Massive Scalar Fields on Black Hole Spacetimes

After a brief discussion of the motivation for the study of massive scalar fields in general relativity and in relation to black holes, I will recall some old results concerning situations where one expects instabilities for massive scalar fields and then discuss a new result concerning the case of a spherically symmetric black hole where one can prove decay for the scalar field. This final decay result is joint with Federico Pasqualotto and Maxime Van de Moortel.