Juan Antonio Valiente Kroon

A conformal approach for the analysis of the non-linear stability of radiation cosmologies

Abstract The conformal Einstein equations for a trace-free (radiation) perfect fluid are derived in terms of the Levi-Civita connection of a conformally rescaled metric. These equations are used to provide a non-linear stability result for de Sitter-like trace-free (radiation) perfect fluid Friedman–Lemaitre–Robertson–Walker cosmological models. The solutions thus obtained exist globally towards the future and are future geodesically complete.

A new class of obstructions to the smoothness of null infinity

Expansions of the gravitational field arising from the development of asymptotically Euclidean, time symmetric, conformally flat initial data are calculated in a neighbourhood of spatial and null infinities up to order 6. To this end a certain representation of spatial infinity as a cylinder is used. This setup is based on the properties of conformal geodesics. It is found that these expansions suggest that null infinity has to be non-smooth unless the Newman-Penrose constants of the spacetime, and some other higher order quantities of the spacetime vanish. As a consequence of these results it is conjectured that similar conditions occur if one were to take the expansions to even higher orders. Furthermore, the smoothness conditions obtained suggest that if time symmetric initial data which are conformally flat in a neighbourhood of spatial infinity yield a smooth null infinity, then the initial data must in fact be Schwarzschildean around spatial infinity.Expansions of the gravitational field arising from the development of asymptotically Euclidean, time symmetric, conformally flat initial data are calculated in a neighbourhood of spatial and null infinities up to order 6. To this end a certain representation of spatial infinity as a cylinder is used. This setup is based on the properties of conformal geodesics. It is found that these expansions suggest that null infinity has to be non-smooth unless the Newman-Penrose constants of the spacetime, and some other higher order quantities of the spacetime vanish. As a consequence of these results it is conjectured that similar conditions occur if one were to take the expansions to even higher orders. Furthermore, the smoothness conditions obtained suggest that if time symmetric initial data which are conformally flat in a neighbourhood of spatial infinity yield a smooth null infinity, then the initial data must in fact be Schwarzschildean around spatial infinity.

Nonexistence of Conformally Flat Slices in Kerr and Other Stationary Spacetimes

It is proved that stationary solutions to the vacuum Einstein field equations with a nonvanishing angular momentum have no Cauchy slice that is maximal, conformally flat, and nonboosted. The proof is based on results coming from a certain type of asymptotic expansion near null and spatial infinity--which also show that the development of Bowen-York-type data cannot have a development admitting a smooth null infinity--and from the fact that stationary solutions do admit a smooth null infinity.

On the Construction of a Geometric Invariant Measuring the Deviation from Kerr Data

This article contains a detailed and rigorous proof of the construction of a geometric invariant for initial data sets for the Einstein vacuum field equations. This geometric invariant vanishes if and only if the initial data set corresponds to data for the Kerr spacetime, and thus, it characterises this type of data. The construction presented is valid for boosted and non-boosted initial data sets which are, in a sense, asymptotically Schwarzschildean. As a preliminary step to the construction of the geometric invariant, an analysis of a characterisation of the Kerr spacetime in terms of Killing spinors is carried out. A space spinor split of the (spacetime) Killing spinor equation is performed, to obtain a set of three conditions ensuring the existence of a Killing spinor of the development of the initial data set. In order to construct the geometric invariant, we introduce the notion of approximate Killing spinors. These spinors are symmetric valence 2 spinors intrinsic to the initial hypersurface and satisfy a certain second order elliptic equation —the approximate Killing spinor equation. This equation arises as the Euler-Lagrange equation of a non-negative integral functional. This functional constitutes part of our geometric invariant —however, the whole functional does not come from a variational principle. The asymptotic behaviour of solutions to the approximate Killing spinor equation is studied and an existence theorem is presented.

Does Asymptotic Simplicity Allow for Radiation Near Spatial Infinity

Abstract.A representation of spatial infinity based on the properties of conformal geodesics is used to obtain asymptotic expansions of the gravitational field near the region where null infinity touches spatial infinity. These expansions show that generic time symmetric initial data with an analytic conformal metric at infinity will give rise to developments with a certain type of logarithmic singularities at the points where null infinity and spatial infinity meet. These logarithmic singularities produce a non-smooth null infinity. The sources of the logarithmic singularities are traced back down to the initial data. It is shown that if the parts of the initial data responsible for the non-regular behaviour of the solutions are not present, then the initial data is static to a certain order. On the basis of these results it is conjectured that the only time symmetric data sets with developments having a smooth null infinity are those which are static in a neighbourhood of infinity. This conjecture generalises a previous conjecture regarding time symmetric, conformally flat data. The relation of these conjectures to Penrose’s proposal for the description of the asymptotic gravitational field of isolated bodies is discussed.

From geometry to numerics: interdisciplinary aspects in mathematical and numerical relativity

This paper reviews some aspects in the current relationship between mathematical and numerical general relativity. Focus is placed on the description of isolated systems, with a particular emphasis on recent developments in the study of black holes. Ideas concerning asymptotic flatness, the initial-value problem, the constraint equations, evolution formalisms, geometric inequalities and quasi-local black hole horizons are discussed in light of the interaction between numerical and mathematical relativists.

Time asymmetric spacetimes near null and spatial infinity: I. Expansions of developments of conformally flat data

The conformal Einstein equations and the representation of spatial infinity as a cylinder introduced by Friedrich are used to analyse the behaviour of the gravitational field near null and spatial infinity for the development of data which are asymptotically Euclidean, conformally flat and time asymmetric. Our analysis allows for initial data whose second fundamental form is more general than the one given by the standard Bowen–York ansatz. The conformal Einstein equations imply, upon evaluation on the cylinder at spatial infinity, a hierarchy of transport equations which can be used to calculate asymptotic expansions for the gravitational field in a recursive way. It is found that the solutions to these transport equations develop logarithmic divergences at the critical sets where null infinity meets spatial infinity. Associated with these, there is a series of quantities expressible in terms of the initial data (obstructions), which if zero, preclude the appearance of some of the logarithmic divergences. The obstructions are, in general, time asymmetric. That is, the obstructions at the intersection of future null infinity with spatial infinity are in general different from those obtained at the intersection of past null infinity with spatial infinity. The latter allows for the possibility of having spacetimes where future and past null infinity have different degrees of smoothness. Finally, it is shown that if both sets of obstructions vanish up to a certain order, then the initial data have to be asymptotically Schwarzschildean in a certain sense.

Conserved quantities for polyhomogeneous spacetimes

The existence of conserved quantities with a structure similar to the Newman-Penrose quantities in a polyhomogeneous spacetime is considered. The most general form for the initial data formally consistent with the polyhomogeneous setting is found. The subsequent study is done for those polyhomogeneous spacetimes where the leading term of the shear contains no logarithmic terms. It is found that for these spacetimes the original NP quantities cease to be constants, but it is still possible to construct a set of ten other quantities that are constant. From these quantities it is possible to obtain, as a particular case, the conserved quantity found by Chrusciel et al.

Geometric invariant measuring the deviation from Kerr data.

A geometrical invariant for regular asymptotically Euclidean data for the vacuum Einstein field equations is constructed. This invariant vanishes if and only if the data correspond to a slice of the Kerr black hole spacetime--thus, it provides a measure of the non-Kerr-like behavior of generic data. In order to proceed with the construction of the geometric invariant, we introduce the notion of approximate Killing spinors.

Time asymmetric spacetimes near null and spatial infinity: II. Expansions of developments of initial data sets with non-smooth conformal metrics

This paper uses the conformal Einstein equations and the conformal representation of spatial infinity introduced by Friedrich to analyse the behaviour of the gravitational field near null and spatial infinity for the development of initial data which are, in principle, non-conformally flat and time asymmetric. The paper is the continuation of the investigation started in Class. Quantum Grav. 21 (2004) 5457–92, where only conformally flat initial data sets were considered. For the purposes of this investigation, the conformal metric of the initial hypersurface is assumed to have a very particular type of non-smoothness at infinity in order to allow for the presence of non-Schwarzschildean stationary initial data sets in the class under study. The calculation of asymptotic expansions of the development of these initial data sets reveals—as in the conformally flat case—the existence of a hierarchy of obstructions to the smoothness of null infinity which are expressible in terms of the initial data. This allows for the possibility of having spacetimes where future and past null infinity have different degrees of smoothness. A conjecture regarding the general structure of the hierarchy of obstructions is presented.