Spacetimes Of Constant Curvature Research Articles

Spacetimes admitting W2-curvature tensor

The object of this paper is to study spacetimes admitting W2-curvature tensor. At first we prove that a W2-flat spacetime is conformally flat and hence it is of Petrov type O. Next, we prove that if the perfect fluid spacetime with vanishing W2-curvature tensor obeys Einstein's field equation without cosmological constant, then the spacetime has vanishing acceleration vector and expansion scalar and the perfect fluid always behaves as a cosmological constant. It is also shown that in a perfect fluid spacetime of constant scalar curvature with divergence-free W2-curvature tensor, the energy-momentum tensor is of Codazzi type and the possible local cosmological structure of such a spacetime is of type I, D or O.

STABILITY OF SPACELIKE HYPERSURFACES WITH HIGHER-ORDER MEAN CURVATURES LINEARLY RELATED IN CONFORMALLY STATIONARY SPACETIMES

In this paper, we establish the notion of (r, s)-stability concerning spacelike hypersurfaces with higher-order mean curvatures linearly related in conformally stationary spacetimes of constant sectional curvature. In this setting, we characterize (r, s)-stable closed spacelike hypersurfaces through the analysis of the first eigenvalue of an operator naturally attached to the higher-order mean curvatures. Moreover, we obtain sufficient conditions which assure the (r, s)-stability of complete spacelike hypersurfaces immersed in the de Sitter space.

Dual fermion condensates in curved space

In this paper we compute the effective action at finite temperature and density for the dual fermion condensate in curved space with the fermions described by an effective field theory with four-point interactions. The approach we adopt refines a technique developed earlier to study chiral symmetry breaking in curved space and it is generalized here to include the U(1)-valued boundary conditions necessary to define the dual condensate. The method we present is general, includes the coupling between the fermion condensate and the Polyakov loop, and applies to any ultrastatic background spacetime with a nonsingular base. It also allows one to include inhomogeneous and anisotropic phases and therefore it is suitable to study situations where the geometry is not homogeneous. We first illustrate a procedure, based on heat kernels, useful to deal with situations where the dual and chiral condensates (as well as any smooth background field eventually present) are slowly or rapidly varying functions in space. Then we discuss a different approach based on the density of states method and on the use of Tauberian theorems to handle the case of arbitrary chemical potentials. As a trial application, we consider the case of constant curvature spacetimes and show how to compute numerically the dual fermion condensate in the case of both homogeneous and inhomogeneous phases.

Probing pure Lovelock gravity by Nariai and Bertotti-Robinson solutions

The product spacetimes of constant curvature describe in Einstein gravity, which is linear in Riemann curvature, Nariai metric which is a solution of Λ-vacuum when curvatures are equal, k1 = k2, while it is Bertotti-Robinson metric describing uniform electric field when curvatures are equal and opposite, k1 = −k2. We probe pure Lovelock gravity by these simple product spacetimes and prove that the same characterization of these solutions is indeed true in general for pure Lovelock gravitational equation of order N in d = 2N + 2 dimension. We also consider these solutions for the conventional setting of Einstein-Gauss-Bonnet gravity.

3d gravity and quantum deformations: a Drinfel'd double approach

The constant curvature spacetimes of 3d gravity and their associated symmetry algebras are shown to arise from the 6d Drinfel'd double that underlies the two-parametric 'hybrid' quantum deformation of the \U0001d530\U0001d529(2, ℛ) algebra. Moreover, the quantum deformation supplies the additional structures (star structure and pairing) that enter in the Chern-Simons formulation of the theory, thus establishing a direct link between quantum \U0001d530\U0001d529(2, ℛ) algebras and 3d gravity models. In this approach the flat spacetimes and Newtonian models arise as Lie algebra contractions that are governed by two dimensionful \U0001d530\U0001d529(2, ℛ) deformation parameters, which are directly related to the cosmological constant and to the speed of light.

Cosmological time versus CMC time in spacetimes of constant curvature

In this paper, we investigate the existence of foliations by constant mean curvature (CMC) hypersurfaces in maximal, globally hyperbolic, spatially compact, spacetimes of constant curvature. In the non-positive curvature case (i.e. for flat and locally anti-de Sitter spacetimes), we prove the existence of a global foliation of the spacetime by CMC Cauchy hypersurfaces. The positive curvature case (i.e. locally de Sitter spacetimes) is more delicate: in general, we are only able to prove the existence of a foliation by CMC Cauchy hypersurfaces in a neighbourhood of the past (or future) singularity. Except in some exceptional and elementary cases, the leaves of the foliation we construct are the level sets of a time function, and the mean curvature of the leaves increases with time. In this case, we say that the spacetime admits a CMC time function. Our proof is based on using the level sets of the cosmological time function as barriers. A major part of the work consists of proving the required curvature estimates for these level sets. One of the difficulties is the fact that the local behaviour of the cosmological time function near one point depends on the global geometry of the spacetime.

On the geometry of maximal spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime

In this paper, we establish new characterizations of totally geodesic spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime, which is supposed to obey the null convergence condition. As applications, we get nonparametric results concerning to entire maximal vertical graphs in a such ambient spacetime. Proceeding, we obtain a lower estimate of the index of relative nullity of complete r-maximal spacelike hypersurfaces immersed in Robertson–Walker spacetimes of constant sectional curvature. In particular, we prove a sort of weak extension of the classical Calabi–Bernstein theorem.

The behaviour of geodesics in constant-curvature spacetimes with expanding impulsive gravitational waves

Motion of free test particles in spacetimes of constant curvature with an expanding impulsive gravitational wave is completely described. Explicit formulae which identify the particle positions and velocities "in front of" and "behind" the impulse are derived for a general impulse of this type. As a particular example, the effect of spherical impulsive wave generated by a snapped cosmic string is analyzed and visualized.

Dynamics of test bodies with spin in de Sitter spacetime

We study the motion of spinning test bodies in the de Sitter spacetime of constant positive curvature. With the help of the 10 Killing vectors, we derive the 4-momentum and the tensor of spin explicitly in terms of the spacetime coordinates. However, in order to find the actual trajectories, one needs to impose the so-called supplementary condition. We discuss the dynamics of spinning test bodies for the cases of the Frenkel and Tulczyjew conditions.

Conserved Killing charges of quadratic curvature gravity theories in arbitrary backgrounds

We extend the Abbott-Deser-Tekin procedure of defining conserved quantities of asymptotically constant-curvature spacetimes, and give an analogous expression for the conserved charges of geometries that are solutions of quadratic curvature gravity models in generic D-dimensions and that have arbitrary asymptotes possessing at least one Killing isometry. We show that the resulting charge expression correctly reduces to its counterpart when the background is taken to be a space of constant curvature and, moreover, is background gauge invariant. As applications, we compute and comment on the energies of two specific examples: the three dimensional Lifshitz black hole and a five dimensional companion of the first, whose energy has never been calculated beforehand.

Higher spin interactions with scalar matter on constant curvature spacetimes: conserved current and cubic coupling generating functions

Cubic couplings between a complex scalar field and a tower of symmetric tensor gauge fields of all ranks are investigated on any constant curvature spacetime of dimension d>2. Following Noether's method, the gauge fields interact with the scalar field via minimal coupling to the conserved currents. A symmetric conserved current, bilinear in the scalar field and containing up to r derivatives, is obtained for any rank r from its flat spacetime counterpart in dimension d+1, via a radial dimensional reduction valid precisely for the mass-square domain of unitarity in (anti) de Sitter spacetime of dimension d. The infinite collection of conserved currents and cubic vertices are summarized in a compact form by making use of generating functions and of the Weyl/Wigner quantization on constant curvature spaces.

Emergent geometry from quantized spacetime

We examine the picture of emergent geometry arising from a mass-deformed matrix model. Because of the mass-deformation, a vacuum geometry turns out to be a constant curvature spacetime such as d-dimensional sphere and (anti-)de Sitter spaces. We show that the mass-deformed matrix model giving rise to the constant curvature spacetime can be derived from the d-dimensional Snyder algebra. The emergent geometry beautifully confirms all the rationale inferred from the algebraic point of view that the d-dimensional Snyder algebra is equivalent to the Lorentz algebra in (d+1)-dimensional {\it flat} spacetime. For example, a vacuum geometry of the mass-deformed matrix model is completely described by a G-invariant metric of coset manifolds G/H defined by the Snyder algebra. We also discuss a nonlinear deformation of the Snyder algebra.

Refraction of geodesics by impulsive spherical gravitational waves in constant-curvature spacetimes with a cosmological constant

We investigate motion of test particles in exact spacetimes with an expanding impulsive gravitational wave which propagates in Minkowski, de Sitter or anti-de Sitter universe. Using the continuous form of these metrics we derive explicit junction conditions and simple refraction formulae for null, timelike and spacelike geodesics crossing a general impulse of this type. In particular, we present a detailed geometrical description of the motion of test particles in a special class of axially symmetric spacetimes in which the impulse is generated by a snapped cosmic string.

Three-dimensional gravity and Drinfel'd doubles: Spacetimes and symmetries from quantum deformations

We show how the constant curvature spacetimes of 3d gravity and the associated symmetry algebras can be derived from a single quantum deformation of the 3d Lorentz algebra sl(2,R). We investigate the classical Drinfel'd double of a “hybrid” deformation of sl(2,R) that depends on two parameters (η,z). With an appropriate choice of basis and real structure, this Drinfel'd double agrees with the 3d anti-de Sitter algebra. The deformation parameter η is related to the cosmological constant, while z is identified with the inverse of the speed of light and defines the signature of the metric. We generalise this result to de Sitter space, the three-sphere and 3d hyperbolic space through analytic continuation in η and z; we also investigate the limits of vanishing η and z, which yield the flat spacetimes (Minkowski and Euclidean spaces) and Newtonian models, respectively.

Charged C-metric with conformally coupled scalar field

We present a generalization of the charged C-metric conformally coupled with a scalar field in the presence of a cosmological constant. The solution is asymptotically flat or a constant curvature spacetime. The spacetime metric has the geometry of a usual charged C-metric with cosmological constant, where the mass and charge are equal. When the cosmological constant is absent it is found that the scalar field only blows up at the angular pole of the event horizon. The presence of the cosmological constant can generically render the scalar field regular where the metric is regular, pushing the singularity beyond the event horizon. For certain cases of enhanced acceleration with a negative cosmological constant, the conical singularity disappears altogether and the scalar field is everywhere regular. The black hole is then rather a black string with its event horizon extending all the way to asymptotic infinity and providing itself the necessary acceleration.

STATIC SPHERICALLY SYMMETRIC SOLUTIONS FOR CONFORMAL GRAVITY IN THREE DIMENSIONS

Static spherically symmetric solutions for conformal gravity in three dimensions are found. Black holes and wormholes are included within this class. Asymptotically the black holes are spacetimes of arbitrary constant curvature, and they are conformally related to the matching of different solutions of constant curvature by means of an improper conformal transformation. The wormholes can be constructed from suitable identifications of a static universe of negative spatial curvature, and it is shown that they correspond to the conformal matching of two black hole solutions with the same mass.

Notes on a paper of Mess

These notes are a companion to the article Lorentz spacetimes of constant curvature by Geoffrey Mess, which was first written in 1990 but never published. Mess' paper will appear together with these notes in a forthcoming issue of Geometriae Dedicata.

Lorentz spacetimes of constant curvature

This paper is unpublished work of Geoffrey Mess written in 1990, which gives a classification of flat and anti-de Sitter domains of dependence in 2+1 dimensions.

Do Killing–Yano tensors form a Lie algebra?

Killing–Yano tensors are natural generalizations of Killing vectors. We investigate whether Killing–Yano tensors form a graded Lie algebra with respect to the Schouten–Nijenhuis bracket. We find that this proposition does not hold in general, but that it does hold for constant curvature spacetimes. We also show that Minkowski and (anti)-deSitter spacetimes have the maximal number of Killing–Yano tensors of each rank and that the algebras of these tensors under the SN bracket are relatively simple extensions of the Poincaré and (A)dS symmetry algebras.

Gravitational lensing from compact bodies: Analytical results for strong and weak deflection limits

We have shown how the quantization of two-dimensional quantum gravity with an action which contains only a positive cosmological constant and boundary cosmological constants leads to the emergence of a spacetime which can be described as a constant negative curvature spacetime with superimposed quantum fluctuations.