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By Roger Blanford, Kip Thorne

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3 Accordingly, we shall regard vectors as unchanged by such transport. This enables us to ignore the issue of where in space a vector actually resides; it is completely determined by its direction and its length. 95 T Fig. 2: A rank-3 tensor T. g. Newton’s F = ma, unchanged. 3 This is not so in curved spaces, as we shall see in Sec. 7. 4 This is a different use of the word rank than for a matrix, whose rank is its number of linearly independent rows or columns. 6 we shall regard T as a box (Fig.

12c) does not say that [Ri¯p ] is a symmetric matrix. , Eq. 14) below. 13a) and correspondingly for the components of a tensor Tp¯q¯r¯ = Rp¯i Rq¯j Rr¯k Tijk , Tijk = Ri¯p Rj q¯Rk¯r Tp¯q¯r¯ . 11) for the bases. To make these laws look natural, we have placed the transformation matrix on the left in the former and on the right in the latter. In Minkowski spacetime (Chap. 2), the placement of indices, up or down, will automatically tell us the order. 13c) xi = Ri¯p xp¯ . The product of two rotation matrices, [Ri¯p Rp¯s¯] is another rotation matrix [Ris¯], which transforms the Cartesian bases es¯ to ei .

Under this product rule, the rotation matrices form a mathematical group: the rotation group, whose group representations play an important role in quantum theory. 6 **Example and Practice: Rotation in x, y Plane7 Consider two Cartesian coordinate systems rotated with respect to each other in the x, y plane as shown in Fig. 4. 14) 0 0 1 7 Exercises marked with double stars are important expansions of the material presented in the text. y y ey ey ex x φ x ex Fig. 4: Two Cartesian coordinate systems {x, y, z} and {¯ x, y¯, z¯} and their basis vectors in Euclidean space, rotated by an angle φ relative to each other in the x, y plane.

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Applications of Classical Physics (web draft april 2013) by Roger Blanford, Kip Thorne

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