Introduction to the mathematics of general relativity

About: Introduction to the mathematics of general relativity is a research topic. Over the lifetime, 2583 publications have been published within this topic receiving 73295 citations.

One-loop divergences of quantized Einstein-Maxwell fields

Abstract: All one-loop divergences of the quantized Einstein-Maxwell system are calculated, using dimensional regularization and the background-field method. The resulting counterterms, including photon and graviton ghost contributions, are quadratic in curvatures when evaluated on the mass shell, and cannot be renormalized away by rescaling. Brans-Dicke theory is also nonrenormalizable; source-free Einstein theory with cosmological term is (formally) renormalizable, as are Weyl models.

An Introduction to General Relativity and Cosmology

Abstract: 1. How the theory of relativity came into being (a brief historical sketch) Part I. Elements of Differential Geometry: 2. A short sketch of two-dimensional differential geometries 3. Tensors, tensor densities 4. Covariant derivatives 5. Parallel transport and geodesic lines 6. Curvature of a manifold: flat manifolds 7. Riemannian geometry 8. Symmetries of Rieman spaces, invariance of tensors 9. Methods to calculate the curvature quickly - Cartan forms and algebraic computer programs 10. The spatially homogeneous Bianchi-type spacetimes 11. The Petrov classification by the spinor method Part II. The Gravitation Theory: 12. The Einstein equations and the sources of a gravitational field 13. The Maxwell and Einstein-Maxwell equations and the Kaluza-Klein theory 14. Spherically symmetric gravitational field of isolated objects 15. Relativistic hydrodynamics and thermodynamics 16. Relativistic cosmology I: general geometry 17. Relativistic cosmology II: the Robertson-Walker geometry 18. Relativistic cosmology III: the Lemaitre-Tolman geometry 19. Relativistic cosmology IV: generalisations of L-T and related geometries 20. The Kerr solution 21. Subjects omitted in this book References.

Topology in general relativity

Abstract: A number of theorems and definitions which are useful in the global analysis of relativistic world models are presented. It is shown in particular that, under certain conditions, changes in the topology of spacelike sections can occur if and only if the model is acausal. Two new covering manifolds, embodying certain properties of the universal covering manifold, are defined, and their application to general relativity is discussed.

3+1 Formalism and Bases of Numerical Relativity

Abstract: These lecture notes provide some introduction to the 3+1 formalism of general relativity, which is the foundation of most modern numerical relativity. The text is rather self-contained, with detailed calculations and numerous examples. Contents: 1. Introduction, 2. Geometry of hypersurfaces, 3. Geometry of foliations, 4. 3+1 decomposition of Einstein equation, 5. 3+1 equations for matter and electromagnetic field, 6. Conformal decomposition, 7. Asymptotic flatness and global quantities, 8. The initial data problem, 9. Choice of foliation and spatial coordinates, 10. Evolution schemes.

Electrodynamics in the general relativity theory

Abstract: The restricted relativity theory resulted mathematically in the introduction of pseudo-euclidean four-dimensional space and the welding together of the electric and magnetic force vectors into the electromagnetic tensor. Einstein's general relativity theory led to the assumption that the fourdimensional space mentioned above is a curved space and the curvTature was made to account for the gravitational phenomena. The Riemann tensor which measures the curvature and the electromagnetic tensor seem thus to play essentially different roles in physics: the former reflects some properties of the space so that gravitation may be said to have been geometricized,-when the space is given all the gravitational features are determined; on the contrary, it seemed that the electromagnetic tensor is superposed on the space, that it is something external with respect to the space, that after space is given the electromagnetic tensor can be given in different ways. Several attempts were made to geometricize the electromagnetic forces, to find a geometric interpretation for the electromagnetic tensor, to inicorporate this tensor into the space in the sense in which the gravitational forces had been incorporated. It seemed that in order to do this it was necessary to change the geometry; to abandon the Riemanii geometry and to adopt a more general space with a more complicated curvature tensor, one part of which would then account for the gravitational properties and the other wouild in the same way account for the electromagnetic phenomena. H. Weyl arrived in a most natural way to such a generalization. His theory always will remaini a brilliant mathematical feat, but it seems that it did not fulfil the expectationis as a physical theory and the same seems to be true with respect to other attempts. The electromagnetic tensor is, however, niot elntirely independelnt of the Riemann tensor in the ordinary genieral relativity theory; these two tenisors are connected by the so called eniergy relation; it seemed to be desirable to try, witliout breaking the frame of the Riemann geometry, to study