New Transformation Equations and the Electric Field Four-vector

19. Covariant Formulation in General Coordinates

Until now, we have limited our study to four-dimensional rectangular coordinates in Euclidean spacetime. However, in order to describe our laws in general four-dimensional coordinate systems in Euclidean spacetime we must make several adjustments. We begin with the relation for the element of spacetime interval, which is the analog of (3.4)

where the are the components of the metric tensor. In order to handle more general coordinate systems, every previous occurrence of must be replaced by . Our formulations of the coordinate transformation equations, (6.2) and (6.4), become

and

respectively. Where previously it was unnecessary to distinguish between covariant and contravariant components of tensors, since there is no distinction in rectangular coordinates, we must now specify which we are using. To comply with convention, superscripts will indicate contravariant components and subscripts will indicate covariant components.

In rectangular coordinates, the derivative of a tensor is, simply, the ordinary derivative. But in general coordinates, we have terms which include nonzero Christoffel symbols. These terms vanish in rectangular coordinates since the components of the metric tensor are constants. In general coordinates, however, the components of the metric tensor are not always constant, thus, the Christoffel symbols do not always vanish. Because of this, all ordinary derivatives of tensors must be replaced by their covariant derivatives, for example, becomes where ";" indicates the covariant derivative of with respect to .

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