The red-shift effect and radiation decay on black hole spacetimes

TY - JOUR

T1 - The red-shift effect and radiation decay on black hole spacetimes

AU - Dafermos, Mihalis

AU - Rodnianski, Igor

PY - 2009/7

Y1 - 2009/7

N2 - 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 Tφ{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 Tφ{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}| ≤ CR E(1 + |u|/-1/2 and |r1/2φ{symbol}| ≤ CR 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.

AB - 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 Tφ{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 Tφ{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}| ≤ CR E(1 + |u|/-1/2 and |r1/2φ{symbol}| ≤ CR 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.

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U2 - 10.1002/cpa.20281

DO - 10.1002/cpa.20281

M3 - Article

SN - 0010-3640

VL - 62

SP - 859

EP - 919

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

IS - 7

ER -