Linear stability of Schwarzschild under perturbations which are non-vanishing on the bifurcation 2-sphere

The authors prove boundedness on an exterior Schwarzschild wedge for C infinity solutions of the covariant Klein-Gordon equation which have compact support on Cauchy surfaces in Kruskal spacetime. Previously used methods enable such boundedness to be proven only for solutions whose initial data satisfy the additional restriction of vanishing at the bifurcation 2-sphere of the horizon. By employing a rarely considered discrete isometry of Kruskal spacetime and the causal propagation property of the equation, they remove this restriction. This also enables them to prove boundedness exterior to the horizon of a spacetime representing the collapse to a black hole of a spherically symmetric compact star for solutions of the same equation having C infinity initial data on a Cauchy surface drawn prior to the collapse.