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

TL;DR

This paper proves an upper bound formulation of Price's law for the decay of gravitational radiation in spherically symmetric gravitational collapse involving scalar and Maxwell fields, advancing understanding of black hole stability.

Contribution

It introduces a new mathematical technique combining conformal geometry, red-shift effect, and energy conservation to analyze long-time behavior of coupled non-linear PDEs in gravitational collapse.

Findings

Established an upper bound for Price's law decay rates.

Applied the results to the strong cosmic censorship conjecture.

Extended analysis to include Maxwell fields in collapse models.

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.