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.

We give a new proof of the global stability of Minkowski space originally established in the vacuum case by Christodoulou and Klainerman. The new approach, which relies on the classical harmonic gauge, shows that the Einstein-vacuum and the Einstein-scalar field equations with asymptotically flat initial data satisfying a global smallness condition produce global (causally geodesically complete) solutions asymptotically convergent to the Minkowski space-time.

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.

A well-known open problem in general relativity, dating back to 1972, has been to prove Price's law for an appropriate model of gravitational collapse. This law postulates inverse-power decay rates for the gravitational radiation flux on the event horizon and null infinity with respect to appropriately normalized advanced and retarded time coordinates. It is intimately related both to astrophysical observations of black holes and to the fate of observers who dare cross the event horizon. In this paper, we prove a well-defined (upper bound) formulation of Price's law for the collapse of a self-gravitating scalar field with spherically symmetric initial data. We also allow the presence of an additional gravitationally coupled Maxwell field. Our results are obtained by a new mathematical technique for understanding the long-time behavior of large data solutions to the resulting coupled non-linear hyperbolic system of p.d.e.'s in 2 independent variables. The technique is based on the interaction of the conformal geometry, the celebrated red-shift effect, and local energy conservation; we feel it may be relevant for the problem of non-linear stability of the Kerr solution. When combined with previous work of the first author concerning the internal structure of charged black holes, which assumed the validity of Price's law, our results can be applied to the strong cosmic censorship conjecture for the Einstein-Maxwell-real scalar field system with complete spacelike asymptotically flat spherically symmetric initial data. Under Christodoulou's C 0 -formulation, the conjecture is proven to be false.

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...

The nonlinear Hartree equation describes the macroscopic dynamics of initially factorized Nboson states, in the limit of large N . In this paper we provide estimates on the rate of convergence of the microscopic quantum mechanical evolution towards the limiting Hartree dynamics. More precisely, we prove bounds on the difference between the one-particle density associated with the solution of the N -body Schrödinger equation and the orthogonal projection onto the solution of the Hartree equation.1 with normalization ϕ L 2 (R 3 ) = 1 (so that ψ N L 2 (R 3N ) = 1) and we study its time-evolution ψ N,t , given by the solution of the N body Schrödinger equationIn (1.2) and in what follows we use the notation x = (x 1 , . . . , x N ) ∈ R 3N .Clearly, because of the interaction among the particles, the factorization of the wave function is not preserved by the time evolution. However, due to the presence of the small constant 1/N in front of the potential energy in (1.1), we may expect the total potential experienced by each particle to be approximated, for large N , by an effective mean field potential, and thus that, in the limit N → ∞, the solution ψ N,t of (1.3) is still approximately (and in an appropriate sense) factorized. We may expect, in other words, that in an appropriate sensewhere ϕ t is a solution of the nonlinear Hartree equation (1.5). The convergence (1.7) holds in the trace norm topology. In particular, (1.7) implies that for arbitrary k and for an arbitrary bounded operator J (k) on L 2 (R 3k ), ψ N,t , J (k) ⊗ 1 (N −k) ψ N,t → ϕ ⊗k t , J (k) ϕ ⊗k t

These lecture notes, based on a course given at the Zürich Clay Summer School (June 23-July 18 2008), review our current mathematical understanding of the global behaviour of waves on black hole exterior backgrounds. Interest in this problem stems from its relationship to the non-linear stability of the black hole spacetimes themselves as solutions to the Einstein equations, one of the central open problems of general relativity. After an introductory discussion of the Schwarzschild geometry and the black hole concept, the classical theorem of Kay and Wald on the boundedness of scalar waves on the exterior region of Schwarzschild is reviewed. The original proof is presented, followed by a new more robust proof of a stronger boundedness statement. The problem of decay of scalar waves on Schwarzschild is then addressed, and a theorem proving quantitative decay is stated and its proof sketched. This decay statement is carefully contrasted with the type of statements derived heuristically in the physics literature for the asymptotic tails of individual spherical harmonics. Following this, our recent proof of the boundedness of solutions to the wave equation on axisymmetric stationary backgrounds (including slowly-rotating Kerr and Kerr-Newman) is reviewed and a new decay result for slowly-rotating Kerr spacetimes is stated and proved. This last result was announced at the summer school and appears in print here for the first time. A discussion of the analogue of these problems for spacetimes with a positive cosmological constant Λ > 0 follows. Finally, a general framework is given for capturing the red-shift effect for nonextremal black holes. This unifies and extends some of the analysis of the previous sections. The notes end with a collection of open problems.

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