Elastic shocks in relativistic rigid rods and balls

We study the free boundary problem for the "hard phase" material introduced by Christodoulou, both for rods in (1+1)-dimensional Minkowski spacetime and for spherically symmetric balls in (3+1)-dimensional Minkowski spacetime. Unlike Christodoulou, we do not consider a "soft phase", and so we regard this material as an elastic medium, capable of both compression and stretching. We prove that shocks must be null hypersurfaces, and derive the conditions to be satisfied at a free boundary. We solve the equations of motion of the rods explicitly, and we prove existence of solutions to the equations of motion of the spherically symmetric balls for an arbitrarily long (but finite) time, given initial conditions sufficiently close to those for the relaxed ball at rest. In both cases we find that the solutions contain shocks if and only if the pressure or its time derivative do not vanish at the free boundary initially. These shocks interact with the free boundary, causing it to lose regularity.Comment: 22 pages, 3 figures; v2: small changes, matches final published version; v3: typos in the references fixe

Mathisson's helical motions demystified

The motion of spinning test particles in general relativity is described by Mathisson-Papapetrou-Dixon equations, which are undetermined up to a spin supplementary condition, the latter being today still an open question. The Mathisson-Pirani (MP) condition is known to lead to rather mysterious helical motions which have been deemed unphysical, and for this reason discarded. We show that these assessments are unfounded and originate from a subtle (but crucial) misconception. We discuss the kinematical explanation of the helical motions, and dynamically interpret them through the concept of hidden momentum, which has an electromagnetic analogue. We also show that, contrary to previous claims, the frequency of the helical motions coincides exactly with the zitterbewegung frequency of the Dirac equation for the electron.Comment: To appear in the Proceedings of the Spanish Relativity Meeting 2011 (ERE2011), "Towards new paradigms", Madrid 29 August - 2 September 201

Spherical linear waves in de Sitter spacetime

We apply Christodoulou's framework, developed to study the Einstein-scalar field equations in spherical symmetry, to the linear wave equation in de Sitter spacetime, as a first step towards the Einstein-scalar field equations with positive cosmological constant. We obtain an integro-differential evolution equation which we solve by taking initial data on a null cone. As a corollary we obtain elementary derivations of expected properties of linear waves in de Sitter spacetime: boundedness in terms of (characteristic) initial data, and a Price law establishing uniform exponential decay, in Bondi time, to a constant.Comment: 9 pages, 1 figure; v2: minor changes, references added, matches final published versio

Strong cosmic censorship: The nonlinear story

A satisfactory formulation of the laws of physics entails that the future evolution of a physical system should be determined from appropriate initial conditions. The existence of Cauchy horizons in solutions of the Einstein field equations is therefore problematic, and expected to be an unstable artifact of General Relativity. This is asserted by the Strong Cosmic Censorship Conjecture, which was recently put into question by an analysis of the linearized equations in the exterior of charged black holes in an expanding universe. Here, we numerically evolve the nonlinear Einstein-Maxwell-scalar field equations with a positive cosmological constant, under spherical symmetry, and provide strong evidence that mass inflation indeed does not occur in the near extremal regime. This shows that nonlinear effects might not suffice to save the Strong Cosmic Censorship Conjecture.Comment: 9 pages, 8 figures. v2: Matches published versio

On the global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant. Part 3: Mass inflation and extendibility of the solutions

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-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 part of this series we established the well posedness of the characteristic problem, whereas in the second part 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 $\Lambda>0$.Comment: 48 pages, 5 figures; v2: some presentation changes, mostly in the Introduction; v3: substantial changes in Section 5; v4: expanded Introduction; some presentation changes; matches final published versio

On the global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant. Part 2: Structure of the solutions and stability of the Cauchy horizon

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.Comment: 44 pages, 13 figures; v2: a few small changes; v3: a paragraph was added in the Introduction, minor clarifications were made thoughout, the list of references was expanded, matches final published versio

Tameness of the pseudovariety LS1

The notion of k-tameness of a pseudovariety was introduced by Almeida and Steinberg and is a strong property which implies decidability of pseudovarieties. In this paper we prove that the pseudovariety LSl, of local semilattices, is k-tame.This work was supported, in part, by FCT through the Centro de Matemática da Universidade do Minho, and by the FCT and POCTI approved project POCTI/32817/MAT/2000 which is comparticipated by the European Community Fund FEDER