Abstract
In this chapter we shall introduce the mathematical tools that are needed for dealing with tensors in non-Cartesian coordinates. These tools are useful in ordinary three-dimensional space and in special relativity, but become essential in general relativity, as we shall see in the next chapter. We begin this chapter by introducing the definitions of the transformation rules of the components of tensors and tensor densities. Further we introduce the concept of affine connection which is important in defining the differentiation of tensors in general coordinates, especially the covariant derivative and the absolute derivative of a four-vector. The definition and the properties of the spatial metric tensor are also given in this chapter. As an application we analyze the equation of motion of a mass point in special relativity by using general coordinates.
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© 2002 Birkhäuser Verlag
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Cercignani, C., Kremer, G.M. (2002). Tensor Calculus in General Coordinates. In: The Relativistic Boltzmann Equation: Theory and Applications. Progress in Mathematical Physics, vol 22. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8165-4_10
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DOI: https://doi.org/10.1007/978-3-0348-8165-4_10
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9463-0
Online ISBN: 978-3-0348-8165-4
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