By Derek F. Lawden
ISBN-10: 047110082X
ISBN-13: 9780471100829
ISBN-10: 0486425401
ISBN-13: 9780486425405
Easy creation can pay distinct recognition to points of tensor calculus and relativity that scholars locate such a lot tricky. Contents contain tensors in curved areas and alertness to normal relativity idea; black holes; gravitational waves; program of normal relativity ideas to cosmology. quite a few routines. answer consultant on hand upon request.
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Extra info for An Introduction to Tensor Calculus, Relativity, and Cosmology
Example text
Suppose a vector field A is defined over this space. Let P At P the vector A is resolved in be a general point (r, 0), as shown in Fig. 4. the radial and transverse directions, with the respective contravariant components A' and A•. Let the components of A parallel to rectangular Cartesian axes Ox and Oy be Ax and AY. 2) Suppose first that A is a constant vector field. That is, at every point its magnitude and direction are the same. This fact is expressed quantitatively by the relations: Ax= constant, AY =constant.
P and Q are two points respectively on p and q 45 SPACE-TIME CURVATURE such that OP=OQ=u. Let arc PQ=v measure the separation between p and qat a distance u from 0. Then we have the simple relation Fig. 6. (a) The deviation between the two straight lines in the Euclidean plane increases at a uniform rate with respect to the length parameter along either of them. ~-~.. v= 'I' . 16) . In Fig. 6b is shown the corresponding situation on the surface of a 2-sphere of unit radius. , great circles) through O, diverging from 0 at a small angle 4> between them.
6 The Levi-Civita Tensor We now define an important antisymmetric tensor. 41) [ijk/]. To show that E;Jkt is a tensor consider a change of coordinates x 1 -J>. x' 1• We then have, using Eq. ' t;xz = ( -g)l/2 [ijkl] . ' ()xz ()x'm ()x'n ()x'P ()x'q ()x'm ()x'n ()x'P ()x'q ( -g)1/ 2 (mnpq) J { X-J>. 42) I E mnpq· There is, however, one word of caution! In Eq. 42) we have used the square root of Eq. 26) and there could be an ambiguity of sign. This ambiguity is related to the orientation of the coordinate system.