Traveling to Gargantua

This talk offers a guided journey through different spacetimes, introducing the geometric and algebraic structures that govern wave propagation in curved backgrounds. The path begins with the familiar Minkowski spacetime and leads to the near-horizon region of rotating black holes. Massless fields, such as electromagnetic or gravitational perturbations, around rotating black holes are described by the linear partial differential equation known as the Teukolsky equation, whose solutions are expressed in terms of Heun functions. For higher order perturbations, the self-interactions arising from non-linear terms require the systematic use of products, derivatives, and integrals of holonomic functions. I will argue that such higher-order analyses are essential for understanding gravitational turbulence in the vicinity of black holes. No prior knowledge of differential geometry or general relativity is assumed.