Linearized solutions of the Einstein equations within a Bondi-Sachs framework, and implications for boundary conditions in numerical simulations

TL;DR

This paper derives exact linearized solutions of Einstein's equations in Bondi-Sachs coordinates for Schwarzschild and Minkowski backgrounds with a dynamic thin shell, aiding numerical relativity boundary conditions and gravitational wave analysis.

Contribution

It provides explicit linearized solutions in Bondi-Sachs form for Schwarzschild and Minkowski backgrounds with matter sources, useful for numerical relativity boundary conditions and gravitational wave comparisons.

Findings

Derived exact solutions for linearized Einstein equations in Bondi-Sachs coordinates.

Reduced the problem to solvable linear ordinary differential equations.

Solutions applicable for boundary conditions at the Schwarzschild horizon.

Abstract

We linearize the Einstein equations when the metric is Bondi-Sachs, when the background is Schwarzschild or Minkowski, and when there is a matter source in the form of a thin shell whose density varies with time and angular position. By performing an eigenfunction decomposition, we reduce the problem to a system of linear ordinary differential equations which we are able to solve. The solutions are relevant to the characteristic formulation of numerical relativity: (a) as exact solutions against which computations of gravitational radiation can be compared; and (b) in formulating boundary conditions on the $r=2M$ Schwarzschild horizon.