Download General Relativity.. An Introduction for Physicists by Hobson, Efstathious, And Lasenby PDF

By Hobson, Efstathious, And Lasenby

ISBN-10: 0511137958

ISBN-13: 9780511137952

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3), we obtain xA = xA − vtA xB = xB − vtB relating the coordinates of the ends of the rod in S to the coordinates in S. The observer in S measures the length of the rod at a fixed time t = tA = tB as = xB − xA = 1 xB − xA = 0 Hence in S the rod appears contracted to the length = 0 1 − v2 /c2 1/2 If a rod is moving relative to S in a direction perpendicular to its length, however, it is straightforward to show that it suffers no contraction. It thus follows that the volume V of a moving object, as measured by simultaneously noting the positions of the boundary points in S, is related to its proper volume V0 by V = V0 1 − v2 /c2 1/2 .

Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated.

For this manifold, we could avoid the degeneracy at the origin by using, for example, Cartesian coordinates. For a general manifold, however, we might have no choice in the matter and might have to work with coordinate systems that cover only a portion of the manifold, called coordinate patches. For example, the set of points making up the surface of a sphere forms a two-dimensional manifold (called S 2 ). This manifold is usually ‘parameterised’ by the coordinates and , but is degenerate at the poles.

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