Abstract:
The study of dynamics in general relativity has been hampered by a lack of coordinate independent measures of chaos. Here we present a variety of invariant measures for quantifying chaotic dynamics in relativity by exploiting the coordinate independence of fractal dimensions. We discuss how preferred choices of time naturally arise in chaotic systems and how the existence of invariant signals of chaos allow us to reinstate standard coordinate dependent measures. As an application, we study the Mixmaster universes and find it to exhibit transient soft chaos.

Abstract:
We clarify some points about the systems considered by Sota, Suzuki and Maeda in Class. Quantum Grav. 13, 1241 (1996). Contrary to the authors' claim for a non-homoclinic kind of chaos, we show the chaotic cases they considered are homoclinic in origin. The power of local criteria to predict chaos is once more questioned. We find that their local, curvature--based criterion is neither necessary nor sufficient for the occurrence of chaos. In fact, we argue that a merit of their search for local criteria applied to General Relativity is just to stress the weakness of locality itself, free of any pathologies related to the motion in effective Riemannian geometries.

Abstract:
The classical notion of center of mass for an isolated system in general relativity is derived from the Hamiltonian formulation and represented by a flux integral at infinity. In contrast to mass and linear momentum which are well-defined for asymptotically flat manifolds, center of mass and angular momentum seem less well-understood, mainly because they appear as the lower order terms in the expansion of the data than those which determine mass and linear momentum. This article summarizes some of the recent developments concerning center of mass and its geometric interpretation using the constant mean curvature foliation near infinity. Several equivalent notions of center of mass are also discussed.

Abstract:
We show that the general theory of relativity can be formulated in the language of Weyl geometry. We develop the concept of Weyl frames and point out that the new mathematical formalism may lead to different pictures of the same gravitational phenomena. We show that in an arbitrary Weyl frame general relativity, which takes the form of a scalar-tensor gravitational theory, is invariant with respect to Weyl tranformations. A kew point in the development of the formalism is to build an action that is manifestly invariant with respect to Weyl transformations. When this action is expressed in terms of Riemannian geometry we find that the theory has some similarities with Brans-Dicke gravitational theory. In this scenario, the gravitational field is not described by the metric tensor only, but by a combination of both the metric and a geometrical scalar field. We illustrate this point by, firstly, discussing the Newtonian limit in an arbitrary frame, and, secondly, by examining how distinct geometrical and physical pictures of the same phenomena may arise in different frames. To give an example, we discuss the gravitational spectral shift as viewed in a general Weyl frame. We further explore the analogy of general relativity with scalar-tensor theories and show how a known Brans-Dicke vacuum solution may appear as a solution of general relativity theory when reinterpreted in a particular Weyl frame. Finally, we show that the so-called WIST gravity theories are mathematically equivalent to Brans-Dicke theory when viewed in a particular frame.

Abstract:
This work reports on the construction of a nonlinear distributional geometry (in the sense of Colombeau's special setting) and its applications to general relativity with a special focus on the distributional description of impulsive gravitational waves.

Abstract:
In this work the late-time evolution of Bianchi type $VIII$ models is discussed. These cosmological models exhibit a chaotic behaviour towards the initial singularity and our investigations show that towards the future, far from the initial singularity, these models have a non-chaotic evolution, even in the case of vacuum and without inflation. These space-time solutions turn out to exhibit a particular time asymmetry. On the other hand, investigations of the late-time behaviour of type $VIII$ models by another author have the result that chaos continues for ever in the far future and that these solutions have a time symmetric behaviour: this result was obtained using the approximation methods of Belinski, Khalatnikov and Lifshitz ($BKL$) and we try to find out a possible reason explaining why the different approaches lead to distinct outcomes. It will be shown that, at a heuristic level, the $BKL$ method gives a valid approximation of the late-time evolution of type $VIII$ models, agreeing with the result of our investigations.

Abstract:
General relativity exhibits a unique feature not represented in standard examples of chaotic systems; it is a spacetime diffeomorphism invariant theory. Thus many characterizations of chaos do not work. It is therefore necessary to develop a definition of chaos suitable for application to general relativity. This presentation will present results towards this goal.

Abstract:
In this preliminary work, we shall reveal the intrinsic geometry of background currents, possibly of electromagnetic origin, in the space-time of General Relativity. Drawing a close analogy between the object of our present study and electromagnetism, we shall show that there exists an inherent, fully non-linear, conservative third-rank radiation current which is responsible for the irregularity in the curvature of the background space(-time), whose potential (generator) is of purely geometric origin.

Abstract:
An attempt is made of giving a self-contained introduction to holomorphic ideas in general relativity, following work over the last thirty years by several authors. The main topics are complex manifolds, spinor and twistor methods, heaven spaces.

Abstract:
We show that the Schwarzschild solution can be embedded in a class of
nonstandard solutions of the vacuum Einstein’s equations with arbitrary
rotation curves. These nonstandard solutions have to be taken as physical, if
dark matter as needed in the standard theory cannot be found. As a consequence
general relativity is considered as a classical field theory in Minkowski space
and not as a geometric theory in the sense of Einstein. Assuming an
asymptotically flat rotation curve and introducing a material disk into this
model we find a matter density in accordance with the Tully- Fisher relation.