Kundt spacetimes as solutions of topologically massive gravity

We obtain new solutions of topologically massive gravity. We find the general Kundt solutions, which in three dimensions are spacetimes admitting an expansion-free null geodesic congruence. The solutions are generically of algebraic type II, but special cases are types III, N or D. Those of type D are the known spacelike-squashed AdS_3 solutions, and of type N are the known AdS pp-waves or new solutions. Those of types II and III are the first known solutions of these algebraic types. We present explicitly the Kundt solutions that are CSI spacetimes, for which all scalar polynomial curvature invariants are constant, whereas for the general case we reduce the field equations to a series of ordinary differential equations. The CSI solutions of types II and III are deformations of spacelike-squashed AdS_3 and the round AdS_3, respectively.Comment: 30 pages. This material has come from splitting v1 of arXiv:0906.3559 into 2 separate papers. v2: minor changes

On higher dimensional Einstein spacetimes with a warped extra dimension

We study a class of higher dimensional warped Einstein spacetimes with one extra dimension. These were originally identified by Brinkmann as those Einstein spacetimes that can be mapped conformally on other Einstein spacetimes, and have subsequently appeared in various contexts to describe, e.g., different braneworld models or warped black strings. After clarifying the relation between the general Brinkmann metric and other more specific coordinate systems, we analyze the algebraic type of the Weyl tensor of the solutions. In particular, we describe the relation between Weyl aligned null directions (WANDs) of the lower dimensional Einstein slices and of the full spacetime, which in some cases can be algebraically more special. Possible spacetime singularities introduced by the warp factor are determined via a study of scalar curvature invariants and of Weyl components measured by geodetic observers. Finally, we illustrate how Brinkmann's metric can be employed to generate new solutions by presenting the metric of spinning and accelerating black strings in five dimensional anti-de Sitter space.Comment: 14 pages, minor changes in the text, mainly in Section 2.

Axial symmetry and conformal Killing vectors

Axisymmetric spacetimes with a conformal symmetry are studied and it is shown that, if there is no further conformal symmetry, the axial Killing vector and the conformal Killing vector must commute. As a direct consequence, in conformally stationary and axisymmetric spacetimes, no restriction is made by assuming that the axial symmetry and the conformal timelike symmetry commute. Furthermore, we prove that in axisymmetric spacetimes with another symmetry (such as stationary and axisymmetric or cylindrically symmetric spacetimes) and a conformal symmetry, the commutator of the axial Killing vector with the two others mush vanish or else the symmetry is larger than that originally considered. The results are completely general and do not depend on Einstein's equations or any particular matter content.Comment: 15 pages, Latex, no figure

Closed cosmologies with a perfect fluid and a scalar field

Closed, spatially homogeneous cosmological models with a perfect fluid and a scalar field with exponential potential are investigated, using dynamical systems methods. First, we consider the closed Friedmann-Robertson-Walker models, discussing the global dynamics in detail. Next, we investigate Kantowski-Sachs models, for which the future and past attractors are determined. The global asymptotic behaviour of both the Friedmann-Robertson-Walker and the Kantowski-Sachs models is that they either expand from an initial singularity, reach a maximum expansion and thereafter recollapse to a final singularity (for all values of the potential parameter kappa), or else they expand forever towards a flat power-law inflationary solution (when kappa^2<2). As an illustration of the intermediate dynamical behaviour of the Kantowski-Sachs models, we examine the cases of no barotropic fluid, and of a massless scalar field in detail. We also briefly discuss Bianchi type IX models.Comment: 15 pages, 10 figure

Lie symmetries for equations in conformal geometries

We seek exact solutions to the Einstein field equations which arise when two spacetime geometries are conformally related. Whilst this is a simple method to generate new solutions to the field equations, very few such examples have been found in practice. We use the method of Lie analysis of differential equations to obtain new group invariant solutions to conformally related Petrov type D spacetimes. Four cases arise depending on the nature of the Lie symmetry generator. In three cases we are in a position to solve the master field equation in terms of elementary functions. In the fourth case special solutions in terms of Bessel functions are obtained. These solutions contain known models as special cases.Comment: 19 pages, To appear in J. Phys.

The stability of cosmological scaling solutions

We study the stability of cosmological scaling solutions within the class of spatially homogeneous cosmological models with a perfect fluid subject to the equation of state p_gamma=(gamma-1) rho_gamma (where gamma is a constant satisfying 0 < gamma < 2) and a scalar field with an exponential potential. The scaling solutions, which are spatially flat isotropic models in which the scalar field energy density tracks that of the perfect fluid, are of physical interest. For example, in these models a significant fraction of the current energy density of the Universe may be contained in the scalar field whose dynamical effects mimic cold dark matter. It is known that the scaling solutions are late-time attractors (i.e., stable) in the subclass of flat isotropic models. We find that the scaling solutions are stable (to shear and curvature perturbations) in generic anisotropic Bianchi models when gamma < 2/3. However, when gamma > 2/3, and particularly for realistic matter with gamma >= 1, the scaling solutions are unstable; essentially they are unstable to curvature perturbations, although they are stable to shear perturbations. We briefly discuss the physical consequences of these results.Comment: AMSTeX, 7 pages, re-submitted to Phys Rev Let

Self-similar spherically symmetric cosmological models with a perfect fluid and a scalar field

Self-similar, spherically symmetric cosmological models with a perfect fluid and a scalar field with an exponential potential are investigated. New variables are defined which lead to a compact state space, and dynamical systems methods are utilised to analyse the models. Due to the existence of monotone functions global dynamical results can be deduced. In particular, all of the future and past attractors for these models are obtained and the global results are discussed. The essential physical results are that initially expanding models always evolve away from a massless scalar field model with an initial singularity and, depending on the parameters of the models, either recollapse to a second singularity or expand forever towards a flat power-law inflationary model. The special cases in which there is no barotropic fluid and in which the scalar field is massless are considered in more detail in order to illustrate the asymptotic results. Some phase portraits are presented and the intermediate dynamics and hence the physical properties of the models are discussed.Comment: 31 pages, 4 figure

Anisotropy in Bianchi-type brane cosmologies

The behavior near the initial singular state of the anisotropy parameter of the arbitrary type, homogeneous and anisotropic Bianchi models is considered in the framework of the brane world cosmological models. The matter content on the brane is assumed to be an isotropic perfect cosmological fluid, obeying a barotropic equation of state. To obtain the value of the anisotropy parameter at an arbitrary moment an evolution equation is derived, describing the dynamics of the anisotropy as a function of the volume scale factor of the Universe. The general solution of this equation can be obtained in an exact analytical form for the Bianchi I and V types and in a closed form for all other homogeneous and anisotropic geometries. The study of the values of the anisotropy in the limit of small times shows that for all Bianchi type space-times filled with a non-zero pressure cosmological fluid, obeying a linear barotropic equation of state, the initial singular state on the brane is isotropic. This result is obtained by assuming that in the limit of small times the asymptotic behavior of the scale factors is of Kasner-type. For brane worlds filled with dust, the initial values of the anisotropy coincide in both brane world and standard four-dimensional general relativistic cosmologies.Comment: 12 pages, no figures, to appear in Class. Quantum Gra