On the Occurrence of Mass Inflation for the Einstein–Maxwell-Scalar Field System with a Cosmological Constant and an Exponential Price Law

Abstract

In this paper we study the spherically symmetric characteristic initial data problem for the Einstein-Maxwell-scalar field system with a positive cosmological constant in the interior of a black hole, assuming an exponential Price law along the event horizon. More precisely, we construct open sets of characteristic data which, on the outgoing initial null hypersurface (taken to be the event horizon), converges exponentially to a reference Reissner-Nördstrom black hole at infinity.We prove the stability of the radius function at the Cauchy horizon, and show that, depending on the decay rate of the initial data, mass inflation may or may not occur. In the latter case, we find that the solution can be extended across the Cauchy horizon with continuous metric and Christoffel symbols in L 2 loc , thus violating the Christodoulou-Chruściel version of strong cosmic censorship. Contents2010 Mathematics Subject Classification. Primary 83C05; Secondary 35Q76, 83C22, 83C57, 83C75. References 491. Introduction 1.1. Strong cosmic censorship and spherical symmetry. Determinism of a physical system, modeled mathematically by evolution equations, is embodied in the questions of existence and uniqueness of solutions for given initial data. The initial value problem (or Cauchy problem) is therefore the appropriate setting for studying these models.Well known examples of equations where the Cauchy problem is quintessential are those of Newtonian mechanics, the Euler and Navier-Stokes systems in hydrodynamics and Maxwell's equations of electromagnetism. Historically, the geometric nature and mathematical complexity of the Einstein equations made it difficult to recognize that they also fit into this framework. It was not until the seminal work of Y. , and her later work with R. Geroch [5], that the central role of the Cauchy problem in general relativity was established. These results relied crucially on recognizing the hyperbolic character of the Einstein equations. Uniqueness of the solutions, as for any hyperbolic PDE, then follows from a domain of dependence property. The essence of [5] consists precisely in showing that given initial data there exists a maximal globally hyperbolic development (MGHD) for the corresponding Cauchy problem, that is, a maximal spacetime where this domain of dependence property holds.For the Einstein equations, global uniqueness fails, and therefore determinism breaks down, if extensions of MGHDs to strictly larger spacetimes can be found. The statement that generically, for suitable Cauchy initial data, * the corresponding MGHD cannot be extended is known as the strong cosmic censorship conjecture (SCCC) [7,9,27].A crucial point in the precise formulation of this conjecture is deciding what exactly is meant by an extension. Various proposals have been advanced, differing on the degree of regularity that is demanded for the larger spacetime. The strongest formulation would correspond to the impossibility of extending the MGHD with a continuous Lorentzian metric. This happens for instance in the Schwa...

Quantum-corrected rotating black holes and naked singularities in ( 2+1 ) dimensions

“…In (3 + 1)-dimensions, it has been shown that classical field perturbations lead to a curvature (non-timelike) singularity at the Cauchy horizon in the case of spherically-symmetric and electrically-charged (Reissner-Nordström) BHs, with [6,7] or without [8][9][10][11] a positive cosmological constant, as well as in the case of rotating (Kerr) BHs [12,13]. These results are in support of strong CCH.…”

Singularities in Reissner–Nordström black holes

“…It follows that the boundary conditions we imposed on M are in fact valid in the interior of M, meaning our results are insensitive to the precise choice of M: we could have equally well chosen M to asymptote to some finite r < r − as v → ∞. From (27) we see that λ derivatives blow up like e κv . Hence the derivative expansion is simply a late time expansion with expansion parameter ≡ e −κv .…”

Strong cosmic censorship in Horndeski theory

“…This occurs if (∂ 2 φ) 2 is integrable at the CH, or equivalently φ ∈ H 2 loc (recall that for the case of η = 0 the extra terms of Θ µν would vanish and one only requires φ ∈ H 1 loc in accordance with (2.12) and [3]). By realizing that the scalar field and spacetime metric share similar regularity requirements [9,10,51,52], it seems adequate to examine the behavior of φ at the CH.…”