□kR)−g. Its gravitational part was obtained by invariance of gravitational action under infinitesimal rigid translations using Noether’s theorem. We also showed that this tensor, in general, is not a covariant object but only an affine object, that is, a pseudo-tensor. Therefore, the pseudo-tensor ταη becomes the one introduced by Einstein if we limit ourselves to General Relativity and its extended corrections have been explicitly indicated. The same method was used to derive the energy–momentum complex in fR gravity both in Palatini and metric approaches. Moreover, in the weak field approximation the pseudo-tensor ταη to lowest order in the metric perturbation h was calculated. As a practical application, the power per unit solid angle Ω emitted by a localized source carried by a gravitational wave in a direction x^ for a fixed wave number k under a suitable gauge was obtained, through the average value of the pseudo-tensor over a suitable spacetime domain and the local conservation of the pseudo-tensor. As a cosmological application, in a flat Friedmann–Lemaître–Robertson–Walker spacetime, the gravitational and matter energy density in f(R) gravity both in Palatini and metric formalism was proposed. The gravitational energy–momentum pseudo-tensor could be a useful tool to investigate further modes of gravitational radiation beyond two standard modes required by General Relativity and to deal with non-local theories of gravity involving □−k terms.

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.

The self-interacting Dirac fields in FLRW spacetime

We prove the existence of global in time solution with the small initial data for the semilinear equation of the spin-\(\frac{1}{2}\) particles in the Friedmann–Lemaître–Robertson–Walker spacetime. Moreover, we also prove that if the initial function satisfies the Lochak–Majorana condition, then the global solution exists for arbitrary large initial value. The solution scatters to free solution for large time. The mass term is assumed to be complex-valued. The conditions on the imaginary part of mass are discussed by proving nonexistence of the global solutions if certain relation between scale function and the mass are fulfilled.

Fluid accretion upon higher-dimensional wormhole and black hole for parameterized deceleration parameter

In this work, we study some parameterizations of the deceleration parameter and investigate the accretion of the fluid in these parameterized models upon the higher-dimensional black hole (BH) and wormholes. For the undergoing analysis, [Formula: see text]-dimensional wormhole (proposed by Morris and Thorne) and Schwarzschild BH were chosen in the backdrop of higher-dimensional Friedmann–Robertson–Walker (FRW) space-time. For these parameterized models, we analyze the change of masses of BH and wormhole in different dimensions.

Dynamics of a Cosmological Model in f(R,T) Gravity: I. On Invariant Planes

Under the background of perfect fluid and flat Friedmann–Lemaître–Robertson–Walker (FLRW) space-time, this paper mainly describes the dynamics of the cosmological model constructed in f(R,T) gravity on three invariant planes, by using the singularity theory and Poincaré compactification in differential equations.

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.

Reconstruction in the Horndeski theory within the scope of the Bianchi I cosmology

In this paper, [R. K. Muharlyamov and T. N. Pankratyeva, Eur. Phys. J. Plus 136, 590 (2021)] we have proposed a reconstruction method for the kinetic gravity braiding theory in the framework of the flat Friedman–Robertson–Walker spacetime. Here, we develop this method in the Bianchi I spacetime model for a subclass of the Horndeski theory: [Formula: see text], [Formula: see text], [Formula: see text]. The Hubble parameter [Formula: see text] and the canonical kinetic term [Formula: see text] are set a priori. The choice of the function [Formula: see text] determines the anisotropic properties of the Universe. This makes it possible to provide believable anisotropy. The presented method allows for a realistic model of the Universe to simply reconstruct some scalar field theory. Reconstruction example is given for anisotropic model of a post-inflationary transition to the radiation-dominated phase. The model is investigated for the absence ghosts and Laplacian instabilities.

New exact solutions to the Eddington-inspired-Born-Infeld equations for a relativistic perfect fluid in a Bianchi type I spacetime

We consider a Bianchi type I physical metric g, an auxiliary metric q and a density matter ρ in Eddington-inspired-Born-Infeld theory. We first derive a system of second order nonlinear ordinary differential equations. Then, by a suitable change of variables, we arrive at a system of first order nonlinear ordinary differential equations. Using both the solution-tube concept for the first order nonlinear ordinary differential equations and the nonlinear analysis tools such as the Arzelá–Ascoli theorem, we prove an existence result for the nonlinear system obtained. The resolution of this last system allows us to obtain new exact solutions for the model considered. Finally, by studying the asymptotic behaviour of the exact solutions obtained, we conclude that this solution is the counterpart of the Friedman–Lemaître–Robertson–Walker spacetime in Eddington-inspired-Born-Infeld theory.

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.