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From the field equations in the linear regime of the characteristicformulation of general relativity, Bishop, for a Schwarzschild's background,and M\"adler, for a Minkowski's background, were able to show that it ispossible to derive a fourth order ordinary differential equation, called masterequation, for the $J$ metric variable of the Bondi-Sachs metric. Once $\beta$,another Bondi-Sachs potential, is obtained from the field equations, and $J$ isobtained from the master equation, the other metric variables are solvedintegrating directly the rest of the field equations. In the past, the masterequation was solved for the first multipolar terms, for both the Minkowski'sand Schwarzschild's backgrounds. Also, M\"adler recently reported ageneralisation of the exact solutions to the linearised field equations when aMinkowski's background is considered, expressing the master equation family ofsolutions for the vacuum in terms of Bessel's functions of the first and thesecond kind. Here, we report new solutions to the master equation for anymultipolar moment $l$, with and without matter sources in terms only of thefirst kind Bessel's functions for the Minkowski, and in terms of the ConfluentHeun's functions (Generalised Hypergeometric) for radiative (nonradiative) casein the Schwarzschild's background. We particularize our families of solutionsfor the known cases for $l=2$ reported previously in the literature and findcomplete agreement, showing the robustness of our results.

Master equation solutions in the linear regime of characteristic formulation of general relativity