A proof of Price's law for the collapse of a self-gravitating scalar field
Mihalis Dafermos, Igor Rodnianski
TL;DR
The paper proves Price's law for the collapse of a self-gravitating scalar field in spherical symmetry, including a Maxwell field, by developing a novel method that fuses conformal geometry, red-shift effects, and local energy conservation to control large-data, nonlinear hyperbolic dynamics. It establishes explicit decay rates on the event horizon $\mathcal{H}^+$ and at null infinity $\mathcal{I}^+$, while also detailing how mass-inflation phenomena and the interior causal structure relate to strong cosmic censorship, including a falsehood of the Christodoulou $C^0$ formulation in the charged setting. The work connects a nonlinear, fully coupled Einstein–Maxwell–scalar system to its linear counterpart on fixed backgrounds (Schwarzschild/Reissner–Nordström) and develops a robust, coordinate-friendly framework based on renormalized Hawking mass and red-shift to derive decay via an induction over spacetime rectangles. These results enhance understanding of black-hole stability, exterior decay rates, and the interior dynamics that drive cosmic censorship scenarios, with potential implications for nonlinear stability analyses of rotating (Kerr) spacetimes. The techniques and conclusions provide a concrete nonlinear Price-type bound and a blueprint for addressing long-time behavior in related self-gravitating, radiating systems.
Abstract
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 on 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 (gr-qc/0307013) concerning the internal structure of charged black holes, which 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 C^0 formulation, the conjecture is proven to be false.