Mathematics

The (linear) Klein-Gordon equation on a Schwarzschild background combines the most fundamental model for massive matter evolving on the simplest type of black hole. Yet, unlike massless fields (the wave equation) which are now well-understood on a black hole, the large-time asymptotics of Klein-Gordon solutions have long remained elusive.Physically, the Klein-Gordon dynamics presents a geometric obstruction: the stable trapping of massive particles, which was conjectured to prevent decay. In this talk, wedisprove this expectation and establish that solutions with localized initial data actually decay polynomially in time. We will highlight the proof's most surprising ingredient: the use of analytic number theory--specifically, bounds on exponential sums similar to the Riemann zeta function.