I will present the first definitive scattering theory for linear waves (\Box_g \psi =0) on the interior of Reissner--Nordström black holes, which can be considered as the most elementary black holes in the context of scattering. These spacetimes admit two Killing horizons, the event and Cauchy horizon, and we investigate how incoming physical data on the event horizon evolve and are scattered to the Cauchy horizon. I will show the heart of the proof of the scattering theory which is the uniform boundedness of the reflection and transmission coefficients of the resulting radial o.d.e. after separation of variables. In the physical space picture, this translates into the fact that the Cauchy evolution from the event horizon to the Cauchy horizon is a Hilbert space isomorphism. The Hilbert spaces are defined by the degenerate T energy fluxes on the horizons.

Finally, I will outline that, in contrast to the above, for a generic set of cosmological constants, there is no T energy scattering theory for either the linear wave equation or the Klein--Gordon equation with conformal mass on the (anti-) de Sitter--Reissner--Nordström interior. This is joint work with Y. Shlapentokh-Rothman.