On the global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant. Part 3: Mass inflation and extendibility of the solutions

摘要

This paper is the third part of a trilogy dedicated to the following problem:given spherically symmetric characteristic initial data for theEinstein-Maxwell-scalar field system with a cosmological constant $\Lambda$,with the data on the outgoing initial null hypersurface given by a subextremalReissner-Nordstrom black hole event horizon, study the future extendibility ofthe corresponding maximal globally hyperbolic development as a "suitablyregular" Lorentzian manifold. In the first part of this series we established the well posedness of thecharacteristic problem, whereas in the second part we studied the stability ofthe radius function at the Cauchy horizon. In this third and final paper we show that, depending on the decay rate ofthe initial data, mass inflation may or may not occur. When the mass iscontrolled, it is possible to obtain continuous extensions of the metric acrossthe Cauchy horizon with square integrable Christoffel symbols. Under slightlystronger conditions, we can bound the gradient of the scalar field. This allowsthe construction of (non-isometric) extensions of the maximal development whichare classical solutions of the Einstein equations. Our results provide evidenceagainst the validity of the strong cosmic censorship conjecture when$\Lambda>0$.