Coordinate Transformations, Tensors and General Relativity

In this chapter, we provide some basic information on coordinate transformations, tensors, partial covariant differentiation, Christoffel symbols and Ricci and Einstein tensors, leading to general relativity. As previously stated, this is necessarily a limited progressive introduction, termed progressive, in the sense that the reader is invited to read on to later sections if more information and further detail are required. Einstein’s general theory of relativity was published over a century ago, and up to this point in time provides the best description of a gravitation field, and is capable of describing a myriad of interesting phenomena in the universe, such as the bending of light through gravitational lensing, the slowing of clocks in gravitational fields and the recently detected ripples in space time due to cataclysmic astrophysical events, such as the coalescence of dense stellar objects. The next generation of GPS systems will require a detailed mapping of the earth’s gravitational field combined with the development of accurate predictive mathematical models. Accordingly for such applications, a superficial understanding may not be sufficient for future space scientists, but rather some prior experience of the actual “gory” details of the discipline may be required.