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

TL;DR

This paper derives simple, explicit, time-dependent solutions to the linearized vacuum Einstein equations on a null cone, useful for numerical relativity and understanding gravitational wave propagation.

Contribution

It introduces two explicit solutions to the master wave equation in the Bondi-Sachs formulation, including an asymptotically flat one, with detailed analysis of their properties and behavior at null infinity.

Findings

One solution is asymptotically flat and regular at the cone vertex.

The solutions describe all multipoles of spin-2 gravitational waves.

The asymptotically flat solution's behavior at null infinity is characterized by the Weyl scalar _4.

Abstract

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 \Psi_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, \Psi_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.