Contra-and Covariant Components

In section 3.1.2 we introduced the contravariant space-time 4-vector whose components are denoted by superscript indices and are given by Eq. (3.7). It is essential to realize that the metric g may be employed in order to lower (or raise) indices of any vector (within one inertial system IS) according to 

The quantities labeled with subscript indices are called the covariant components of the 4-vector X and are given by 

The effect of lowering the index of (the components of) a vector from superscript to subscript is to change the sign of the spatial components. We may thus express the four-dimensional distance between two infinitesimal neighboring events as 

For later convenience we introduce the four-dimensional scalar product between any two 4-vectors a and b by 

As a consequence, the four-dimensional distance between two infinitesimal neighboring events may now also be expressed as 

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Similarly to the nonrelativistic situation [cf. Eq. (2.29)], the components of the Lorentz transformation matrix A may be expressed as derivatives of the new coordinates with respect to the old ones or vice versa. However, since we have to distinguish contra- and covariant components of vectors in the relativistic framework, there are now four different possibilities to express these derivatives ... 

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