Strong Cosmic Censorship for Surface‐Symmetric Cosmological Spacetimes with Collisionless Matter

This paper addresses strong cosmic censorship for spacetimes with self‐gravitating collisionless matter, evolving from surface‐symmetric compact initial data. The global dynamics exhibit qualitatively different features according to the sign of the curvature k of the symmetric surfaces and the cosmological constant Λ. With a suitable formulation, the question of strong cosmic censorship is settled in the affirmative if Λ=0 or k≤0, Λ > 0. In the case Λ > 0, k=1, we give a detailed geometric characterization of possible “boundary” components of spacetime; the remaining obstruction to showing strong cosmic censorship in this case has to do with the possible formation of extremal Schwarzschild–de Sitter‐type black holes. In the special case that the initial symmetric surfaces are all expanding, strong cosmic censorship is shown in the past for all k,Λ. Finally, our results also lead to a geometric characterization of the future boundary of black hole interiors for the collapse of asymptotically flat data: in particular, in the case of small perturbations of Schwarzschild data, it is shown that these solutions do not exhibit Cauchy horizons emanating from i ⁺ with strictly positive limiting area radius.© 2016 Wiley Periodicals, Inc.     

A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds

We consider Kerr spacetimes with parameters a and M such that |a|≪M, Kerr-Newman spacetimes with parameters |Q|≪M, |a|≪M, and more generally, stationary axisymmetric black hole exterior spacetimes (M,g)(\mathcal{M},g) which are sufficiently close to a Schwarzschild metric with parameter M>0 and whose Killing fields span the null generator of the event horizon. We show uniform boundedness on the exterior for solutions to the wave equation □ _g ψ=0. The most fundamental statement is at the level of energy: We show that given a suitable foliation Σ _τ , then there exists a constant C depending only on the parameter M and the choice of the foliation such that for all solutions ψ, a suitable energy flux through Σ _τ is bounded by C times the initial energy flux through Σ₀. This energy flux is positive definite and does not degenerate at the horizon, i.e. it agrees with the energy as measured by a local observer. It is shown that a similar boundedness statement holds for all higher order energies, again without degeneration at the horizon. This leads in particular to the pointwise uniform boundedness of ψ, in terms of a higher order initial energy on Σ₀. Note that in view of the very general assumptions, the separability properties of the wave equation or geodesic flow on the Kerr background are not used. In fact, the physical mechanism for boundedness uncovered in this paper is independent of the dispersive properties of waves in the high-frequency geometric optics regime.     

The red‐shift effect and radiation decay on black hole spacetimes

We consider solutions to the linear wave equation □_g? = 0 on a (maximally extended) Schwarzschild spacetime with parameter M > 0, evolving from sufficiently regular initial data prescribed on a complete Cauchy surface Σ, where the data are assumed only to decay suitably at spatial infinity. (In particular, the support of ? may contain the bifurcate event horizon.) It is shown that the energy flux F(??) of the solution (as measured by a strictly timelike T? that asymptotically matches the static Killing field) through arbitrary achronal subsets ?? of the black hole exterior region satisfies the bound F(??) ≤ C E(v + u), where v and u denote the infimum of the Eddington‐Finkelstein advanced and retarded time of ??, v₊ denotes max{1, v}, and u₊ denotes max{1, u}, where C is a constant depending only on the parameter M, and E depends on a suitable norm of the solution on the hypersurface t ? u + v = 1. (The bound applies in particular to subsets ?? of the event horizon or null infinity.) It is also shown that ? satisfies the pointwise decay estimate |?| ≤ C Ev in the entire exterior region, and the estimates |r?| ≤ C_R?E(1 + |u|)^?1/2 and |r^1/2?| ≤ C_R?Eu in the region {r ≥ R?} ∩ J⁺(Σ) for any R? > 2M. The estimates near the event horizon exploit an integral energy identity normalized to local observers. This estimate can be thought to quantify the celebrated red‐shift effect. The results in particular give an independent proof of the classical result |?| ≥ C E of Kay and Wald without recourse to the discrete isometries of spacetime. © 2009 Wiley Periodicals, Inc.     

Hyperbolic balance laws with dissipative source relaxing to adiabatic conservation laws

The paper discusses the long time behavior of BV solutions to the Cauchy problem for hyperbolic systems of balance laws with partial dissipation, when the relaxed system is adiabatic.

     

The modulation equations of nonlinear geometric optics

The purpose of this paper is to present an introduction to the central aspects of the modulation equations of nonlinear geometric optics. That means regularity properties and existence in the large.     

Partitionable variational inequalities with applications to network and economic equilibria

In this paper, we describe a useful class of finite-dimensional variational inequalities which we call partitionable. These variational inequalities are characterized by state functions which can be thought of as nonlinear separable functions added to antisymmetric linear functions. In the case of partitionable variational inequalities, questions of the monotonicity and coercivity of the state function can be addressed by considering the monotonicity and coercivity of a series of lower-dimensional functions. These functions are generally simpler to investigate than the state function. In the applications, these lower-dimensional functions are usually the natural functions to consider. To demonstrate, we conclude the paper by reviewing several models in the recent literature which give rise to partitionable variational inequalities.     

Black Hole Formation from a Complete Regular Past

An open problem in general relativity has been to construct an asymptotically flat solution to a reasonable Einstein-matter system containing a black hole and yet causally geodesically complete to the past, containing no white holes. We construct such a solution in this paper–in fact a family of such solutions, stable in a suitable sense–where matter is described by a self-gravitating scalar field.     

The interior of charged black holes and the problem of uniqueness in general relativity

We consider a spherically symmetric, double characteristic initial value problem for the (real) Einstein‐Maxwell‐scalar field equations. On the initial outgoing characteristic, the data is assumed to satisfy the Price law decay widely believed to hold on an event horizon arising from the collapse of an asymptotically flat Cauchy surface. We establish that the heuristic mass inflation scenario put forth by Israel and Poisson is mathematically correct in the context of this initial value problem. In particular, the maximal future development has a future boundary over which the space‐time is extendible as a C⁰ metric but along which the Hawking mass blows up identically; thus, the space‐time is inextendible as a C¹ metric. In view of recent results of the author in collaboration with I. Rodnianski, which rigorously establish the validity of Price's law as an upper bound for the decay of scalar field hair, the C⁰ extendibility result applies to the collapse of complete, asymptotically flat, spacelike initial data where the scalar field is compactly supported. This shows that under Christodoulou's C⁰ formulation, the strong cosmic censorship conjecture is false for this system. © 2005 Wiley Periodicals, Inc.