text: We prove the global non-linear stability, without symmetry assumptions, of slowly rotating charged black holes in de Sitter spacetimes in the context of the initial value problem for the Einstein-Maxwell equations: If one perturbs the initial data of a slowly rotating Kerr-Newman-de Sitter (KNdS) black hole, then in a neighborhood of the exterior region of the black hole, the metric and the electromagnetic field decay exponentially fast to their values for a possibly different member of the KNdS family. This is a continuation of recent work of the author with Vasy on the stability of the Kerr-de Sitter family for the Einstein vacuum equations. Our non-linear iteration scheme automatically finds the final black hole parameters as well as the gauge in which the global solution exists; we work in a generalized wave coordinate Lorenz gauge, with gauge source functions lying in a suitable finite-dimensional space. We include a self-contained proof of the linear mode stability of Reissner-Nordstrom-de Sitter black holes, building on work by Kodama-Ishibashi. In the course of our non-linear stability argument, we also obtain the first proof of the linear (mode) stability of slowly rotating KNdS black holes using robust perturbative techniques.
  @cite_14: Energy and decay estimates for the wave equation on the exterior region of slowly rotating Kerr spacetimes are proved. The method used is a generalisation of the vector-field method that allows the use of higher-order symmetry operators. In particular, our method makes use of the second-order Carter operator, which is a hidden symmetry in the sense that it does not correspond to a Killing symmetry of the spacetime.
  @cite_107: 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 through 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 had 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 C0-formulation, the conjecture is proven to be false.
  @cite_78: The authors prove boundedness on an exterior Schwarzschild wedge for Cinfinity 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 Cinfinity initial data on a Cauchy surface drawn prior to the collapse.
  @cite_60: 
  @cite_28: 
  @cite_48: We study a semilinear equation with derivatives satisfying a null condition on slowly rotating Kerr spacetimes. We prove that given sufficiently small initial data, the solution exists globally in time and decays with a quantitative rate to the trivial solution. The proof uses the robust vector field method. It makes use of the decay properties of the linear wave equation on Kerr spacetime, in particular the � !
  @cite_22: 
  @cite_20: 
  @cite_17: 
  @cite_39: We construct a large class of dynamical vacuum black hole spacetimes whose exterior geometry asymptotically settles down to a fixed Schwarzschild or Kerr metric. The construction proceeds by solving a backwards scattering problem for the Einstein vacuum equations with characteristic data prescribed on the event horizon and (in the limit) at null infinity. The class admits the full "functional" degrees of freedom for the vacuum equations, and thus our solutions will in general possess no geometric or algebraic symmetries. It is essential, however, for the construction that the scattering data (and the resulting solution spacetime) converge to stationarity exponentially fast, in advanced and retarded time, their rate of decay intimately related to the surface gravity of the event horizon. This can be traced back to the celebrated redshift effect, which in the context of backwards evolution is seen as a blueshift.
  @cite_5: The symmetric tensor spherical harmonics (STSH’s) on the N‐sphere (SN), which are defined as the totally symmetric, traceless, and divergence‐free tensor eigenfunctions of the Laplace–Beltrami (LB) operator on SN, are studied. Specifically, their construction is shown recursively starting from the lower‐dimensional ones. The symmetric traceless tensors induced by STSH’s are introduced. These play a crucial role in the recursive construction of STSH’s. The normalization factors for STSH’s are determined by using their transformation properties under SO(N+1). Then the symmetric, traceless, and divergence‐free tensor eigenfunctions of the LB operator in the N‐dimensional de Sitter space‐time which are obtained by the analytic continuation of the STSH’s on SN are studied. Specifically, the allowed eigenvalues of the LB operator under the restriction of unitarity are determined. Our analysis gives a group‐theoretical explanation of the forbidden mass range observed earlier for the spin‐2 field theory in de Sit...
  @cite_72: 
  @cite_80: 
  @cite_16: 
  @cite_13: 
  @cite_105: 
  @cite_94: 
  @cite_82: 
summary: There is a large amount of literature on asymptotically flat black hole spacetimes; we only give a brief account here. The seminal papers by Wald @cite_60 and Kay--Wald @cite_78 proved the boundedness of linear scalar waves on the Schwarzschild spacetime; Dafermos and Rodnianski @cite_94 gave a more robust proof, exploiting the red-shift effect, and established polynomial decay. Polynomial decay for linear scalar waves on Kerr was proved by Andersson--Blue @cite_14 and in the aforementioned @cite_13 ; see also @cite_82 . The precise decay rates (Price's law @cite_16 ) were obtained by Tataru @cite_105 , see also @cite_72 , as well as @cite_107 in a spherically symmetric but non-linear context. Strichartz estimates were proved by Tataru and Tohaneanu @cite_28 , also in collaboration with Marzuola and Metcalfe @cite_80 . For non-linear results, we refer to Luk's work on Kerr @cite_48 , Ionescu--Klainerman @cite_22 and Stogin @cite_5 for a wave map equation, and Dafermos--Holzegel--Rodnianski @cite_39 for a scattering construction of dynamical Kerr black holes. Many of these works rely on the very influential vector field method' developed by Klainerman @cite_17 ; see also @cite_20 for a discussion of recent developments.