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Some of the difficulties of interpretation of general relativity theory can be avoided by employing the tetrad formalism. The timelike vector of any tetrad is identified physically with the velocity vector of an observer, the spacelike vectors with the directions of local Cartesian coordinate axes used by him. It is found that the Newtonian concept of nonrotation is represented most closely if the spacelike vectors undergo Fermi propagation along the observer's world-line.The behaviour of free particles is investigated by referring to tetrads the equation of geodesic deviation, which provides an immediate physical interpretation for the Riemann tensor. A simple model of a gyroscope is constructed by modifying Papapetrou's equations of a spinning test particle. The behaviour of this model supports the interpretation of Fermi propagation.The interpretation is developed by investigating the discontinuity in the Riemann tensor across the boundary of a world-tube of matter, using Lichnerowicz's conditions. * Present address: King's College, University of London, London, W.C.2. England. * c = 1 here and throughout. The range and summation conventions are assumed for Greek indices over 0, 1, 2, 3 and for Latin indices over 1, 2, 3. * * Cosmological space-times and other space-times admitting groups of motions may have preferred coordinate systems defined by their special symmetry properties, but the present argument is about general space-times which cannot be assumed to have such symmetries.

This is an introduction to geometrical topics that are useful in applied mathematics and theoretical physics, including manifolds, metrics, connections, Lie groups, spinors and bundles, preparing readers for the study of modern treatments of mechanics, gauge fields theories, relativity and gravitation. The order of presentation corresponds to that used for the relevant material in theoretical physics: the geometry of affine spaces, which is appropriate to special relativity theory, as well as to Newtonian mechanics, is developed in the first half of the book, and the geometry of manifolds, which is needed for general relativity and gauge field theory, in the second half. Analysis is included not for its own sake, but only where it illuminates geometrical ideas. The style is informal and clear yet rigorous; each chapter ends with a summary of important concepts and results. In addition there are over 650 exercises, making this a book which is valuable as a text for advanced undergraduate and postgraduate students.

The string-model analog of general relativity is shown to be unphysically dependent on an embedding gauge. Moreover

Plane gravitational waves are here defined to be non-flat solutions of Einstein’s empty spacetime field equations which admit as much symmetry as do plane electromagnetic waves, namely, a 5-parameter group of motions. A general plane-wave metric is written down and the properties of plane wave space-times are studied in detail. In particular, their characterization as 4 plane ’ is justified further by the construction of 4 sandwich waves ’ bounded on both sides by (null) hyperplanes in flat space-time. It is shown that the passing of a sandwich wave produces a relative acceleration in free test particles, and inferred from this that such waves transport energy.

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