A student-friendly style, over 100 illustrations, and numerous exercises are brought together in this textbook for advanced undergraduate and beginning graduate students in physics and mathematics. Lewis Ryder develops the theory of general relativity in detail. Covering the core topics of black holes, gravitational radiation, and cosmology, he provides an overview of general relativity and its modern ramifications. The book contains chapters on gravitational radiation, cosmology, and connections between general relativity and the fundamental physics of the microworld. It explains the geometry of curved spaces and contains key solutions of Einstein's equations - the Schwarzschild and Kerr solutions. Mathematical calculations are worked out in detail, so students can develop an intuitive understanding of the subject, as well as learn how to perform calculations. The book also includes topics concerned with the relation between general relativity and other areas of fundamental physics. Selected solutions for instructors are available under Resources.

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Introduction to General Relativity

We introduce a model designed to describe charged particles as stable topological solitons of a field with values on the internal space S^3. These solitons behave like particles with relativistic properties like Lorentz contraction and velocity dependence of mass. This mass is defined by the energy of the soliton. In this sense this model is a generalisation of the sine-Gordon model from 1+1 dimensions to 3+1 dimensions, from S^1 to S^3. (We do not chase the aim to give a four-dimensional generalisation of Coleman's isomorphism between the Sine-Gordon model and the Thirring model which was shown in 2-dimensional space-time.) For large distances from the center of solitons this model tends to a dual U(1)-theory with freely propagating electromagnetic waves. Already at the classical level it describes important effects, which usually have to be explained by quantum field theory, like particle-antiparticle annihilation and the running of the coupling.

A Model for Topological Fermions

Starting with El Naschie's ideas of complex conjugate time the inverse Lorentz transformation of time rate is extended onto an internal gravity (artificially induced accelerations) and then expanded onto an external gravitational field. This result raises the total number of physically identifiable distinct dimensions to six. It also implies that the physical reality we live in looks like abstract multispatial geometric structure composed of several single 3D spaces. This conclusion suggests that there are few more physical dimensions to be uncovered. In such a multispatial hyperspace any particle pictured as a point in linear vector space could be viewed as a string or even as a surface in a space that is dual to the primary one.

On conjugate complex time III: Superstrings and complex Lorentz transformation

Nonstandard Lagrangians entitled ‘nonnaturals’ by Arnold have recently gained increasing importance both in applied mathematics and in physical theories. These types of Lagrangians appear in some group of dissipative dynamical systems, and they play an important role in a number of field theories. However, the role of nonstandard Lagrangians in geometric theories like general relativity is still absent. In this communication, we would like to discuss the relevance of nonstandard Lagrangians in general relativity using the principles of calculus of variations. In fact, nonstandard Lagrangians came in different forms, features, and characteristics, depending on the nature of the dynamical problem under study. In this work, we will be concerned with time-dependent Lagrangians of the form L 1+ γ(t) . After deriving the modified geodesic equation using the basic techniques of Riemannian differential geometry which will be used to axiomatize a large part of our work, we show that many interesting consequences will be raised accordingly when applied to FRW cosmology.

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Some consequences of nonstandard Lagrangians with time-dependent coefficients in general relativity

In the complex plane thep-harmonic equation div(|∇u|p−2∇u) = 0, 1 <p < ∞, exhibits some features reminiscent of Function Theory. Our results about curvature in this structure complement known facts about minimal surfaces and harmonic functions. Quasiregular mappings are used.

Richard L. Faber

Papers

Differential Geometry and Relativity Theory: An Introduction