The linear stability of the Schwarzschild solution to gravitational perturbations

Understand

We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to a linearised Kerr metric.

• We express the equations in a suitable double null gauge.
• To obtain decay, one must in fact add a residual pure gauge solution which we prove to be itself quantitatively controlled from initial data.
• Our result a fortiori includes decay statements for general solutions of the Teukolsky equation (satisfied by gauge-invariant null-decomposed curvature components).