On the linear regime of the characteristic formulation of general relativity in the Minkowski and Schwarzschild's backgrounds - Carlos Eduardo Cedeño Montaña (2016)
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 Minkowskis background, and in terms of the Heuns function in the Schwarzschilds 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 Minkowskis background are shown as particular cases of this formalism.