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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.
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01 Jan 1990TL;DR: The authors provided an introduction to general relativity for mathematics undergraduates or graduate physicists, focusing on an intuitive grasp of the subject and a calculational facility rather than on a rigorous mathematical exposition.
Abstract: This textbook provides an introduction to general relativity for mathematics undergraduates or graduate physicists. After a review of Cartesian tensor notation and special relativity the concepts of Riemannian differential geometry are introducted. More emphasis is placed on an intuitive grasp of the subject and a calculational facility than on a rigorous mathematical exposition. General relativity is then presented as a relativistic theory of gravity reducing in the appropriate limits to Newtonian gravity or special relativity. The Schwarzchild solution is derived and the gravitational red-shift, time dilation and classic tests of general relativity are discussed. There is a brief account of gravitational collapse and black holes based on the extended Schwarzchild solution. Other vacuum solutions are described, motivated by their counterparts in linearised general relativity. The book ends with chapters on cosmological solutions to the field equations. There are exercises attached to each chapter, some of which extend the development given in the text.
99 citations
TL;DR: In this article, a field theory representing a natural generalization of the theory of relativity is constructed by using a tetrad-space, and a unique set of field equations exactly equal in number (16) to the unknowns used, and having the same strength as those of general relativity, is obtained.
Abstract: A field theory representing a natural generalization of the theory of relativity is being constructed by using a tetrad-space. A unique set of field equations exactly equal in number (16) to the unknowns used, and having the same strength as those of general relativity, is obtained. All physical elements of interest are related directly to the members of the geometrical structure.
99 citations
TL;DR: The eigenvalues of the Dirac operator are diffeomorphism-invariant functions of the geometry as discussed by the authors, namely, ''observables'' for general relativity.
Abstract: The eigenvalues of the Dirac operator are diffeomorphism-invariant functions of the geometry, namely, ``observables'' for general relativity. Recent work by Chamseddine and Connes suggests taking them as gravity's dynamical variables. We compute their Poisson brackets, find that these can be expressed in terms of energy momenta $T$ of the eigenspinors, and show that $T$ is the Jacobian matrix of the transformation from metric to eigenvalues. We consider a small modification of the spectral action that gets rid of the cosmological term, and derive its equations of motion. These are solved if $T$ scales linearly. We show that such a scaling law yields Einstein equations.
96 citations
TL;DR: In this paper, the Cauchy problem for obtaining an independent complete set of such quantities ("observables") is discussed, and it is also pointed out that the observables obtained may alternatively be viewed as the metric tensor in a special "gauge" (i.e., with a special coordinate condition).
Abstract: The construction of a complete set of quantities in general relativity, whose functional form is invariant under coordinate transformations, is indicated. The set obtained is highly redundant. The Cauchy problem for obtaining an independent complete set of such quantities ("observables") is therefore discussed. It is also pointed out that the observables obtained may alternatively be viewed as the metric tensor in a special "gauge" (i.e., with a special coordinate condition). This latter viewpoint may facilitate the quantization of general relativity.
95 citations
TL;DR: In this paper, the well known method ofNewman andPenrose is used to find solutions of the Einstein empty space field equations, which are algebraically special and where the degenerate principal null vectors are not hypersurface orthogonal.
Abstract: The well known method ofNewman andPenrose is used to find solutions of the Einstein empty space field equations, which are algebraically special and where the degenerate principal null vectors are not hypersurface orthogonal. As is to be expected the method systematically yields the results obtained byKerr. An explanation is given of the complex coordinate transformation technique of generating new metrics from Schwarzschild's; also a generalisation of Kerr and Schild type metrics is investigated.
95 citations