On the Linear Regime of the Characteristic formulation of General relativity in the Minkowski and Schwarzschild's Backgrounds

TL;DR

This paper develops a linear approximation of Einstein's equations using the characteristic formulation, solving for metric variables in Minkowski and Schwarzschild backgrounds with applications to binary systems.

Contribution

It introduces a simplified decoupling method for the equations, deriving explicit solutions in terms of Bessel and Heun functions for different backgrounds.

Findings

Derived a master equation for metric variable J.

Solved the master equation using Bessel and Heun functions.

Applied the formalism to binary systems in Minkowski background.

Abstract

We present here the linear regime of the Einstein's field equations in the characteristic formulation. Through a simple decomposition of the metric variables in spin-weighted spherical harmonics, the field equations are expressed as a system of coupled ordinary differential equations. The process for decoupling them leads to a simple equation for $J$ - one of the Bondi-Sachs metric variables - known in the literature as the master equation. Then, this last equation is solved in terms of Bessel's functions of the first kind for the Minkowski's background, and in terms of the Heun's function in the Schwarzschild's case. In addition, when a matter source is considered, the boundary conditions across the time-like world tubes bounding the source are taken into account. These boundary conditions are computed for all multipole modes. Some examples as the point particle binaries in circular and eccentric orbits, in the Minkowski's background are shown as particular cases of this formalism.