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General theory of Relativity by S. P. Puri

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2

Tensor Analysis and Riemannian Geometry

Part I Line Element

2.1 RIEMANNIAN SPACE

In the Euclidean space of three dimensions, each point is specified by three coordinates (x1, x2, x3). The distance ds between two neighbouring points (x1, x2, x3) and (x1 + dx1, x2 + dx2, x3 + dx3) is given by

 

ds2 = (dx1)2 + (dx2)2 + (dx3)2         (2.1)

 

We may extend the concept of Cartesian space in three dimensions to n-dimensional space. Each point will be designated by n coordinates (x1,x2,…,xn), which are shown collectively by (x). Further, we assume that the distance between any two neighbouring points is given by

Equation

where gμν (x) are functions of the coordinates ...

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