José M. M. Senovilla

Singularity Theorems and Their Consequences

A detailed study of the singularity theorems is presented. I discuss the plausibility and reasonability of their hypotheses, the applicability and implications of the theorems, as well as the theorems themselves. The consequences usually extracted from them, some of them without the necessary rigour, are widely and carefully analysed with many clarifying examples and alternative views.

Causal structures and causal boundaries

We give an up-to-date perspective with a general overview of the theory of causal properties, the derived causal structures, their classification and applications and the definition and construction of causal boundaries and of causal symmetries, mostly for Lorentzian manifolds but also in more abstract settings.

Trapped surfaces and symmetries

We prove that strictly stationary spacetimes cannot contain closed trapped or marginally trapped surfaces. The result is purely geometric and holds in arbitrary dimension. Other results concerning the interplay between (generalized) symmetries and trapped submanifolds are also presented.

Some Properties of the Bel and Bel± Robinson Tensors

The properties of the Bel and Bel-Robinson tensors seem to indicate that they are closely related to the gravitational energy-momentum. We present some new properties of these tensors which might throw some light onto this relationship. First, for any spacetime we find a decomposition of the Bel tensor in terms of the Bel-Robinson tensor and two other tensors, which we call the “pure matter” super-energy tensor and the “matter-gravity coupling” super-energy tensor. We show that the pure matter super-energy tensor of any Einstein-Maxwell field is simply the “square” of the usual energy-momentum tensor. This, together with the fact that the Bel-Robinson tensor has dimensions of energy density square, leads us to the definition of square root for the Bel-Robinson tensor: a two-covariant symmetric traceless tensor with dimensions of energy density and such that its “square” gives the Bel-Robinson tensor. We prove that this square root exists if and only if the spacetime is of Petrov type O, N or D, and its general expression is explicitly presented. The properties of this new tensor are examined and some interesting explicit examples are analyzed. Of particular interest are an invariant function that appears in the spherically symmetric metrics and an expression for the energy carried out by pure plane gravitational waves. We also examine the decomposition of the whole Bel tensor for Vaidyas radiating metric and Kerr-Newmans solution. Finally, we generalize the definition of square root to a factorization of the Bel-Robinson tensor and get the general solution for all Petrov types.

Causal relationship: a new tool for the causal characterization of Lorentzian manifolds

We define and study a new kind of relation between two diffeomorphic Lorentzian manifolds called a causal relation, which is any diffeomorphism characterized by mapping every causal vector of the first manifold onto a causal vector of the second. We perform a thorough study of the mathematical properties of causal relations and prove in particular that two given Lorentzian manifolds (say V and W) may be causally related only in one direction (say from V to W, but not from W to V). This leads us to the concept of causally equivalent (or isocausal in short) Lorentzian manifolds as those mutually causally related and to a definition of causal structure over a differentiable manifold as the equivalence class formed by isocausal Lorentzian metrics upon it. Isocausality is a more general concept than the conformal relationship, because we prove the remarkable result that a conformal relation is characterized by the fact of being a causal relation of the particular kind in which both and −1 are causal relations. Isocausal Lorentzian manifolds are mutually causally compatible, they share some important causal properties, and there are one-to-one correspondences, which are sometimes non-trivial, between several classes of their respective future (and past) objects. A more important feature is that they satisfy the same standard causality constraints. We also introduce a partial order for the equivalence classes of isocausal Lorentzian manifolds providing a classification of all the causal structures that a given fixed manifold can have. By introducing the concept of causal extension we put forward a new definition of causal boundary for Lorentzian manifolds based on the concept of isocausality, and thereby we generalize the traditional Penrose constructions of conformal infinity, diagrams and embeddings. In particular, the concept of causal diagram is given. Many explicit clarifying examples are presented throughout the paper.

Trapped surfaces, horizons and exact solutions in higher dimensions

A very simple criterion to ascertain if (D − 2)-surfaces are trapped in arbitrary D-dimensional Lorentzian manifolds is given. The result is purely geometric, independent of the particular gravitational theory, of any field equations or of any other conditions. Many physical applications arise, a few are shown here: a definition of general horizon, which reduces to the standard one in black holes/rings and other known cases; the classification of solutions with a (D − 2)-dimensional Abelian group of motions and the invariance of the trapping under simple dimensional reductions of the Kaluza–Klein/string/M-theory type. Finally, a stronger result involving closed trapped surfaces is presented. It provides in particular a simple sufficient condition for their absence.

The 1965 Penrose singularity theorem

We review the first modern singularity theorem, published by Penrose in 1965. This is the first genuine post-Einstenian result in General Relativity, where the fundamental and fruitful concept of closed trapped surface was introduced. We include historical remarks, an appraisal of the theorems impact, and relevant current and future work that belongs to its legacy.

Junction conditions for F(R)-gravity and their consequences

I present the junction conditions for F(R) theories of gravity and their implications: the generalized Israel conditions and equations. These junction conditions are necessary to construct global models of stars, galaxies, etc., where a vacuum region surrounds a finite body in equilibrium, as well as to describe shells of matter and braneworlds, and they are stricter than in General Relativity in both cases. For the latter case, I obtain the field equations for the energy-momentum tensor on the shell/brane, and they turn out to be, remarkably, the same as in General Relativity. An exceptional case for quadratic F(R), previously overlooked in the literature, is shown to arise allowing for a discontinuous R, and leading to an energy-momentum content on the shell with unexpected properties, such as non-vanishing components normal to the shell and a new term resembling classical dipole distributions. For the former case, they do not only require the agreement of the first and second fundamental forms on both sides of the matching hypersurface, but also that the scalar curvature R and its first derivative agree there too. I argue that, as a consequence, matched solutions in General Relativity are not solutions of F(R)-models generically. Several relevant examples are analyzed.

Models of regular Schwarzschild black holes satisfying weak energy conditions

We prove the existence of regular Schwarzschild black holes satisfying the weak energy conditions everywhere by presenting two explicit models. One of these models is explicitly seen to be complete (and therefore regular) by giving a maximal extension across the horizons.

A NOTE ON TRAPPED SURFACES IN THE VAIDYA SOLUTION

The Vaidya solution describes the gravitational collapse of a finite shell of incoherent radiation falling into flat spacetime and giving rise to a Schwarzschild black hole. There has been a question whether closed trapped surfaces can extend into the flat region (whereas closed outer trapped surfaces certainly can). For the special case of self-similar collapse we show that the answer is yes, if and only if the mass function rises fast enough.