Differential Geometry and the Derivation of the Kerr Metric

This book developed from a personal determination to understand how the Kerr metric can be derived from Einstein's field equation of general relativity. The first half of the book contains a detailed account of the methods of differential geometry suitable for physicists. The aim was to explain the methods used in more advanced approaches to general relativity in a way which would be accessible to undergraduates. The book tries to strike a balance between the abstract rigour of pure mathematics and the needs of the student of general relativity who is mainly interested in applications. The topics covered include: manifolds, tangent vectors, one-forms, tensors, alternating tensors, and differential forms. The book also gives a thorough account of the related ideas of the covariant derivative and connections on a manifold. Also covered are the topics: tensor fields, the Lie derivative, the Lie bracket, the exterior derivative, the torsion and curvature of an affine connection, the Riemann and Ricci tensors, the metric tensor, isometries, Cartan's equations of structure, and tetrad frames. The first half of the book concludes with an explanation of how Cartan's equations and tetrad frames can be used to solve Einstein's field equation.The second half of the book is devoted to a complete and thorough derivation of the Kerr metric which models the spacetime surrounding a rotating black hole. This important solution of Einstein's field equation is often quoted in books on general relativity, but none seem to include a full derivation of it. This book sets out to rectify the deficiency. Indeed, the author hopes that the book will be found to be useful for anyone who wants to see how the Kerr metric can be derived, as well as those wanting a digestible account of the methods of differential geometry. Read more