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

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.

Characteristic extraction in numerical relativity: binary black hole merger waveforms at null infinity

“…4.3 of ref. [47] for the case of a dynamic spacetime on a Minkowski background with = 2, m = 0. We write…”

Simple, explicitly time-dependent, and regular solutions of the linearized vacuum Einstein equations in Bondi-Sachs coordinates

“…Bishop and collaborators [18,19] proposed a procedure to find linearized perturbations on a Schwarzschild background. However, in their approach "...some of the expressions get very complicated and in order to simplify the presentation we now specialize to the case l = 2..." [18] to obtain the solution is overly complicated, although the calculation of linearized perturbations is achievable by a simpler method. Nevertheless, Reisswig et al [20] determined a l = 3 multipole using Bishop's approach.…”

The Transformation of a Schwarzschild Black Hole Linear Perturbations to Bondi Frame

“…The above construction was not made in [16] because in that work it was possible to neglect the imaginary component in J and U. However, we shall see that it is not the case for odd-parity perturbations.…”