Christian Lübbe

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.

An extension theorem for conformal gauge singularities

We analyze conformal gauge, or isotropic, singularities in cosmological models in general relativity. Using the calculus of tractors, we find conditions in terms of tractor curvature for a local extension of the conformal structure through a cosmological singularity and prove a local extension theorem along a congruence of timelike conformal geodesics.

On de Sitter-like and Minkowski-like spacetimes

Friedrichs proofs for the global existence results of de Sitter-like spacetimes and of semi-global existence of Minkowski-like spacetimes (Friedrich 1986 Commun. Math. Phys. 107 587) are re-examined and discussed, making use of the extended conformal field equations and a gauge based on conformal geodesics. In this gauge, the location of the conformal boundary of the spacetimes is known a priori once the initial data have been prescribed. Thus, it provides an analysis which is conceptually and calculationally simpler.

The extended Conformal Einstein field equations with matter: the Einstein-Maxwell field

Abstract A discussion is given of the conformal Einstein field equations coupled with matter whose energy–momentum tensor is trace-free. These resulting equations are expressed in terms of a generic Weyl connection. The article shows how in the presence of matter it is possible to construct a conformal gauge which allows to know a priori the location of the conformal boundary. In vacuum this gauge reduces to the so-called conformal Gaussian gauge. These ideas are applied to obtain (i) a new proof of the stability of Einstein–Maxwell de Sitter-like spacetimes; (ii) a proof of the semi-global stability of purely radiative Einstein–Maxwell spacetimes.

A Class of Conformal Curves in the Reissner–Nordström Spacetime

A class of curves with special conformal properties (conformal curves) is studied on the Reissner–Nordström spacetime. It is shown that initial data for the conformal curves can be prescribed so that the resulting congruence of curves extends smoothly to future and past null infinity. The formation of conjugate points on these congruences is examined. The results of this analysis are expected to be of relevance for the discussion of the Reissner–Nordström spacetime as a solution to the conformal field equations and for the global numerical evaluation of static black hole spacetimes.A class of curves with special conformal properties (conformal curves) is studied on the Reissner–Nordstrom spacetime. It is shown that initial data for the conformal curves can be prescribed so that the resulting congruence of curves extends smoothly to future and past null infinity. The formation of conjugate points on these congruences is examined. The results of this analysis are expected to be of relevance for the discussion of the Reissner–Nordstrom spacetime as a solution to the conformal field equations and for the global numerical evaluation of static black hole spacetimes.

A STABILITY RESULT FOR PURELY RADIATIVE SPACETIMES

A semiglobal stability result for a class of purely radiative vacuum spacetimes arising from hyperboloidal data is given. This result generalizes semiglobal existence results for Minkowski-like spacetimes to the case where the reference solution contains gravitational radiation. The analysis makes use of the extended conformal field equations and a gauge based on conformal geodesics so that the location and structure of the conformal boundary of the perturbed solutions is known a priori.

Spherically symmetric anti–de Sitter-like Einstein-Yang-Mills spacetimes

The conformal field equations are used to discuss the local existence of spherically symmetric solutions to the Einstein-Yang-Mills system which behave asymptotically like the anti-de Sitter spacetime. By using a gauge based on conformally privileged curves we obtain a formulation of the problem in terms of an initial boundary value problem on which a general class of maximally dissipative boundary conditions can be discussed. The relation between these boundary conditions and the notion of mass on asymptotically anti-de Sitter spacetimes is analysed.

A conformal extension theorem based on null conformal geodesics

In this article we describe the formulation of null geodesics as null conformal geodesics and their description in the tractor formalism. A conformal extension theorem through an isotropic singularity is proven by requiring the boundedness of the tractor curvature and its derivatives to sufficient order along a congruence of null conformal geodesic. This article extends earlier work by Lubbe and Tod, J. Math. Phys. 50, 112501 (2009) (“An extension theorem for conformal gauge singularities,” e-print arXiv:0710.5552).In this article we describe the formulation of null geodesics as null conformal geodesics and their description in the tractor formalism. A conformal extension theorem through an isotropic singularity is proven by requiring the boundedness of the tractor curvature and its derivatives to sufficient order along a congruence of null conformal geodesic. This article extends earlier work by Tod and Lübbe [9].

On the conformal structure of the extremal Reissner-Nordstrom spacetime

We analyse various conformal properties of the extremal Reissner?Nordstr?m spacetime. In particular, we obtain conformal representations of the neighbourhoods of spatial infinity, time-like infinity and the cylindrical end?the so-called cylinders at spatial infinity and at the horizon, respectively?which are regular with respect to the conformal Einstein field equations and their associated initial data sets. We discuss possible implications of these constructions for the propagation of test fields and non-linear perturbations of the gravitational field close to the horizon.

The Conformal Einstein Field Equations for Trace-free Perfect Fluids

A nonlinear stability analysis is carried out for the trace-free (radiation) perfect fluid Friedmann-Lemaitre-Robertson-Walker models with a de Sitter-like cosmological constant. It is shown that the solutions close to the above FLRW spacetimes exist globally towards the future and are future geodesically complete. For this analysis we formulate the conformal Einstein field equations for a trace-free (radiation) perfect fluid in terms of the Levi-Civita connection of a conformally rescaled metric.