A proof of Price’s law for the collapse of a self-gravitating scalar field

By Mihalis Dafermos, Igor Rodnianski. 2003

231 citations recorded by the source corpus.

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.

Subfields: Astronomy and Astrophysics, Finance

View Paper: https://www.semanticscholar.org/paper/7e8e912bc8a5024543901c22760430ce24be6316

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Citations in the corpus

Local decay of waves on asymptotically flat stationary space-times (2009, methodology): “This also includes the higher order bounds k≥ 1, following an approach similar to that of Dafermos-Rodnianski [16].”
The interior of dynamical vacuum black holes I: The $C^0$-stability of the Kerr Cauchy horizon (2017, background): “–Rodnianski [35]) Let (M, g, Fμν , ψ) be the maximal Cauchy development arising from spherically symmetric, admissible(23), regular asymptotically flat two-ended initial data for the system (1.”
The 1965 Penrose singularity theorem (2014, background): “3), it has also become an object of interest for mathematicians —see for instance the use of trapped surfaces to prove the decay rate of gravitational radiation flux [80]—and it has evolved into a richer fauna of interesting “trapped-like” submanifolds with many geometrical and physical implications”
Black Holes Without Spacelike Singularities (2012, methodology): “Examining the proofs of [17, 20], we easily see”
Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations I (2011, background): “For results on the coupled wave equation see [20].”
The linear stability of the Schwarzschild solution to gravitational perturbations (2016, background): “In the case of Schwarzschild, on the other hand, even for solutions arising from compactly supported initial data, though the decay rate (64) for φ can be indeed improved to t−3 [18, 28, 50], it cannot, for generic initial data, be further improved [45].”
Stability and Instability of Extreme Reissner–Nordström Black Hole Spacetimes for Linear Scalar Perturbations II (2011, methodology): “I also thank Igor Rodnianski for sharing useful insights.”
A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds (2008, methodology): “(In fact, this was a byproduct of the main result of [11], which concerns decay rates for spherically symmetric solutions to the coupled Einstein-(Maxwell)-scalar field system.”
Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov) (2010, result): “(There was also recent work by Marzuola, Metcalf, Tataru and Tohaneanu [36] on Strichartz estimates, and by Donninger, Schlag and Soffer [18] on L∞ estimates on Schwarzschild black holes, following L∞ estimates of Dafermos and Rodnianski [12, 10], of Blue and Soffer [4] on non-rotating charged black”
Decay and Non-Decay of the Local Energy for the Wave Equation on the De Sitter–Schwarzschild Metric (2007, methodology): “We refer to the papers of Blue–Soffer [4], Blue–Sterbenz [5], and Dafermos–Rodnianski [12] and references therein for an overview.”
Strichartz Estimates on Schwarzschild Black Hole Backgrounds (2008, background): “The same authors, in [5,6], have provided studies that give improved estimates near the photon sphere r = 3M .”