Determining the extremality of Reissner Nordstrom Black-Holes using late-time tails at null infinity.

Extreme Reissner-Nordström black holes perturbed by a massless scalar field have permanent scalar hair that serve as candidates for black hole (BH) hair, in possible violation of the no-hair conjecture in linearized theory. This scalar hair for extreme Reissner-Nordström (ERN) BHs involves a certain quantity s[ψ] evaluated at future null infinity (I+) and equals a nonvanishing constant quantity H[ψ]—so-called Aretakis charge. This is calculated on the BH's event horizon (EH), but vanishes if the BH is nonextreme. We write the 1+1 dimensional scalar wave equation in ERN backgrounds for azimuthal (m = 0) modes in compactified hyperboloidal coordinates such that I+ is included in the computational domain at a finite radial coordinate . We re-write the second-order hyperbolic partial differential equation as a coupled system of two first-order hyperbolic equations and solve this system numerically. I also describe how we can improve on these results by implementing a high accuracy discontinuous galerkin (DG) scheme to solve the wave equation.