Abstract
This paper is the second 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-Nordstrom black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development as a "suitably regular" Lorentzian manifold. In the first paper of this sequence, we established well posedness of the characteristic problem with general initial data. In this second paper, we generalize the results of Dafermos on the stability of the radius function at the Cauchy horizon by including a cosmological constant. This requires a considerable deviation from the strategy followed by Dafermos, focusing on the level sets of the radius function instead of the red-shift and blue-shift regions. We also present new results on the global structure of the solution when the free data is not identically zero in a neighborhood of the origin. In the third and final paper, we will consider the issue of mass inflation and extendibility of solutions beyond the Cauchy horizon.
Highlights
As in [6], we introduce a certain generic element in the formulation of our problem by perturbing a subextremal Reissner-Nordström black hole by arbitrary characteristic data along the ingoing null direction
The study of the conditions under which the solutions can be extended across the Cauchy horizon is left to
Our first objective is to obtain good upper bounds for −λ in the different regions of spacetime. These will enable us to show that the radius function r is bounded below by a positive constant
Summary
We take many ideas from [6] and [7] and build on these works. In particular, we borrow the following three very important techniques. (i) The partition of the spacetime domain of the solution into four regions and the construction of a carefully chosen spacelike curve to separate the last two. When we pass to Bondi coordinates this function disappears, making a simple application of Gronwall’s inequality, such as the one we present, possible This would not work in the double null coordinate system (u, v). Our first objective is to obtain good upper bounds for −λ in the different regions of spacetime These will enable us to show that the radius function r is bounded below by a positive constant. As soon as the initial data field is not identically zero, the curve {r = r−} is contained in P (Theorem 8.1). This is an interesting geometrical condition and it is conceptually relevant given the importance that we confer to the curves of constant r. The affine parameter of any outgoing null geodesic inside the event horizon is finite at the Cauchy horizon (Corollary 8.3)
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
