Handbook of Normal Frames and Coordinates (Progress in Mathematical Physics)

B. Z. Iliev, «Handbook of Normal Frames and Coordinates (Progress in Mathematical Physics) »

Birkhäuser Basel | ISBN: 376437618X | ( October 19 , 2006 ) | 441 pages | PDF | 3.57 MB

“The main subject of this book is an up-to-date and in-depth survey of the theory of normal frames and coordinates in differential geometry. The existing results, as well as new ones obtained lately by the author, on the theme are presented. The text is so organized that it can serve equally well as a reference manual, introduction to and review of the current research on the topic. Correspondingly, the possible audience ranges from graduate and post-graduate students to scientists working in differential geometry and theoretical/mathematical physics. This

is reflected in the bibliography which consists mainly of standard (text)books and journal articles.

The present monograph is the first attempt for collecting the known facts concerting normal frames and coordinates into a single publication. For that reason, the considerations and most of the proofs are given in details.

Conventionally local coordinates or frames, which can be holonomic or not, are called normal if in them the coefficients of a linear connection vanish on some subset, usually a submanifold, of a differentiable manifold. Until recently the existence of normal frames was known (proved) only for symmetric linear connections on submanifolds of a manifold. Now the problems concerning normal frames for derivations of the tensor algebra over a differentiable manifold are well investigate; in particular they completely cover the exploration of normal frames for arbitrary linear connections on a manifold. These rigorous results are important in connection with some physical applications. They may be applied for rigorous analysis of the equivalence principle. This results in two general conclusions: the (strong) equivalence principle (in its ‘conventional’ formulations) is a provable theorem and

the normal frames are the mathematical realization of the physical concept of ‘inertial’ frames. The normal frames find other important physical application in the bundle formulation of quantum mechanics. It turns out that in a normal frame the bundle Heisenberg and Schr¨odinger pictures of motion coincide. Applying some freedom of language, we can state the general physical idea: the normal frames are the most suitable ones for describing free objects and events, i.e., such that on them do not act any forces. Regardless of the different realizations of that idea in general relativity and its generalizations, quantum mechanics, gauge theories etc., there is an underlying mathematical background for the general xiv Preface description of such situations: the existence (or on-existence) of normal frames in vector bundles. This observation fixes to a great extend the mathematical tools required for the description of some fundamental physical theories.

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