We study imbedded general hypersurfaces in spacetime i.e. hypersurfaces whose timelike, spacelike or null character can change from point to point.Inherited geometrical structures on these hypersurfaces are defined by two distinct methods: the first one, in which a rigging vector (a vector not tangent to the hypersurface anywhere) induces the standard rigged connection; and the other one, more adapted to physical aspects, where each observer in spacetime induces a completely new type of connection that we call the rigged metric connection which is volume preserving. The generalisation of the Gauss and Codazzi equations are also given. With the above machinery, we attack the problem of matching two spacetimes across a general hypersurface. It is seen that the preliminary junction conditions allowing for the correct definition of Einstein's equations in the distributional sense reduce to the requirement that the first fundamental form of the hypersurface be continuous, because then, there exists a maximal C 1 atlas in which the metric is continuous. The Bianchi identities are then proven to hold in the distributional sense. Next, we find the proper junction conditions which forbid the appearance of singular parts in the curvature. These are shown equivalent to the existence of coordinate systems where the metric is C 1 . Finally, we derive the physical implications of the junction conditions: only six independent discontinuities of the Riemann tensor are allowed. These are six matter discontinuities at non-null points of the hypersurface. For null points, the existence of two arbitrary discontinuities of the Weyl tensor (together with four in the matter tensor) are also allowed. The classical results for timelike, spacelike or null hypersurfaces are trivially recovered.

In this contribution I intend to give a summary of the new relevant results obtained by using the general superenergy tensors. After a quick review of the definition and properties of these tensors, several of their mathematical and physical applications are presented. In particular, their interest and usefulness is mentioned or explicitly analyzed in 1) the study of causal propagation of general fields; 2) the existence of an infinite number of conserved quantities in Ricci-flat spacetimes; 3) the different gravitational theories, such as Einstein's General Relativity or, say, n = 11 supergravity; 4) the appearance of some scalars possibly related to entropy or quality factors; 5) the possibility of superenergy exchange between different physical fields and the appearance of mixed conserved currents.

Rece ived Septem be r 2 4, 1 997. Rev . ve rsion Febr u ar y 26 , 19 98A detailed study of the singu larity t heorem s is present ed . I discuss the plausibility and reason ability of t heir hy pot heses, t he ap plicability an d im plicat ions of t he theorem s, as well as the t heorem s t hem selves. T he consequences usually ex t ract ed from t hem , som e of t hem w it hout the necessa ry rigour, are widely and carefu lly an aly sed w it h m any clarify ing ex am ples an d alternat ive view s.

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 ∇R 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.

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 theorem's impact, and relevant current and future work that belongs to its legacy.Singularity theorems 2 these will be discussed in sections 2 and 3-Penrose's theorem is, without a doubt, the first such theorem in its modern form containing new important ingredients and fruitful ideas that immediately led to many new developments in theoretical relativity, and to devastating physical consequences concerning the origin of the Universe and the collapse of massive stars, see sections 5 and 6. In particular, Penrose introduced geodesic incompleteness to characterize singularities (see subsection 4.1), used the notion of a Cauchy hypersurface (and thereby of global hyperbolicity, see section 4) and, more importantly, he presented the gravitational community with a precious gift in the form of a novel concept: closed trapped surfaces, see subsections 4.2 and 7.2.The fundamental, germinal and very fruitful notion of closed trapped surface is a key central idea in the physics of Black Holes, Numerical Relativity, Mathematical Relativity, Cosmology and Gravity Analogues. It has had an enormous influence as explained succinctly in section 5 and, in its refined contemporary versions -see subsections 7.2 and 7.3-, keeps generating many more advances (sections 7 and 8) of paramount importance and will probably maintain such prolific legacy with some unexpected applications in gravitational physics.As argued elsewhere [288], the singularity theorems constitute the first genuine post-Einstenian content of classical GR, not foreseen in any way by Einstein -as opposed to many other "milestones" discussed in this issue. § The global mathematical developments needed for the singularity theorems, and the ideas on incompleteness or trapping -and thus also their derived inferences -were not treated nor mentioned, neither directly nor indirectly, in any of Einstein's writings. In 1965 GR left adolescence behind, emancipated from its creator, and became a mature physical theory full of vitality and surprises. Before 1955Prior to 1965 there were many indications that the appearance of some kind of catastrophic irregularities, say "singularities", was common in GR. We give a succinct summary of some selected and instructive cases, with side historical remarks. Friedman closed models and the de Sitter solutionAs an early example, in 1922 Friedman [120] looked for solutions of (1) with the formwhere a(t) is a function of time t, F is an arbitrary function, and dΩ 2 represents the standard metric for a round sphere of unit radius. This Ansatz followed previous discussions by Einstein and de Sitter [97,87] where the space (for each t =const.) was § As a side historical remark, Einstein himself wrote a paper on "singularities" [99], later generalized to higher di...

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We consider the region T in spacetime containing future-trapped closed surfaces and its boundary B, and derive some of their general properties. We then concentrate on the case of spherical symmetry, but the methods we use are general and applicable to other situations. We argue that closed trapped surfaces have a non-local property, "clairvoyance", which is inherited by B. We prove that B is not a marginally trapped tube in general, and that it can have portions in regions whose whole past is flat. For asymptotically flat black holes, we identify a general past barrier, well inside the event horizon, to the location of B under physically reasonable conditions. We also define the core Z of the trapped region as that part of T which is indispensable to sustain closed trapped surfaces. We prove that the unique spherically symmetric dynamical horizon is the boundary of such a core, and we argue that this may serve to single it out. To illustrate the results, some explicit examples are discussed, namely Robertson-Walker geometries and the imploding Vaidya spacetime.