Inside the event horizon of a black hole there is its trapping boundary --- outside of which no trapped surfaces pass --- and its core --- into which any trapped surface must extend. For an evolving black hole this gives three distinct candidates for the "surface" of the black hole (except that we do not know if the core is unique). I  will review what we do know about these boundaries in the case of spherically symmetric black holes.

The Riquier-Janet theory dating back to early 1900's gives the existence of solutions for systems of overdetermined partial differential equations based on an ordering of partial derivatives.  We apply this method to the system of Newman-Penrose equations using the symbolic programming language REDUCE;  we obtain "systems in involution" for certain special cases including conformally flat,  Petrov Type-D and  Ricci tensor of rank 1 metrics and discuss the situations where the dynamics of the physical
sources are determined by the inegrability conditions.

We will present a sketch of how to obtain the local line-element of second-order symmetric spacetimes (i.e., those which satisfy nabla^2 Rie=0) in arbitrary dimension, and how the local version yields the global one.

In this talk I will present a construction of a geometric invariant for initial data sets for the Einstein vacuum field equations. This geometric invariant vanishes if and only if the initial data set corresponds to data for the Kerr spacetime, and thus, it characterises this type of data. In order to construct the geometric invariant, we introduce the notion of approximate Killing spinors. These spinors are symmetric valence 2 spinors intrinsic to the initial hypersurface and satisfy a certain second order elliptic equation ---the approximate Killing spinor equation. This equation arises as the Euler-Lagrange equation of a non-negative integral functional. This functional constitutes part of the geometric invariant ---however, the whole functional does not come from a variational principle. If time allows I will also present a similar construction measuring the deviation from Minkowski data as well as recent applications of the tools.

Recently the well-known Petrov classification of the Weyl tensor in four dimensional Lorentzian geometry has been generalised to any dimension higher than four. An interesting problem is to find and classify exact solutions in dimensions higher than four according to their Petrov type. In this work we study and classify a set of five dimensional vacuum type D solutions characterised by the invariance of the Riemann tensor on a 2-dimensional space-like plane. To achieve this we use an extension of the G.H.P. formalism used in dimension four to dimension five. Our extension is a natural application of the spinor calculus in dimension five, developed by us which is also briefly discussed.

In the standard conformal completion of an asymptotically flat (at spacelike and null infinity) spacetime, where spacelike infinity, i^0, is added as a point, with future and past null infinity forming the null cone at i^0, one can show that the Weyl tensor, suitably rescaled, has a direction dependent limit. This rescaled limit can  be decomposed into an electric and a magnetic part, and for physical reasons, one expects the magnetic part to be zero. I will give a condition when this will follow, a condition which is only expressed in terms of quantities on future null infinity.

Trapped surfaces are the precursors of singularities in gravitational collapse. The behaviour of trapped surfaces can be studied by constructing a model of a black hole in 2+1 dimensional anti-de Sitter space. Letting a point particle in this model fall into the black hole shows how the trapped surfaces can make sudden "jumps" from one location to another.

H-space is a complex 4-D space with zero Ricci curvature and self-dual Weyl curvature.  Remarkably, such spaces can be constructed from real, (asymptotically flat) spacetimes and completely describe their emitted gravitational radiation. The H-space construction was developed in the late 70's by Newman et. al. and, in a different form (the twister non-linear gravition construction), by Penrose, and led to hopes of a new approach to quantum gravity. However, in spite of producing many intriguing and surprising results, some of which will be described in this talk, H-space theory soon encountered a number of fundamental difficulties and its initial promise was not realised.  In this talk I shall take a new look at these difficulties and show that they may actually shed light on the nature of quantum (non-linear) gravitons.

The Ricci Flow procedure and its recent application to the solution of the Poincaré Conjecture are reviewed and shown to be incompatible with Einstein's equations. A more general and relativistic geometric flow condition is derived from Nash's theorem, including the emergence of a topological term in Einstein's equation. The result is applied an explanation of the accelerated expansion of the universe. It is also shown that a cosmological constant emerge only when Nash's geometric flow ceases to be. 

In this talk I will discuss information geometric methods in particular the Ruppeiner and Weinhold geometries, their applications to black holes, and our results. It is known that the Ruppeiner geometry is physically significant for many classes of black holes in that it addresses phase transitions, extremal limits and specific heat signatures as encoded information in the metric. 

Despite the successes of the Ruppeiner and Weinhold approaches there remains some important issues to be addressed, in particular the way the coordinate basis for the metric is defined. Recent results show that a free-energy based (Fisher-Rao) metric might be more suitable for some black holes. These puzzling
anomalies are related to how phase transitions appear in the curvature scalar of the information metric. Having a clear idea to these incongruities will lead to a better understanding of critical phenomena (phase transitions) in black holes.

It is important to recognize, derive or know identities satisfied by tensors in many independent fields of mathematics, computation and physics. Pioneering work by Lovelock uncovered a set of dimensionally dependent identities (ddi's) which are both surprising at first sight and very useful. These results were thoroughly analyzed and generalized by Edgar and Höglund, who introduced the notion of "fddi" (fundamental dimensionally-dependent identity) from which other ddi's can be derived, but itself is not deducible from any other ddi.

I will present a systematic and very simple way to derive tensor identities of any degree and order by means of double Hodge dualizations. The method works in any semi-Riemannian manifold of arbitrary dimension and signature. Even though the identities thereby produced are dimensionally dependent, the method is fully independent of the dimension  ---so that identities depending explicitly on the dimension can be easily derived. These identities seem to be new but, according to Edgar (unpublished), they can actually be derived ---via a very complicated way--- from the fddis above, so that both methods would eventually be mathematically equivalent. Nevertheless, the use of the double-dual method reveals identities which would have never been known by the sole application of fddis. Some relevant examples will be shown.

We obtain a purely local characterisation that singles out the Majumdar-Papapetrou class, the near-orizon Bertotti-Robinson geometry and the Reissner-Nordström exterior solution, together with its plane and hyperbolic counterparts, among the static electrovacuum spacetimes. These five classes are found to form the whole set of static Einstein-Maxwell fields without sources and conformally flat space of orbits, this is, the conformastat electrovacuum spacetimes. The main part of the proof consists on showing that a functional relationship between the gravitational and electromagnetic potentials must always exist. The classification procedure provides also an improved characterisation of Majumdar-Papapetrou, by only requiring a conformally flat space of orbits with a vanishing Ricci scalar of the usual conveniently normalised 3-metric. A simple global consideration allows us to state that the asymptotically flat subset of the Majumdar-Papapetrou class and the Reissner-Nordström exterior solution are the only asymptotically flat conformastat electrovacuum spacetimes.

In this talk I look at the limits of using classical linear distributional geometry. I then go on to look at a non-linear theory of distributional geometry based on Colombeau algebras and show that this is compatible with the linear theory in situations where both may be used. For both the linear and non-linear theory of distributional geometry I will use a geometric coordinate free description. I end by looking at applications of these ideas to the description of singularities in the theory of general relativity. In particular I give examples of both distributional curvature tensors and energy-momentum tensors which are well defined within the Colombeau theory but not using the linear theory. In some of these examples the Colombeau result is associated to a classical distribution but in other examples it can only be described using nonlinear distributions.

I will explain the null alignment `boost weight' classification of tensors over a vector space with Lorentzian metric, and mention some recent related results. When applied to the Weyl tensor of an n-dimensional space-time, this yields a natural generalization of the well-known Petrov classification for n=4, based on its Debever principal null directions; this will serve as a guiding theme for this talk. 

Classi finds Petrov type and Segre type and calculates the Riemaniann and its covariant derivatives in the form of the Newman-Penrose spinors and their covariant derivatives. Equating these components to each other in the same frame is used to determine equivalence between different line elements.

Classi, including sources, is available for free. It requires a Lisp system, PSL (Portable Standard Lisp) or CSL (Codemist Standard Lisp), they are included in the free distribution of Reduce at sourceforge.net. It runs on most operating systems including Unix/Linux and Windows with Cygwin installed.