Doubt is cast on the much quoted result of Yakupov that the torsion vector in embedding class two vacuum space-times is necessarily a gradient vector. This result is equivalent to the fact that the two second fundamental forms associated with the embedding necessarily commute. This result greatly simplifies the integrability conditions (Codazzi equations) and has been assumed in most later investigations of class 2 vacuum space-times.

Yakupov stated the result without proof, but hinted that it followed purely algebraically from his identity: Rˆkl_ij C_kl = 0, where C_ij is the commutator of the two second fundamental forms associated with the embedding.

A new simplified proof of Yakupov’s identity is presented in which it is shown to follow purely algebraically from the Gauss equations and vacuum conditions. Then several examples are presented of non-commuting second fundamental forms that satisfy Yakupov's identity and the vacuum condition. Thus it appears unlikely that Yakupov's results can follow purely algebraically. These examples are all necessarily of Petrov type I, N or type III.

Similar techniques are employed to investigate Einstein spaces (R_ab = κg_ab) of embedding class two. Here it appears that there exist very many examples in which the commutator of the second fundamental forms is non-zero contrary to Yakupov’s assertion. All Petrov types seem to be possible except types III, N & O.

The integrability conditions arising from the Codazzi and Ricci embedding equations turn out to be quite complex and these are still currently being investigated. Preliminary results indicate that any vacuum solutions must be of Petrov type N and belong to Kundt's class with twist-free and expansion-free rays.

Ingemar Bengtsson

Trapped surfaces in spherical symmetry

Abstract:

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.

Ayse Humeyra Bilge

The application of the Riquier-Janet theory to the construction of "Complete tables"

Abstract:

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.

Oihane Fernández Blanco

Characterization and classification of second-order symmetric Lorentzian manifolds

Abstract:

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.

Michael Bradley

Classification of the conformally flat pure radiation metrics

Abstract:

A comparison between the Cartan-Karlhede classification of the conformally flat pure radiation metrics, found by Edgar and Ludwig, and the construction of these metrics in terms of the Generalised Invariant Formalism (GIF), developed by Ramos and Vickers, is made.

Thomas Bäckdahl

How to measure deviation from the Kerr and Minkowski spacetimes on a slice

Abstract:

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.

Vitalij Chatyrko

Brian's activity for Africa

Alfonso Garsía-Parrado

A new class of five-dimensional Petrov type D space-times

Abstract:

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.

Magnus Herberthson

On the magnetic part of the Weyl tensor at spacelike infinity

Abstract:

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.

Emma Jakobsson

Trapped surfaces in 2+1 dimensions

Abstract:

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.

Malcolm Ludvigsen

H-space: its achievements and its goals

Abstract:

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.

Marcos Maia

The Poincaré Conjecture and the Cosmological Constant

Abstract:

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.

Narit Pidokrajt

Geometry of black hole thermodynamics: Energy vs Entropy representation

Abstract:

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.

José Senovilla

A systematic approach to tensor identities

Abstract:

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.

Estelita Vaz

Elasticity and cylindrical symmetry

Abstract:

Due to recent developments in astrophysics, with the research on neutron stars, which have found to be in elastic states, the theory of relativistic elasticity reveals to be important for many significant applications in this context, where spherically symmetric and axially symmetric elastic space-times are used to model the astrophysical objects.

Following a study on the EFE´s for spherically symmetric elastic configurations, we pursued the study of elasticity for axially symmetric spacetimes. Here some cylindrically symmetric static configurations are considered, the corresponding energy-momentum tensor and Einstein field equations being investigated.

Raül Vera

A local characterisation for static charged black holes

Abstract:

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.

James Vickers

Nonlinear distributional geometry

Abstract:

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.

Lode Wylleman

Classification of tensors in space-times of arbitrary dimension

Abstract:

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.

Jan Åman

Classification of space-times with the computer algebra program Classi

Abstract:

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.