Regularity of a double null coordinate system for Kerr–Newman–de Sitter spacetimes

We construct a double null coordinate system \((u,v,\theta _\star ,\phi _\star )\) $(u,v,{\theta }_{\star },{\varphi }_{\star })$ for Kerr–Newman–de Sitter black hole interior spacetimes and prove that the two dimensional spheres given by the intersection of the hypersurfaces \(u=\hbox {constant}\) $u=\text{constant}$ and \(v=\hbox {constant}\) $v=\text{constant}$ are \(C^\infty\) $C^{\infty }$ in Boyer–Lindquist coordinates (including at the “poles"). The null coordinates allow one to immediately extend some results previously proven for Kerr. As an example, we illustrate how Sbierski’s result in (On the initial value problem in general relativity and wave propagation in black-hole spacetimes. Doctoral thesis, 2014. https://doi.org/10.17863/CAM.16140), for the wave equation on the black hole interior, for Reissner–Nordström and Kerr spacetimes, applies to Kerr–Newman–de Sitter spacetimes.