By Ehlers J. (ed.)
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Extra resources for Relativity theory and astrophysics. Relativity and cosmology
Example text
XN ) (k = 1, 2, . . 2) we may write dy k = ∂y k dxm ∂xm (k, m = 1, 2, . . 3) and the metric form can be written as follows: ds2 = gmn dxm dxn (m, n = 1, 2, . . 5) where gmn = δjk is a symmetric second-order system. The square of the infinitesimal line element ds is by definition an invariant in all coordinate systems and dxm is an absolute vector. Thus, using the criteria for the tensor character of systems and the fact that gmn is a symmetric system, we conclude that gmn is an absolute second-order covariant tensor.
N). 3) The components ∂G/∂zk are linear and homogeneous functions of the components ∂G/∂xm . The components of the system ∂G/∂xm are by definition treated as the components of a special type of tensors, called covariant vectors. 2 Contravariant Vectors Generally speaking, any system of quantities, defined with respect to the systems of coordinates {xk } and {zk } by N quantities Ak and A¯ k respectively, which is transformed according to the transformation law Tensors, Relativity, and Cosmology.
Therefore, it was important to introduce a new class of mathematical objects which are defined by their transformation laws with respect to the transformations from one system of coordinates to another. Such mathematical objects are called tensors. The systems, that we call tensors, have linear and homogeneous transformation laws with respect to the transformations of coordinates. In order to define the main types of tensors, let us consider an arbitrary transformation of variables xk into some new variables zk , defined by zk = zk (x1 , x2 , .