General Relativity An Introduction For Physicists by Kirby I. Bland, Michael G. Sarr, Markus W. Büchler, Attila

By Kirby I. Bland, Michael G. Sarr, Markus W. Büchler, Attila Csendes, Oliver James Garden, John Wong

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3 Curves and surfaces 27 The association of points with the values of their parameters can be thought of as a mapping of the points of a manifold into points of the Euclidean space of the same dimension. This means that ‘locally’ a manifold looks like the corresponding Euclidean space: it is ‘smooth’ and has a certain number of dimensions. 1 These N coordinates are entirely general and are denoted collectively by xa , where it is understood that a = 1 2 N. As a technical point, we should mention that in general it may not be possible to cover the whole manifold with only one non-degenerate coordinate system, namely, one which ascribes a unique set of N coordinate values to each point, so that the correspondence between points and sets of coordinate values (labels) is one-to-one.

The index a appearing on each side of this equation is said to be a free index and may take on separately any value from 1 to N . 6 Geometry of manifolds 31 appears in the denominator of a partial derivative as a subscript (and vice versa). Thus the index b on the right-hand side in effect appears once as a subscript and once as a superscript, and hence there is an implied summation from 1 to N . An index that is summed over in this way is called a dummy index, because it can be replaced by any other index not already in use.

11 A path in the t x -plane can be specified by giving one coordinate in terms of the other, for example x = x t , or alternatively by giving both coordinates as functions of a parameter along the curve: t = t x=x . For massive particles the natural parameter to use is the proper time . is the worldline of a particle, moving at constant speed v along the x-axis of S, which passes through the origin of S at t = 0. 11 The Doppler effect A useful illustration of particle worldlines and the concept of proper time is provided by deriving the Doppler effect in a transparently four-dimensional manner.

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