Annals of PDE

, 3:8 | Cite as

On the Global Uniqueness for the Einstein–Maxwell-Scalar Field System with a Cosmological Constant: Part 3. Mass Inflation and Extendibility of the Solutions

  • João L. Costa
  • Pedro M. Girão
  • José Natário
  • Jorge Drumond Silva


This paper is the third part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the Einstein–Maxwell-scalar field system with a cosmological constant \(\Lambda \), with the data on the outgoing initial null hypersurface given by a subextremal Reissner–Nordström black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development as a “suitably regular” Lorentzian manifold. In the first part [7] of this series we established the well posedness of the characteristic problem, whereas in the second part [8] we studied the stability of the radius function at the Cauchy horizon. In this third and final paper we show that, depending on the decay rate of the initial data, mass inflation may or may not occur. When the mass is controlled, it is possible to obtain continuous extensions of the metric across the Cauchy horizon with square integrable Christoffel symbols. Under slightly stronger conditions, we can bound the gradient of the scalar field. This allows the construction of (non-isometric) extensions of the maximal development which are classical solutions of the Einstein equations. Our results provide evidence against the validity of the strong cosmic censorship conjecture when \(\Lambda >0\).


Einstein equations Black holes Strong cosmic censorship Cauchy horizon Scalar field Spherical symmetry 

Mathematics Subject Classification

Primary 83C05 Secondary 35Q76 83C22 83C57 83C75 



The authors thank M. Dafermos for bringing the Epilogue of [11] to their attention.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.ISCTE - Instituto Universitário de LisboaLisbonPortugal
  2. 2.Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal

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