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

Abstract

This paper is the third part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the EinsteinMaxwell-scalar field system with a cosmological constant Λ, with the data on the outgoing initial null hypersurface given by a subextremal ReissnerNordström 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 Λ > 0.

Boundedness of Massless Scalar Waves on Kerr Interior Backgrounds

“…A full linear stability result for slowly rotating Kerr-de Sitter spacetime was obtained by Hintz and Vasy, see Theorem 1.2 in [38]. 13 A function ψ belonging to the Sobolev space W 1,2 loc would have the properties that locally ψ and all of its first weak derivatives exist and are square integrable. Taking (1) seriously as a model for the full Einstein field equations and therefore considering ψ as an agent for the metric two tensor g, the result of Luk and Oh suggests that in the full theory the Christoffel symbols would fail to be square integrable.…”

“…Generic smooth and compactly supported initial data to the wave equation on fixed subextremal Reissner-Nordström background on a two-ended asymptotically flat Cauchy surface Σ give rise to solutions that are not in W 1,p loc , for all 1 < p < 2, for the subextremal range 0 < log r+ r− < 1; 12 and not in W 1,2 loc , for p ≥ 2 for By non-degeneracy we mean that the multiplier is constructed such that it does not become null on the hypersurfaces of interest 12. Although Reissner-Nordström spacetimes with subextremal range approaching the extremal range are expected to show a more stable behavior on CH + in comparison to black holes with a small charge and mass ratio, it remains open to show that solutions are not in W 1,p loc , for all 1 < p < 2, for the subextremal range 1 ≤ logr + r − .the entire subextremal range, in a neighborhood of any point on the future Cauchy horizon CH + 13. A version of the Strong Cosmic Censorship Conjecture demanding that solutions should be inextendible in W 1,p loc beyond CH + could be interpreted as generically the Lorentzian manifold cannot be extended even in a weak sense such that the Einstein equations are still satisfied.…”

Strong cosmic censorship for a scalar field in a logarithmic-de Sitter black hole

“…The decay of the scalar field will be exponential rather than power-law near the Cauchy horizon [17][18][19][20][21][22][23][24]. So the validity of SCC depends on the competition between the exponential decay and blue-shift effect, which can be characterized by [25][26][27][28][29][30]…”

A Possible Failure of Determinism in General Relativity

“…Will it still, upon perturbation, become a "strong enough" singularity in order to uphold SCC? A convenient way to measure the strength of such a (Cauchy horizon) singularity is in terms of the regularity of the spacetime metric extensions it allows [15][16][17]. For instance, mass inflation is related to inextendibility in (the Sobolev space) H 1 which turns out to be enough to guarantee the nonexistence of extensions as (weak) solutions of the Einstein equations [18], i.e., the complete breakdown of the field equations.…”

“…Moreover, for spherically symmetric self-gravitating scalar fields, the control of both the Hawking mass and the gradient of the field is enough to control the Kretschmann scalar [16]. We will, henceforth, relate β < 1 to the blow up of curvature components.…”