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

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 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.

accepted for publication in Phys. Rev. D

References in corpus (4)

• Strategies for the Characteristic Extraction of Gravitational Waveforms
• Explicit solution of the linearized Einstein equations in TT gauge for all multipoles
• Quasi-Normal Modes of a Schwarzschild White Hole
• The Bondi-Sachs metric at the vertex of a null cone: axially symmetric vacuum solutions

Cited by in corpus (9)

• Gravitational waves in a de Sitter universe
• Affine-null metric formulation of General Relativity at two intersecting null hypersurfaces
• Characteristic Formulation for Metric $f(R)$ Gravity
• Slowly rotating Kerr metric derived from the Einstein equations in affine-null coordinates
• Master equation solutions in the linear regime of characteristic formulation of general relativity
• Quasinormal modes of a Schwarzschild black hole within the Bondi-Sachs framework
• Characteristic initial value problems for the Einstein-Maxwell-scalar field equations in spherical symmetry
• Gravitational radiation by point particle eccentric binary systems in the linearised characteristic formulation of general relativity
• Plane fronted electromagnetic waves and an asymptotic limit of Liénard--Wiechert fields