Introduction

In this chapter, we will develop further the mathematical formalism required to understand different aspects of curved spacetime, focusing on the description of spacetime curvature. It uses extensively the concepts developed in Chapter 4, especially the idea of parallel transport. Most of the topics described here will be used in the subsequent chapters, except for the ideas discussed in Section 5.6 related to the classification of spacetime curvature which fall somewhat outside the main theme of development.

Three perspectives on the spacetime curvature

The discussion in the previous chapter used only the fact that the metric tensor depended on the coordinates. This dependence can arise either due to the use of curvilinear coordinates in flat spacetime or due to genuine curvature of the spacetime. The gravitational field generated by matter manifests itself as the curvature of the spacetime and hence, to study the gravitational effects, we need to develop the mathematical machinery capable of describing and analysing the curvature of spacetime. This will be the aim of the current chapter and we shall begin by introducing the concept of spacetime curvature from three different – but closely related – perspectives. At a fundamental level, these three perspectives stem from the same source, viz. behaviour of vectors under parallel transport; however, we will discuss them separately in the next three subsections for greater clarity.

Parallel transport around a closed curve

The first perspective on curvature originates from the changes induced in a vector when it is parallel transported around a small, closed, curve in the spacetime.