#### Bill_K

Science Advisor

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Continue reading...The Absolute Derivative

In relativity we typically deal with two types of quantities: fields, which are defined everywhere, and particle properties, which are defined only along a curve or world line. The familiar covariant derivative is appropriate when we need to differentiate a field. A field is a function of all four coordinates, and the covariant derivative of ##\varphi(x)## consists of the four partial derivatives ##\partial \varphi /\partial x^\mu## plus correction terms involving the Christoffel symbols, one for each tensor index on ##\varphi##.

A particle property ##\varphi(s)##, on the other hand, is a function only of a single parameter ##s## running along the curve. In this situation, the partial derivatives of ##\varphi## with respect to the four coordinates do not exist. (Unfortunately many references miss this point!) Writing partial derivatives would require that ##\varphi## be defined everywhere in a neighborhood of the curve, which is not the case. The...