In this chapter, the geometric description of generic branes in Yang–Mills matrix model is elaborated, and structures familiar from gravity are identified. The dynamics resulting from the classical model is interpreted as pre-gravity.

With the form of the target theory built up over the previous two chapters, we move to a geometric description of gravitational motion. By recasting the relative dynamics of a pair of falling objects as the deviation of nearby geodesic trajectories in a spacetime with a metric, Einstein’s equation is motivated. To describe geodesic deviation quantitatively, the Riemann tensor is introduced, and its role in characterizing spacetime structure is developed. With the full field equation of general relativity in place, the linearized limit is carefully developed and compared with the gravito-electro-magnetic theory from the first chapter.

The mathematics required to analyse higher dimensional curved spaces and space-times is developed in this chapter. General coordinate transformations, tangent spaces, vectors and tensors are described. Lie derivatives and covariant derivatives are motivated and defined. The concepts of parallel transport and a connection is introduced and the relation between the Levi-Civita connection and geodesics is elucidated. Christoffel symbols the Riemann tensor are defined as well as the Ricci tensor, the Ricci scalar and the Einstein tensor, and their algebraic and differential properties are described (though technical details of the derivationa of the Rimeann tensor are let to an appendix).