### Abstract

We consider solutions to the linear wave equation _{g} φ{symbol} = 0 on a (maximally extended) Schwarzschild spacetime with parameter M > 0, evolving from sufficiently regular initial data prescribed on a complete Cauchy surface Σ, where the data are assumed only to decay suitably at spatial infinity. (In particular, the support of φ{symbol} may contain the bifurcate event horizon.) It is shown that the energy flux F ^{T}_{φ{symbol}} (S) of the solution (as measured by a strictly timelike T that asymptotically matches the static Killing field) through arbitrary achronal subsets S of the black hole exterior region satisfies the bound F ^{T}_{φ{symbol}} (S) ≤ CE(υ^{-2}_{+} + u^{-2}), where v and u denote the infimum of the Eddington-Finkelstein advanced and retarded time of S, v_{+} denotes max{1, υ}, and u_{+} denotes max{1 u}, 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 = u + ν = 1. (The bound applies in particular to subsets S of the event horizon or null infinity.) It is also shown that φ{symbol} satisfies the pointwise decay estimate |φ{symbol}| ≤ CEν^{-1}_{+} in the entire exterior region, and the estimates |rφ{symbol}| ≤ C_{R} E(1 + |u|/^{-1/2} and |^{r1/2φ{symbol}}| ≤ C_{R} Eu_{+}^{-1} C in the region {r ≥ R} ∩ J^{+} (Σ) for any 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 |φ{symbol}| ≤ CE of Kay and Wald without recourse to the discrete isometries of spacetime.

Original language | English (US) |
---|---|

Pages (from-to) | 859-919 |

Number of pages | 61 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 62 |

Issue number | 7 |

DOIs | |

State | Published - Jul 1 2009 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics