Hyperboloidal Approach to Quasinormal Modes

Oscillations of black hole spacetimes exhibit divergent behavior toward the bifurcation sphere and spatial infinity. This divergence can be understood as a consequence of the geometry in these spacetime regions. In contrast, black-hole oscillations are regular when evaluated toward the event horizon and null infinity. Hyperboloidal surfaces naturally connect these regions, providing a geometric regularization of time-harmonic oscillations called quasinormal modes (QNMs). This review traces the historical development of the hyperboloidal approach to QNMs. We discuss the physical motivation for the hyperboloidal approach and highlight current developments in the field.