Maxwell Equations Solution Using Spin-weighted Spherical Harmonics

The electric and magnetic fields in vacuum, in a source-free region, are divergenceless and, if it is assumed that they have a harmonic time dependence with frequency ω, they satisfy the vector Helmholtz equation. We can then solve the Helmholtz equation using the spin-weighted spherical harmonics. However a lot of constants produced in the process have to be determined. In this work we find these constants corresponding to the electric and magnetic fields, i.e. we obtain the multipole solution of Maxwell equations in a rigorous manner. In addition we give an Expression for the vector potential in terms of the multipole moments