arXiv subjects

Thomas Mädler

Publications and source records attributed to Thomas Mädler.

20 records · Page 2Linked to original sources

arXiv2013

Simple, explicitly time-dependent and regular solutions of the linearized vacuum Einstein equations on a null cone

Perturbations of the linearized vacuum Einstein equations on a null cone in the Bondi-Sachs formulation of General Relativity can be derived from a single master function with spin weight two, which is related to the Weyl scalar Ψ_0, and which is determined by a simple wave equation. Utilizing a standard spin representation of the tensors on a sphere and two different approaches to solve the master equation, we are able to determine two simple and explicitly time-dependent solutions. Both solutions, of which one is asymptotically flat, comply with the regularity conditions at the vertex of the null cone. For the asymptotically flat solution we calculate the corresponding linearized perturbations, describing all multipoles of spin-2 waves that propagate on a Minkowskian background spacetime. We also analyze the asymptotic behavior of this solution at null infinity using a Penrose compactification, and calculate the Weyl scalar, Ψ_4. Because of its simplicity, the asymptotically flat solution presented here is ideally suited for testbed calculations in the Bondi-Sachs formulation of numerical relativity. It may be considered as a sibling of the well-known Teukolsky-Rinne solutions, on spacelike hypersurfaces, for a metric adapted to null hypersurfaces.

gr-qc↗

arXiv2013

The Bondi-Sachs metric at the vertex of a null cone: axially symmetric vacuum solutions

In the Bondi-Sachs formulation of General Relativity space-time is foliated via a family of null cones. If these null cones are defined such that their vertices are traced by a regular world-line then the metric tensor has to obey regularity conditions at the vertices. We explore these regularity conditions when the world line is a time-like geodesic. In particular, we solve the Einstein equations for the Bondi-Sachs metric near the vertices for axially symmetric vacuum space- times. The metric is calculated up to third order corrections with respect to a flat metric along the time-like geodesic, as this is the lowest order where non- linear coupling of the metric coefficients occurs. We also determine the boundary conditions of the metric to arbitrary order of these corrections when a linearized and axially symmetric vacuum space-time is assumed. In both cases we find that (i) the initial data on the null cone must have a very rigid angular structure for the vertex to be a regular point, and (ii) the initial data are determined by functions depending only on the time of a geodesic observer tracing the vertex. The latter functions can be prescribed freely, but if the vertex is assumed to be regular they must be finite and have finite derivatives along the geodesic.

gr-qc↗