The Nonlinear Stability of the Schwarzschild Metric Without Symmetry - Mihalis Dafermos

Explore a comprehensive lecture on the nonlinear stability of the Schwarzschild metric without symmetry, delivered by Mihalis Dafermos from Princeton University at the Institute for Advanced Study. Delve into advanced topics in mathematical physics and analysis, including the Schwarzschild family, Kerr spacetime, and linear stability of Schwarzschild. Examine the main theorems, gauge invariant quantities, and the Einstein equations in double null gauge. Gain insights into the boundedness and decay for Teukolsky, wave equations on Kerr, and the proof of decay for Teukolsky on Schwarzschild. Investigate the transition from linear to nonlinear stability, reformulation of the linear stability theorem, and the challenges associated with nonlinear difficulties.

Intro
The nonlinear stability of the Schwarzschild metric without symmetry
Plan of the Lecture
The Schwarzschild family
The stability question
The main theorem (first version)
Kerr spacetime
The main theorem (second version)
Linear stability of Schwarzschild
The linear theorem in double null gauge
Aside: the linear theorem in (generalised) harmonic gauge
Step 1: Gauge invariant quantities
Boundedness and decay for Teukolsky
Wave equation on Kerr
Aside: mode stability for wave and Teukolsky on la s M
Proof of decay for Teukolsky on Schwarzschild
Aside: proof of decay for Teukolsky on Kerr
Step 2: from Teukolsky to the full system
The Einstein equations in double null gauge
The linearised equations
Recall: What does "linear stability" mean?
The Schwarzschild and fixed mass Kerr modes
Reformulation of the linear stability theorem
Nonlinear difficulties
role of double null gauge