We consider solutions to the linear wave equation g D 0 on a (maximally extended) Schwarzschild spacetime with parameter M > 0, evolving from sufficiently regular initial data prescribed on a complete Cauchy surface †, where the data are assumed only to decay suitably at spatial infinity. (In particular, the support of may contain the bifurcate event horizon.) It is shown that the energy flux F Q T .S/ of the solution (as measured by a strictly timelike Q T that asymptotically matches the static Killing field) through arbitrary achronal subsets S of the black hole exterior region satisfies the bound, where v and u denote the infimum of the Eddington-Finkelstein advanced and retarded time of S, v C denotes maxf1; vg, and u C denotes maxf1; ug, where C is a constant depending only on the parameter M , and E depends on a suitable norm of the solution on the hypersurface t :(The bound applies in particular to subsets S of the event horizon or null infinity.) It is also shown that satisfies the pointwise decay estimate j j Ä CEv 1 C in the entire exterior region, and the estimates jr j Ä C Q R E.1 C juj/ 1=2 and jr 1=2 j Ä C Q R Eu 1 C in the region fr Q Rg \ J C . †/ for any Q R > 2M . The estimates near the event horizon exploit an integral energy identity normalized to local observers. This estimate can be thought to quantify the celebrated red-shift effect. The results in particular give an independent proof of the classical result j j Ä CE of Kay and Wald without recourse to the discrete isometries of spacetime.

This paper concludes the series begun in [M. Dafermos and I. Rodnianski Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: the cases a ≪ M or axisymmetry, arXiv:1010.5132], providing the complete proof of definitive boundedness and decay results for the scalar wave equation on Kerr backgrounds in the general subextremal a < M case without symmetry assumptions. The essential ideas of the proof (together with explicit constructions of the most difficult multiplier currents) have been announced in our survey [M. Dafermos and I. Rodnianski The black hole stability problem for linear scalar perturbations, in Proceedings of the 12th Marcel Grossmann Meeting on General Relativity, T. Damour et al (ed.), World Scientific, Singapore, , pp. 132189, arXiv:1010]. Our proof appeals also to the quantitative mode-stability proven in [Y. Shlapentokh-Rothman Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime, arXiv:1302.6902, to appear, Ann. Henri Poincaré], together with a streamlined continuity argument in the parameter a, appearing here for the first time. While serving as Part III of a series, this paper repeats all necessary notations so that it can be read independently of previous work.

Note that the mode analysis corresponding to formulation 1. described above yields necessary but not sufficient conditions for either statements (2a) and (2b) of true linear stability.( 1 ) In the case of the linear scalar wave equation g ϕ = 0, which can be thought of as a "poor man's" version of linearised gravity, the analogue of (2a) for Schwarzschild was proven by , and the analogue of (2a) and ( 2b) are shown now for the full subextremal Kerr family in [26], following a host of recent activity [24], [73], [1], [21]. See [25], [23] for a survey. See [6], [2] for generalisations to the Maxwell equations and [3] for a discussion of the extremal case |a|=M . Concerning the linearised Einstein equations themselves, work on the wave equation easily generalises to establish physical space decay on certain quantities, for instance those gauge-invariant quantities satisfying the Regge-Wheeler equation on Schwarzschild [32], [7], [29]. For the full system of linearised gravity however, both problems (2a) and ( 2b) have remained open until today. We note explicitly that even the question of uniform boundedness, let alone decay, for the gauge-invariant quantities satisfying the Teukolsky equation on Schwarzschild has remained open.(3) The full non-linear stability of Schwarzschild as a solution to the Cauchy problem for the non-linear Einstein vacuum equations (2). This is the definitive formulation of the fundamental question. See our previous [18] for a precise statement of the conjecture in the language of the Cauchy problem for (2). In analogy with 2. above, one could distinguish between questions of (3a) orbital stability and (3b) asymptotic stability. Experience from non-linear problems, however, in particular the proof of the non-linear stability of Minkowski space [14] referred to earlier (see also [47], [5]), indicates that (3a) and (3b) are naturally coupled.( 2 ) Since non-linear stability is thus necessarily a question of asymptotic stability, the "Schwarzschild" problem is more correctly re-phrased as the non-linear asymptotic stability of the Kerr family in a neighbourhood of Schwarzschild. For even if one restricts to small perturbations of Schwarzschild, it is expected that generically, spacetime dynamically asymptotes to a very slowly rotating Kerr solution with a =0. Since in the context of a non-linear stability proof, one effectively must "linearise" around the solution one expects to approach, this suggests that to resolve the full ( 1 ) Thus, the mode analysis can be an effective tool to show instability, but never, on its own, stability. For instability results for related problems proven via the existence of unstable modes, see [70], [28] and references therein. See also discussion in [78].( 2 ) This coupling arises from the super-criticality of the Einstein vacuum equations (2). Note that under spherical symmetry (where the vacuum equations must be replaced, however, by a suitable Einstein-matter system to restore a dynamical degree of freedom) this super-criticality is broken in the presence o...