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

By Mihalis Dafermos, Igor Rodnianski. 2005

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We consider solutions to the linear wave equation □gϕ = 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 may contain the bifurcate event horizon.) It is shown that the energy flux F T̃ϕ (𝒮) of the solution (as measured by a strictly timelike T̃ that asymptotically matches the static Killing field) through arbitrary achronal subsets 𝒮 of the black hole exterior region satisfies the bound F T̃ϕ (𝒮) ≤ C E(v −2+ + u −2+ ), where v and u denote the infimum of the Eddington‐Finkelstein advanced and retarded time of 𝒮, v+ denotes max{1, v}, 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 + v = 1. (The bound applies in particular to subsets 𝒮 of the event horizon or null infinity.) It is also shown that ϕ satisfies the pointwise decay estimate |ϕ| ≤ C Ev −1+ in the entire exterior region, and the estimates |rϕ| ≤ CR̃E(1 + |u|)−1/2 and |r1/2ϕ| ≤ CR̃Eu −1+ in the region {r ≥ R̃} ∩ J+(Σ) for any R̃ > 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| ≥ C E of Kay and Wald without recourse to the discrete isometries of spacetime. © 2009 Wiley Periodicals, Inc.

Subfields: Astronomy and Astrophysics, Mathematical Physics, Nuclear and High Energy Physics

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Citations in the corpus

A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds (2008, methodology): “In the course of the above study, the work [ 19 ] introduced the use of a vector field multiplier current J Y associated to a vector field Y which becomes null on (and transversal to) the horizon H + , such that the flux of J N where”
Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov) (2010, result): “On perturbations of Minkowski space itself, the full non-linear analysis was done in the path-breaking work of Christodoulou and Klainerman [9], and Lindblad and Rodnianski simplified the analysis [34, 35], Bieri [2, 3] succeeded in relaxing the decay conditions, while Wang [59] obtained additional,”
Slowly decaying waves on spherically symmetric spacetimes and ultracompact neutron stars (2014, background): “On the other hand, if we take one more angular derivative of the initial data (or even an ò of a derivative—see [36] and also [1]), then we can prove a local decay statement—see [35].”
Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations I (2011, methodology): “8). Moreover, in view of the lack of redshift along H + , the divergence of the energy current J N μ associated to the redshift vector field N , first introduced in [ 20 ], is not positive definite nearH + (see Sect.”
Hidden symmetries and decay for the wave equation on the Kerr spacetime (2009, methodology): “In the terminology of [17], this is a Cauchy stability argument.”
Local decay of waves on asymptotically flat stationary space-times (2009, methodology): “The estimates in the Schwarzschild case are the end result of a series of papers by LabaSoffer [27], Blue-Soffer [3], [4],[38], Blue-Sterbenz [5] Dafermos-Rodnianski [16], [13] and Marzuola-Metcalfe-Tataru-Tohaneanu [30].”
Strichartz Estimates on Schwarzschild Black Hole Backgrounds (2008, background): “We remark that a related form of a nondegenerate energy expression was introduced in [14] and proved to be bounded in the exterior region on surfaces t = const .”
The global non-linear stability of the Kerr-de Sitter family of black holes (2016, background): “There is also work by Donninger, Schlag and Soffer [49] on L∞ estimates on Schwarzschild black holes, following L∞ estimates of Dafermos and Rodnianski [40], and of Blue and Soffer [16] on non-rotating charged black holes giving L(6) estimates.”
Price's Law on Nonstationary Space-Times (2011, unknown): “it is known to hold for Schwarzschild (see [24] and [45]) as well as for Kerr with small angular momentum (see [29], [23], [66]) and for a class of small stationary perturbations of Schwarzschild (see [23]).”