Marc Mars

Axially symmetric Einstein-Straus models

The existence of static and axially symmetric regions in a Friedman-Lemaitre cosmology is investigated under the only assumption that the cosmic time and the static time match properly on the boundary hypersurface. It turns out that the most general form for the static region is a two-sphere with arbitrarily changing radius which moves along the axis of symmetry in a determined way. The geometry of the interior region is completely determined in terms of background objects. When any of the most widely used energy-momentum contents for the interior region is imposed, both the interior geometry and the shape of the static region must become exactly spherically symmetric. This shows that the Einstein-Straus model, which is the generally accepted answer for the null influence of the cosmic expansion on the local physics, is not a robust model and it is rather an exceptional and isolated situation. Hence, its suitability for solving the interplay between cosmic expansion and local physics is doubtful and more adequate models should be investigated.

The 2m ≤ r property of spherically symmetric static space-times

Abstract We prove that all spherically symmetric static space-times which are both regular at r = 0 and satisfy the single energy condition ϱ + pr + 2pt ≥ 0 cannot contain any black hole region (equivalently, they must satisfy 2m r ≤ 1 everywhere). This result holds even when the space-time is allowed to contain a finite number of matching hypersurfaces. This theorem generalizes a result by Baumgarte and Rendall when the matter contents of the space-time is a perfect fluid and also complements their results in the general non-isotropic case.

Space–time averages in macroscopic gravity and volume-preserving coordinates

The definition of the covariant space-time averaging scheme for the objects (tensors, geometric objects, etc.) on differentiable metric manifolds with a volume n-form, which has been proposed for the formulation of macroscopic gravity, is analyzed. An overview of the space-time averaging procedure in Minkowski space-time is given and comparison between this averaging scheme and that adopted in macroscopic gravity is carried out throughout the paper. Some new results concerning the algebraic structure of the averaging operator are precisely formulated and proved, the main one being that the averaging bilocal operator is idempotent iff it is factorized into a bilocal product of a matrix-valued function on the manifold, taken at a point, by its inverse at another point. The previously proved existence theorems for the averaging and coordination bilocal operators are revisited with more detailed proofs of related results. A number of new results concerning the structure of the volume-preserving averaging oper...

A self-similar inhomogeneous dust cosmology

A detailed study of an inhomogeneous dust cosmology contained in a -law family of perfect-fluid metrics recently presented by Mars and Senovilla is performed. The metric is shown to be the most general orthogonally transitive, Abelian, on solution admitting an additional homothety such that the self-similar group is of Bianchi type VI and the fluid flow is tangent to its orbits. The analogous cases with Bianchi types I, II, III, V, VIII and IX are shown to be impossible thus making this metric privileged from a mathematical viewpoint. The differential equations determining the metric are partially integrated and the line-element is given up to a first-order differential equation of Abel type of first kind and two quadratures. The solutions are qualitatively analysed by investigating the corresponding autonomous dynamical system. The spacetime is regular everywhere except for the big bang and the metric is complete both into the future and in all spatial directions. The energy density is positive, bounded from above at any instant of time and with an spatial profile (in the direction of inhomogeneity) which is oscillating with a rapidly decreasing amplitude. The generic asymptotic behaviour at spatial infinity is a homogeneous plane wave. Well known dynamical system results indicate that this metric is very likely to describe the asymptotic behaviour in time of a much more general class of inhomogeneous dust cosmologies.

Non-diagonal separable perfect-fluid spacetimes

We consider non-diagonal inhomogeneous cosmologies admitting an Abelian orthogonally transitive two-dimensional group of isometries under the assumption of separation of variables. We find all the sets of compatible systems of ordinary equations which give rise to reasonable solutions (satisfying energy conditions) and study them. It turns out that five of these families exist. Two of them represent stiff fluids with the density equal to the pressure. A third family satisfies an equation of state of the linear type . Regarding the fourth family we can integrate the field equations so that there only remains an ordinary differential equation for an unknown to be solved. We also give some explicit particular solutions in this case. The fifth family contains an explicit singularity-free stiff fluid solution which has already been studied by Mars in 1995 and an explicit singularity-free family satisfying the dominant energy condition (but not the strong energy condition) and which is eternally expanding and contracting without passing any big bang or big crunch.

perfect-fluid cosmologies with a proper conformal Killing vector

We study the Einstein field equations for spacetimes admitting a maximal two-dimensional Abelian group of isometries acting orthogonally transitively on spacelike surfaces and, in addition, with at least one conformal Killing vector. The three-dimensional conformal group is restricted to the case when the two-dimensional Abelian isometry subalgebra is an ideal and it is also assumed to act on non-null hypersurfaces (both spacelike and timelike cases are studied). We consider both diagonal and non-diagonal metrics, and find all the perfect-fluid solutions under these assumptions (except those already known). We find four families of solutions, each one containing arbitrary parameters for which no differential equations remain to be integrated. We write the line-elements in a simplified form and perform a detailed study for each of these solutions, giving the kinematical quantities of the fluid velocity vector, the energy density and pressure, values of the parameters for which the energy conditions are fulfilled everywhere, the Petrov type, the singularities in the spacetimes and the Friedmann - Lemaitre - Robertson - Walker metrics contained in each family.

Stationary and Axisymmetric Perfect Fluids with one Conformal Killing Vector

We study the stationary and axisymmetric non-convective differentially rotating perfect-fluid solutions of Einsteins field equations admitting one conformal symmetry. We analyse the two inequivalent Lie algebras not exhaustively considered in our previous work and show that the general solution for each Lie algebra depends on one arbitrary function of one of the coordinates while a set of three ordinary differential equations for four unknowns remains to be solved. The conformal Killing vector of these solutions is necessarily homothetic. We summarize in a table all the possible solutions for all the allowed Lie algebras and also add a corrigendum to an erroneous statement in our previous paper concerning the differentially rotating character of one of the solutions presented.

LETTER TO THE EDITOR: General Non-Rotating Perfect-Fluid Solution with an Abelian Spacelike C3 Including Only One Isometry

The general solution for non-rotating perfect-fluid spacetimes admitting one Killing vector and two conformal (non-isometric) Killing vectors spanning an abelian three-dimensional conformal algebra (C3) acting on spacelike hypersurfaces is presented. It is of Petrov type D; some properties of the family such as matter contents are given. This family turns out to be an extension of a solution recently given in [9] using completely different methods. The family contains Friedman-Lemaître-Robertson-Walker particular cases and could be useful as a test for the different FLRW perturbation schemes. There are two very interesting limiting cases, one with a non-abelian G2 and another with an abelian G2 acting non-orthogonally transitively on spacelike surfaces and with the fluid velocity non-orthogonal to the group orbits. No examples are known to the authors in these classes.