1,721,048 research outputs found
Introducing Einstein's Relativity: A deeper understanding
No full textThere is little doubt that Einstein's theory of relativity captures the imagination. Not only has it radically altered the way we view the universe, but the theory also has a considerable number of surprises in store. This is especially so in the three main topics of current interest that this book reaches, namely: black holes, gravitational waves, and cosmology. The main aim of this textbook is to provide students with a sound mathematical introduction coupled to an understanding of the physical insights needed to explore the subject. Indeed, the book follows Einstein in that it introduces the theory very much from a physical point of view. After introducing the special theory of relativity, the basic field equations of gravitation are derived and discussed carefully as a prelude to first solving them in simple cases and then exploring the three main areas of application. This new edition contains a substantial extension content that considers new and updated developments in the field. Topics include coverage of the advancement of observational cosmology, the detection of gravitational waves from colliding black holes and neutron stars, and advancements in modern cosmology. Einstein's theory of relativity is undoubtedly one of the greatest achievements of the human mind. Yet, in this book, the author makes it possible for students with a wide range of abilities to deal confidently with the subject. Based on both authors' experience teaching the subject this is achieved by breaking down the main arguments into a series of simple logical steps. Full details are provided in the text and the numerous exercises while additional insight is provided through the numerous diagrams. As a result this book makes an excellent course for any reader coming to the subject for the first time while providing a thorough understanding for any student wanting to go on to study the subject in depth<br/
A global theory of algebras of generalized functions II: tensor distributions
No full textWe extend the construction of [19] by introducing spaces of generalized tensor fields on smooth manifolds that possess optimal embedding and consistency properties with spaces of tensor distributions in the sense of L. Schwartz. We thereby obtain a universal algebra of generalized tensor fields canonically containing the space of distributional tensor fields. The canonical embedding of distributional tensor fields also commutes with the Lie derivative. This construction provides the basis for applications of algebras of generalized functions in nonlinear distributional geometry and, in particular, to the study of spacetimes of low differentiability in general relativity
Generalised hyperbolicity in spacetimes with string-like singularities
No full textIn this paper we present well-posedness results for H^1 solutions of the wave equation for spacetimes that contain string-like singularities. These results extend a framework in which one characterises gravitational singularities as obstruction to the dynamics of test fields rather than point particles. In particular,we discuss spacetimes with cosmic strings
Quantised vortices and mutual friction in relativistic superfluids
No full textWe consider the detailed dynamics of an array of quantised superfluid vortices in the framework of general relativity, as required for quantitative modelling of realistic neutron star cores. Our model builds on the variational approach to relativistic (multi-) fluid dynamics, where the vorticity plays a central role. The description provides a natural extension of, and a better insight into, existing Newtonian models. In particular, we account for the mutual friction associated with scattering of a second ‘normal’ component in the mixture off of the superfluid vortices. This is an important step which facilitates the connection with the involved microphysics
Synthetic versus distributional lower Ricci curvature bounds
No full textWe compare two standard approaches to defining lower Ricci curvature bounds for Riemannian metrics of regularity below C2. These are, on the one hand, the synthetic definition via weak displacement convexity of entropy functionals in the framework of optimal transport, and the distributional one based on non-negativity of the Ricci-tensor in the sense of Schwartz. It turns out that distributional bounds imply entropy bounds for metrics of class C1 and that the converse holds for C1,1-metrics under an additional convergence condition on regularizations of the metric
Generalised hyperbolicity in spacetimes with Lipschitz regularity
No full textIn this paper we obtain general conditions under which the wave equation is wellposed in spacetimes with metrics of Lipschitz regularity. In particular, the results can be applied to spacetimes where there is a loss of regularity on a hypersurface such as shell-crossing singularities, thin shells of matter, and surface layers. This provides a framework for regarding gravitational singularities not as obstructions to the world lines of point-particles, but rather as obstruction to the dynamics of test fields
Generalised connections and curvature
No full textThe concept of generalised (in the sense of Colombeau) connection on a principal fibre bundle is introduced. This definition is then used to extend results concerning the geometry of principal fibre bundles to those that only have a generalised connection. Some applications to singular solutions of Yang–Mills theory are given
Combining Cauchy and characteristic codes III: the interface problem in axial symmetry
No full textThis paper is part of a long term program to develop combined Cauchy and characteristic codes as investigative tools in numerical relativity. In this, the third stage of the program, attention is devoted to axisymmetric systems possessing two spatial degrees of freedom. The method relies on being able to pass information backwards and forwards across an interface exterior to any central source present. Formulas are obtained which show how it is possible to relate the canonical forms of the interior Cauchy and exterior characteristic metric functions, and their derivatives, on the interface
Nonlinear generalised functions on manifolds
No full textThis paper lays the foundations for a nonlinear theory of differential geometry that is developed in a subsequent paper which is based on Colombeau algebras of tensor distributions on manifolds. We adopt a new approach and construct a global theory of algebras of generalised functions on manifolds based on the concept of smoothing operators. This produces a generalisation of previous theories in a form which is suitable for applications to differential geometry. The generalised Lie derivative is introduced and shown to commute with the embedding of distributions. It is also shown that the covariant derivative of a generalised scalar field commutes with this embedding at the level of association
Green operators for low regularity spacetimes
No full textIn this paper we define and construct advanced and retarded Green operators for the wave operator on spacetimes with low regularity. In order to do so we require that the spacetime satisfies the condition of generalised hyperbolicity which is equivalent to well-posedness of the classical inhomogeneous problem with zero initial data where weak solutions are properly supported. Moreover, we provide an explicit formula for the kernel of the Green operators in terms of an arbitrary eigenbasis of H 1 and a suitable Green matrix that solves a system of second order ODEs.</p
- …
