Spacetimes Of Constant Curvature Research Articles

Cut-and-paste for impulsive gravitational waves with Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}: the mathematical analysis

Impulsive gravitational waves are theoretical models of short but violent bursts of gravitational radiation. They are commonly described by two distinct spacetime metrics, one of local Lipschitz regularity and the other one even distributional. These two metrics are thought to be ‘physically equivalent’ since they can be formally related by a ‘discontinuous coordinate transformation’. In this paper we provide a mathematical analysis of this issue for the entire class of nonexpanding impulsive gravitational waves propagating in a background spacetime of constant curvature. We devise a natural geometric regularisation procedure to show that the notorious change of variables arises as the distributional limit of a family of smooth coordinate transformations. In other words, we establish that both spacetimes arise as distributional limits of a smooth sandwich wave taken in different coordinate systems which are diffeomorphically related.

Hermitian realizations of the Yang model

The Yang model is an example of noncommutative geometry on a background spacetime of constant curvature. We discuss the Hermitian realizations of its associated algebra on phase space in a perturbative expansion up to sixth order. We also discuss its realizations on extended phase spaces, that include additional tensorial and/or vectorial degrees or freedom.

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.

Some curvature properties of perfect fluid spacetimes

In this paper we assume that a perfect fluid is the source of the gravitational field while analyzing the solutions to the Einstein field equations. With this new and innovative approach, we investigate some curvature properties of perfect fluid spacetimes. Firstly, we provide a necessary condition for a Lorentzian manifold to be a perfect fluid spacetime. We prove that a conformally flat perfect fluid space-time is a spacetime of quasi constant sectional curvature. Also, we acquire that a conharmonically flat perfect fluid spacetime represents the radiation era. Next, we show that if a stiff matter fluid obeys Yang’s equations, then the vorticity of the fluid vanishes. Moreover, we find that if the Ricci tensor is Killing, then the perfect fluid spacetime is (i) expansion-free and shear-free, and its flow is geodesic, however, not necessarily vorticity-free, and (ii) σ and p are constant. Besides, we establish that Ricci semi-symmetric or Ricci symmetric perfect fluid spacetimes represent either dark matter era, or phantom era. Finally, we acquire that if a perfect fluid is Ricci symmetric, then either the spacetime represents a dark matter era, or a static spacetime.

Characterizations of a Spacetime of Quasi‐Constant Sectional Curvature and F(R)$\mathcal {F}(\mathcal {R})$‐Gravity

AbstractThe main aim of this article is to investigate a spacetime of quasi‐constant sectional curvature. At first, the existence of such a spacetime is established by several examples. We have shown that a spacetime of quasi‐constant sectional curvature agrees with the present state of the universe and it represents a Robertson Walker spacetime. Moreover, if the spacetime is Ricci semi‐symmetric or Ricci symmetric, then either the spacetime represents a spacetime of constant sectional curvature, or the spacetime represents phantom era. Also, we prove that a Ricci symmetric spacetime of quasi‐constant sectional curvature represents a static spacetime and the spacetime under consideration is of Petrov type I, D or O. Finally, we concentrate on a quasi‐constant sectional curvature spacetime solution in ‐gravity. As a result, various energy conditions are studied and analyzed our obtained outcomes in terms of a ‐gravity model.

Codazzi tensors and their space-times and Cotton gravity

We study the geometric properties of certain Codazzi tensors for their own sake, and for their appearance in the recent theory of Cotton gravity. We prove that a perfect-fluid tensor is Codazzi if and only if the metric is a generalized Stephani universe. A trace condition restricts it to a warped space-time, as proven by Merton and Derdziński. We also give necessary and sufficient conditions for a space-time to host a current-flow Codazzi tensor. In particular, we study the static and spherically symmetric cases, which include the Nariai and Bertotti-Robinson metrics. The latter are a special case of Yang Pure space-times, together with spatially flat FRW space-times with constant curvature scalar. We apply these results to the recent Cotton gravity by Harada. We show that the equation of Cotton gravity is Einstein’s equation modified by the presence of a Codazzi tensor, which can be chosen freely and constrains the space-time where the theory is staged. In doing so, the tensor (chosen in forms appropriate for physics) implies the form of the Ricci tensor. The two tensors specify the energy-momentum tensor, which is the source in the equation of Cotton gravity for the metric implied by the Codazzi tensor. For example, we show that the Stephani, Nariai and Bertotti-Robinson space-times are characterized by a “current flow” Codazzi tensor. Because of it, they solve Cotton gravity with physically sensible energy-momentum tensors. Finally, we discuss Cotton gravity in constant curvature space-times.

Axisymmetric Stationary Space-Times of Constant Scalar Curvature in Four Dimensions

Axisymmetric Stationary Space-Times of Constant Scalar Curvature in Four Dimensions

Exact scalar (quasi-)normal modes of black holes and solitons in gauged SUGRA

In this paper we identify a new family of black holes and solitons that lead to the exact integration of scalar probes, even in the presence of a non-minimal coupling with the Ricci scalar which has a non-trivial profile. The backgrounds are planar and spherical black holes as well as solitons of SU (2) × SU (2) mathcal{N} = 4 gauged supergravity in four dimensions. On these geometries, we compute the spectrum of (quasi-)normal modes for the non-minimally coupled scalar field. We find that the equation for the radial dependence can be integrated in terms of hypergeometric functions leading to an exact expression for the frequencies. The solutions do not asymptote to a constant curvature spacetime, nevertheless the asymptotic region acquires an extra conformal Killing vector. For the black hole, the scalar probe is purely ingoing at the horizon, and requiring that the solutions lead to an extremum of the action principle we impose a Dirichlet boundary condition at infinity. Surprisingly, the quasinormal modes do not depend on the radius of the black hole, therefore this family of geometries can be interpreted as isospectral in what regards to the wave operator non-minimally coupled to the Ricci scalar. We find both purely damped modes, as well as exponentially growing unstable modes depending on the values of the non-minimal coupling parameter. For the solitons we show that the same integrability property is achieved separately in a non-supersymmetric solutions as well as for the supersymmetric one. Imposing regularity at the origin and a well defined extremum for the action principle we obtain the spectra that can also lead to purely oscillatory modes as well as to unstable scalar probes, depending on the values of the non-minimal coupling.

A model of non-minimally coupled gravitation and electromagnetism in (1+2) dimensions

Following earlier works of Dereli and collaborators, we study a three dimensional toy model where we extend the topologically massive gravity with electrodynamics by the most general RF 2-type non-minimal coupling terms. Here R denotes the possible curvature terms and F denotes the electromagnetic 2-form. We derive the variational feld equations and look for exact solutions on constant negative curvature space-times with a constant, self-dual electromagnetic feld. The notion of self-dual electromagnetic felds in three dimensions is introduced by Dereli and collaborators in the study of exact solutions of models with gravity-electromagnetism couplings. We note the conditions that the parameters of the model have to satisfy for these self-dual solutions to exist.

The Kerr-Schild double copy in Lifshitz spacetime

The Kerr-Schild double copy is a map between exact solutions of general relativity and Maxwell’s theory, where the nonlinear nature of general relativity is circumvented by considering solutions in the Kerr-Schild form. In this paper, we give a general formulation, where no simplifying assumption about the background metric is made, and show that the gauge theory source is affected by a curvature term that characterizes the deviation of the background spacetime from a constant curvature spacetime. We demonstrate this effect explicitly by studying gravitational solutions with non-zero cosmological constant. We show that, when the background is flat, the constant charge density filling all space in the gauge theory that has been observed in previous works is a consequence of this curvature term. As an example of a solution with a curved background, we study the Lifshitz black hole with two different matter couplings. The curvature of the background, i.e., the Lifshitz spacetime, again yields a constant charge density; however, unlike the previous examples, it is canceled by the contribution from the matter fields. For one of the matter couplings, there remains no additional non-localized source term, providing an example for a non-vacuum gravity solution corresponding to a vacuum gauge theory solution in arbitrary dimensions.

Riemannian manifolds dual to static spacetimes

We establish a one-to-one correspondence between static spacetimes and Riemannian manifolds that maps causal geodesics to geodesics, as suggested by L. C. Epstein. We then explore constant curvature spacetimes - such as the de Sitter and the anti-de Sitter universes - and find that they map to constant curvature Riemannian manifolds, namely the Euclidean space, the sphere and the hyperbolic space. By imposing the conditions required to map to the sphere, we obtain the spherically symmetric metrics for which there is radial oscillatory motion with a period independent of the amplitude. We then consider the case of a perfect fluid and an Einstein cluster and determine the pressure and density profiles required to find this type of motion. Finally, we give examples of surfaces corresponding to certain types of motion for metrics that do not exhibit constant curvature, such as the Schwarzschild, Schwarzschild de Sitter and Schwarzschild anti-de Sitter solutions, and even for a simplified model of a wormhole.

Determinantal invariant gravity

Einstein–Hilbert action with a determinantal invariant has been considered. The obtained field equation contains the inverse Ricci tensor, [Formula: see text]. The linearized solution of invariant has been examined, and it is concluded that the constant curvature spacetime metric solution of the field equation gives different curvature constant for each values of [Formula: see text]. The [Formula: see text] gives a trivial solution for constant curvature, [Formula: see text].

Cayley--Klein Poisson Homogeneous Spaces

The nine two-dimensional Cayley-Klein geometries are firstly reviewed by following a graded contraction approach. Each geometry is considered as a set of three symmetrical homogeneous spaces (of points and two kinds of lines), in such a manner that the graded contraction parameters determine their curvature and signature. Secondly, new Poisson homogeneous spaces are constructed by making use of certain Poisson-Lie structures on the corresponding motion groups. Therefore, the quantization of these spaces provides noncommutative analogues of the Cayley-Klein geometries. The kinematical interpretation for the semi-Riemannian and pseudo-Riemannian Cayley-Klein geometries is emphasized, since they are just Newtonian and Lorentzian spacetimes of constant curvature.

Null hypersurfaces in de Sitter and anti-de Sitter cosmologies

The study of gravitational waves in the presence of a cosmological constant has led to interesting forms of the de Sitter and anti-de Sitter line elements based on families of null hypersurfaces. The forms are interesting because they focus attention on the geometry of null hypersurfaces in spacetimes of constant curvature. Two examples are worked out in some detail. The first originated in the study of collisions of impulsive gravitational waves in which the post-collision spacetime is a solution of Einstein’s field equations with a cosmological constant, and the second originated in the generalization of plane fronted gravitational waves with parallel rays to include a cosmological constant.

IDEAL characterization of isometry classes of FLRW and inflationary spacetimes

In general relativity, an IDEAL (Intrinsic, Deductive, Explicit, ALgorithmic) characterization of a reference spacetime metric g0 consists of a set of tensorial equations T[g] = 0, constructed covariantly out of the metric g, its Riemann curvature and their derivatives, that are satisfied if and only if g is locally isometric to the reference spacetime metric g0. The same notion can be extended to also include scalar or tensor fields, where the equations are allowed to also depend on the extra fields ϕ. We give the first IDEAL characterization of cosmological FLRW spacetimes, with and without a dynamical scalar (inflaton) field. We restrict our attention to what we call regular geometries, which uniformly satisfy certain identities or inequalities. They roughly split into the following natural special cases: constant curvature spacetime, Einstein static universe, and flat or curved spatial slices. We also briefly comment on how the solution of this problem has implications, in general relativity and inflation theory, for the construction of local gauge invariant observables for linear cosmological perturbations and for stability analysis.

Asymptotic behavior of Cauchy hypersurfaces in constant curvature space–times

We study the asymptotic behavior of convex Cauchy hypersurfaces on maximal globally hyperbolic spatially compact space–times of constant curvature. We generalise the result of Belraouti (Annales de l’institut Fourier 64(2):457–466, 2015) to the (2+1) de Sitter and anti de Sitter cases. We prove that in these cases the level sets of quasi-concave times converge in the Gromov equivariant topology, when time goes to 0, to a real tree. Moreover, this limit does not depend on the choice of the time function. We also consider the problem of asymptotic behavior in the flat \((n+1)\) dimensional case. We prove that the level sets of quasi-concave times converge in the Gromov equivariant topology, when time goes to 0, to a CAT(0) metric space. Moreover, this limit does not depend on the choice of the time function.

New improved massive gravity and three-dimensional spacetimes of constant curvature and constant torsion

We derive the field equations for topologically massive gravity coupled with the most general quadratic curvature terms using the language of exterior differential forms and a first order constrained variational principle. We find variational field equations both in the presence and absence of torsion. We then show that spaces of constant negative curvature (i.e. the anti de-Sitter space $AdS_3$) and constant torsion provide exact solutions.

Hidden symmetries and Lie algebra structures from geometric and supergravity Killing spinors

We consider geometric and supergravity Killing spinors and the spinor bilinears constructed out of them. The spinor bilinears of geometric Killing spinors correspond to the antisymmetric generalizations of Killing vector fields which are called Killing–Yano forms. They constitute a Lie superalgebra structure in constant curvature spacetimes. We show that the Dirac currents of geometric Killing spinors satisfy a Lie algebra structure up to a condition on 2-form spinor bilinears. We propose that the spinor bilinears of supergravity Killing spinors give way to different generalizations of Killing vector fields to higher degree forms. It is also shown that those supergravity Killing forms constitute a Lie algebra structure in six- and ten-dimensional cases. For five- and eleven-dimensional cases, the Lie algebra structure depends on an extra condition on supergravity Killing forms.

Geometry and topology of complete Lorentz spacetimes of constant curvature

We study proper, isometric actions of nonsolvable discrete groups Γ on the 3-dimensional Minkowski space R as limits of actions on the 3-dimensional anti-de Sitter space AdS. To each such action is associated a deformation of a hyperbolic surface group Γ0 inside O(2, 1). When Γ0 is convex cocompact, we prove that Γ acts properly on R if and only if this group-level deformation is realized by a deformation of the quotient surface that everywhere contracts distances at a uniform rate. We give two applications in this case. (1) Tameness: A complete flat spacetime is homeomorphic to the interior of a compact manifold. (2) Geometric transition: A complete flat spacetime is the rescaled limit of collapsing AdS spacetimes.

Lorentz invariant vacuum solutions in general relativity

All Lorentz invariant solutions of vacuum Einstein's equations are found. It is proved that these solutions describe space-times of constant curvature.