The metric tensor is a rank 2 tensor which may also be called the fundamental

The metric tensor is a rank 2 tensor which may also

This preview shows page 150 - 153 out of 171 pages.

functionalities permeate the whole discipline of tensor calculus. The metric tensor is a rank-2 tensor which may also be called the fundamental ten- sor . The main purpose of the metric tensor is to generalize the concept of distance to general curvilinear coordinate frames and maintain the invariance of distance in different coordinate systems. In orthonormal Cartesian coordinate systems the distance element squared, ( ds ) 2 , between two infinitesimally neighboring points in space, one with coordinates x i and the other with coordinates x i + dx i , is given by: ( ds ) 2 = dx i dx i = δ ij dx i dx j (240) This definition of distance is the key to introducing a rank-2 tensor, g ij , called the metric tensor which, for a general coordinate system, is defined by: ( ds ) 2 = g ij dx i dx j (241) The above defined metric tensor is of covariant type. The metric tensor has also a contravariant type which is usually notated with g ij . 149
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6 METRIC TENSOR 150 The components of the covariant and contravariant metric tensor are given by: g ij = E i · E j g ij = E i · E j (242) where the indexed E are the covariant and contravariant basis vectors as defined in § 2.6.1. The metric tensor has also a mixed type which is given by: g i j = E i · E j = δ i j g j i = E i · E j = δ j i (243) and hence it is the same as the unity tensor . For a coordinate system in which the metric tensor can be cast in a diagonal form where the diagonal elements are ± 1 the metric is called flat . For Cartesian coordinate systems, which are orthonormal flat-space systems, we have: g ij = δ ij = g ij = δ ij (244) The metric tensor is symmetric in its two indices, that is: g ij = g ji g ij = g ji (245) This can be easily explained by the commutativity of the dot product of vectors in reference to the above equations involving the dot product of the basis vectors. The contravariant metric tensor is used for raising covariant indices of covariant and mixed tensors, e.g. A i = g ik A k A ij = g ik A j k (246) Similarly, the covariant metric tensor is used for lowering contravariant indices of con-
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6 METRIC TENSOR 151 travariant and mixed tensors, e.g. A i = g ik A k A ij = g ik A k j (247) In these raising and lowering operations the metric tensor acts, like a Kronecker delta, as an index replacement operator as well as shifting the position of the index. Because it is possible to shift the index position of a tensor by using the covariant and contravariant types of the metric tensor, a given tensor can be cast into a covariant or a contravariant form, as well as a mixed form in the case of tensors of rank > 1 . However, it should be emphasized that the order of the indices must be respected in this process, because two tensors with the same indicial structure but with different indicial order are not equal in general, as stated before. For example: A i j = g jk A ik 6 = A i j = g jk A ki (248) Some authors insert dots (e.g. A i · j and A · i j ) to remove any ambiguity about the order of the indices.
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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