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

TL;DR

This paper derives new solutions to the master equation in the linear characteristic formulation of general relativity, applicable to any multipolar order and background, expanding previous specific solutions and confirming their validity.

Contribution

It provides generalized solutions for the master equation for all multipolar moments in Minkowski and Schwarzschild backgrounds, including matter sources, using Bessel and Heun functions.

Findings

New solutions for all multipolar moments $l$ in Minkowski background.

Solutions expressed in terms of Bessel functions for Minkowski case.

Solutions expressed in terms of Confluent Heun functions for Schwarzschild case.

Abstract

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