Starting from the definition of tensorial objects by their response to coordinate transformation, this chapter builds the flat space vector calculus machinery needed to understand the role of the metric and its associated geodesic curves in general. The emphasis here is on using tensors to build equations that are “generally covariant,” meaning that their content is independent of the coordinate system used to express them. Motivated by the transformation of gravitational energy sources, the gravitational field should be a second-rank tensor, and given the way in which that tensor must show up in a particle motion Lagrangian, it is natural to interpret that tensor as a metric.

One of the prominent advantages of the gauge formalism as a field theory is its sophisticated mathematical structure, being based on analytical mechanics. Everything about the system dynamics that we need can be derived by rote out of a Lagrangian density of the system, which itself can be determined uniquely based on the prescribed symmetry underlying in the physical phenomenon we want to describe. In our case, we can find how the dislocation and defect fields should be incorporated into the continuum theory of elasticity, with direct correspondences to the differential geometrical (DG) counterparts introduced in Chapter 6. Also the formalism can provide us with a bridge between the DG pictures and the method of quantum field theory (QFT) discussed in Chapter 8 via the Lagrangian density.

This chapter introduces how we can use the quantum fields introduced in the previous chapter to access amplitudes and, thus, measurable quantities, such as the cross sections and the particle lifetime. More specifically, an educational tour of quantum electrodynamics (QED), which describes the interaction of electrons (or any charged particles) with photons, is proposed. Although this chapter uses concepts from quantum field theory, it is not a course on that topic. Rather, the aim here is to expose the concepts and prepare the reader to be able to do simple calculations of processes at the lowest order. The notions of gauge invariance and the S-matrix are, however, explained. Many examples of Feynman diagrams and the calculation of the corresponding amplitudes are detailed. Summation and spin averaging techniques are also presented. Finally, the delicate concept of renormalisation is explained, leading to the notion of the running coupling constant.

In this chapter, we will discuss Riemannian metrics on infinite-dimensional spaces. Particular emphasis will be placed on the new challenges which arise on infinite-dimensional spaces. One new feature is that Riemannian metrics comes in several flavours on infinite-dimensional spaces. These are not present in the finite dimensional setting. The strongest flavour (as we shall see) is the notion of a strong Riemannian metric which is treated in classical monographs on infinite-dimensional geometry. It is also the most restrictive setting as it forces one to work on Hilbert manifolds. Of greater interest are for this reason the weak Riemannian metrics which are however possibly ill behaved. As an example we will discuss at length geodesics for Riemannian metrics on infinite-dimensional spaces. The aim is to exhibit examples of Riemannian manifolds for which the finite dimensional theory breaks down and the geodesic distance vanishes.

We equip differentiable manifolds with a metric and introduce differential geometry, which provides the mathematical formalism underlying the theory of general relativity and many other applications in different areas of physics, science, and engineering.

This last part of the book introduces the Einstein equation – the basic equation of general relativity, in much the same way that Maxwell’s equations are the basic equations of electromagnetism. Geometries such as the Schwarzschild geometry, or those of the FRW cosmological models, are particular solutions of the Einstein equation. Just three new mathematical ideas are needed to give an efficient and standard discussion of the Einstein equation: a more precise definition of vectors in terms of directional derivatives; the notion of dual vectors as a linear map from vectors to real numbers; and the covariant derivative of a vector field in curved spacetime. These mathematical concepts are introduced in this chapter.

This Chapter describes, in concise manner, aspects of differential geometry that are necessary to follow the developments of this book. We give several definitions of the concept of the manifold, illustrated by a number of examples. We then define differential forms, which are viewed as the most primitive objects one can put on a manifold. We define their wedge product and the operation of exterior differentiation. We then define the notions necessary to define the integration of differential forms. After this we define vector fields, their Lie bracket, interior product, then tensors. We then describe the Lie derivative. We briefly talk about distributions and their integrability conditions. Define metrics and isometries. Then define Lie groups, discuss their action on manifolds, then define Lie algebras. Describe main Cartan's isomoprhisms. Define fibre bundles and the Ehresmann connections. Define principal bundles and connections in them. Describe the Hopf fibration. Define vector bundles and give some canonical examples of the latter. Describe covariant differentiation. Briefly reivew Riemannian geometry and the affine connection. We end this Chapter with a description of spinors and their relation to differential forms.