Showing papers by "José Natário published in 2014"

Gravito-electromagnetic analogies

Abstract: We reexamine and further develop different gravito-electromagnetic analogies found in the literature, and clarify the connection between them. Special emphasis is placed in two exact physical analogies: the analogy based on inertial fields from the so-called “1+3 formalism”, and the analogy based on tidal tensors. Both are reformulated, extended and generalized. We write in both formalisms the Maxwell and the full exact Einstein field equations with sources, plus the algebraic Bianchi identities, which are cast as the source-free equations for the gravitational field. New results within each approach are unveiled. The well known analogy between linearized gravity and electromagnetism in Lorentz frames is obtained as a limiting case of the exact ones. The formal analogies between the Maxwell and Weyl tensors are also discussed, and, together with insight from the other approaches, used to physically interpret gravitational radiation. The precise conditions under which a similarity between gravity and electromagnetism occurs are discussed, and we conclude by summarizing the main outcome of each approach.

On the global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant. Part 1: Well posedness and breakdown criterion

Abstract: This paper is the first 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 (MGHD) as a "suitably regular" Lorentzian manifold. In this first part we establish well posedness of the Einstein equations for characteristic data satisfying the minimal regularity conditions leading to classical solutions. We also identify the appropriate notion of maximal solution, from which the construction of the corresponding MGHD follows, and determine breakdown criteria. This is the unavoidable starting point of the analysis; our main results will depend on the detailed understanding of these fundamentals. In the second part of this series we study the stability of the radius function at the Cauchy horizon. In the third and final paper we show that, depending on the decay rate of the initial data, mass inflation may or may not occur; in fact, it is even possible to have (non-isometric) extensions of the spacetime across the Cauchy horizon as classical solutions of the Einstein equations.

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

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.

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 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 $\Lambda>0$.

Center of mass, spin supplementary conditions, and the momentum of spinning particles

Abstract: We discuss the problem of defining the center of mass in general relativity and the so-called spin supplementary condition. The different spin conditions in the literature, their physical significance, and the momentum-velocity relation for each of them are analyzed in depth. The reason for the non-parallelism between the velocity and the momentum, and the concept of "hidden momentum", are dissected. It is argued that the different solutions allowed by the different spin conditions are equally valid descriptions for the motion of a given test body, and their equivalence is shown to dipole order in curved spacetime. These different descriptions are compared in simple examples.

Relativistic elasticity of rigid rods and strings

Abstract: We show that the equation of motion for a rigid one-dimensional elastic body (i.e. a rod or string whose speed of sound is equal to the speed of light) in a two-dimensional spacetime is simply the wave equation. We then solve this equation in a few simple examples: a rigid rod colliding with an unmovable wall, a rigid rod being pushed by a constant force, a rigid string whose endpoints are simultaneously set in motion (seen as a special case of Bell’s spaceships paradox), and a radial rigid string that has partially crossed the event horizon of a Schwarzschild black hole while still being held from the outside.

Solutions to Selected Exercises

Abstract: In this chapter we present the solutions to 140 selected exercises, chosen among the 333 exercises in the previous chapters.

Relativistic elasticity of rigid rods and strings

Abstract: We show that the equation of motion for a rigid one-dimensional elastic body (i.e. a rod or string whose speed of sound is equal to the speed of light) in a two-dimensional spacetime is simply the wave equation. We then solve this equation in a few simple examples: a rigid rod colliding with an unmovable wall, a rigid rod being pushed by a constant force, a rigid string whose endpoints are simultaneously set in motion (seen as a special case of Bell's spaceships paradox), and a radial rigid string that has partially crossed the event horizon of a Schwarzschild black hole while still being held from the outside.