Stationary solutions and asymptotic flatness I

In this article and its sequel we discuss the asymptotic structure of space-times representing isolated bodies in General Relativity. Such space-times are usually required to be asymptotically flat (AF), and thus to have a prescribed type of asymptotic. Despite all the "reasonable" that the requirement is, it seems to be against the spirit of General Relativity where the global structure of the space-time should be also considered as a variable. It is shown here that, even eliminating from the definition any a priori reference or assumption about the asymptotic, the space-times of isolated bodies are unavoidably and a posteriori AF. In precise terms, between the two articles it is proved that any vacuum strictly stationary space-time end whose (quotient) manifold is diffeomorphic to R^3 minus a ball and whose Killing field has its norm bounded away from zero is necessarily AF with Schwarzschidian fall off. The "excised" ball would contain (if any) the actual material body, but this information or any other is not necessary to reach the conclusion. Physical and mathematical implications are also discussed.Comment: The original submission was revised and divided in two: Stationary solutions and asymptotic flatness I & Stationary solutions and asymptotic flatness I

On the shape of bodies in General Relativistic regimes

The analysis of axisymmetric spacetimes, dynamical or stationary, is usually made in the reduced space. We prove here a stability property of the quo- tient space and use it together with minimal surface techniques to constraint the shape of General Relativistic bodies in terms of their energy and rotation. These constraints are different in nature to the mechanical limitations that a particular material body can have and which can forbid, for instance, rotation faster than a certain rate, (after which the body falls apart). The relations we are describing instead are fundamental and hold for all bodies, albeit they are useful only in General Relativistic regimes. For Neutron stars they are close to be optimal, and, although precise models for these stars display tighter con- straints, our results are significative in that they do not depend on the equation of state.Comment: 20 pages, 3 figure

General K=-1 Friedman-Lema\^itre models and the averaging problem in cosmology

We introduce the notion of general K=-1 Friedman-Lema\^itre (compact) cosmologies and the notion of averaged evolution by means of an averaging map. We then analyze the Friedman-Lema\^itre equations and the role of gravitational energy on the universe evolution. We distinguish two asymptotic behaviors: radiative and mass gap. We discuss the averaging problem in cosmology for them through precise definitions. We then describe in quantitative detail the radiative case, stressing on precise estimations on the evolution of the gravitational energy and its effect in the universe's deceleration. Also in the radiative case we present a smoothing property which tells that the long time H^{3} x H^{2} stability of the flat K=-1 FL models implies H^{i+1} x H^{i} stability independently of how big the initial state was in H^{i+1} x H^{i}, i.e. there is long time smoothing of the space-time. Finally we discuss the existence of initial "big-bang" states of large gravitational energy, showing that there is no mathematical restriction to assume it to be low at the beginning of time.Comment: Revised version. 32 pages, 1 figur

Stationary solutions and asymptotic flatness II

This is the second part of the investigation started in [Stationary solutions and asymptotic flatness I]. We prove here that Strongly Stationary ends having cubic volume growth are Weakly Asymptotically Flat. Combined with the results of the previous paper this shows that Strongly Stationary ends are Asymptotically Flat with Schwarzschidian fall off

On static solutions of the Einstein - Scalar Field equations

In this article we study self-gravitating static solutions of the Einstein-ScalarField system in arbitrary dimensions. We discuss the existence and the non-existence of geodesically complete solutions depending on the form of the scalar field potential $V(\phi)$, and provide full global geometric estimates when the solutions exist. Our main results are summarised as follows. For the Klein-Gordon field, namely when $V(\phi)=m^{2}|\phi|^{2}$, it is proved that geodesically complete solutions have Ricci-flat spatial metric, have constant lapse and are vacuum, (that is $\phi$ is constant and equal to zero if $m\neq 0$). In particular, when the spatial dimension is three, the only such solutions are either Minkowski or a quotient thereof (no nontrivial solutions exist). When $V(\phi)=m^{2}|\phi|^{2}+2\Lambda$, that is, when a vacuum energy or a cosmological constant is included, it is proved that no geodesically complete solution exists when $\Lambda>0$, whereas when $\Lambda<0$ it is proved that no non-vacuum geodesically complete solution exists unless $m^{2}<-2\Lambda/(n-1)$, ($n$ is the spatial dimension) and the spatial manifold is non-compact. The proofs are based on techniques in comparison geometry \'a la Backry-Emery.Comment: Introduction changed and small application to geons remove