By Kirby I. Bland, Michael G. Sarr, Markus W. Büchler, Attila Csendes, Oliver James Garden, John Wong
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Extra info for General Relativity An Introduction For Physicists
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.