ABSTRACT
This chapter shows that Christoffel’s symbols allow to compute the derivative of vectors, one-forms, and tensors of any rank, and that they coincide with the quantities which describe the effects of a gravitational field on moving bodies. It shows how to construct a coordinate frame adapted to an observer. This frame is especially useful to describe physical experiments. The chapter discusses the covariant derivative of vectors, the spacetime metric in a LIF, the covariant derivative of scalars and one-forms, and transformation rules for christoffel’s symbols. It shows that the covariant derivative of the metric tensor vanishes in any coordinate frame.
