In this chapter some empty space solutions of Einstein's are presented. The form of the Ricci tensor for a general spherical spherically symmetric static metric is given, from which the Schwarzschild solution is derived. Gravitational waves are presented as a solution of Einstein’s equations in empty space in a linear approximation.

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).

The concept to the metric is introduced. Various geometries, both flat and curved, are described including Euclidean space; Minkowski space-time; spheres; hyperbolic planes and expanding space-times. Lorentz transformations and relativistic time dilation in flat space-time is discussed as well as gravitational red-shift and the Global Positioning System. Hubble expansion and the cosmological red-shift are also explained.

One occasionally encounters the misconception that two-component spinor notation is somehow inherently ill-suited or unwieldy for practical use. Perhaps this is due in part to a lack of examples of calculations using two-component language in the pedagogical literature. In this chapter, we seek to dispel this idea by presenting Feynman rules for external fermions using two-component spinor notation, intended for practical calculations of cross sections, decays, and radiative corrections.

Consider a collection of $N$ two-component left-handed $(\frac{1}{2},0)$ fermions. The corresponding free-field Lagrangian is invariant under a global $\text{U}\left(N\right)$ symmetry. When a mass term and interactions are added to the theory, the global $\text{U}\left(N\right)$ symmetry is broken down to a discrete ${\mathbb{Z}}_2$ symmetry that reflects the fact that any term in the Lagrangian must contain an even number of fermion fields.

In Chapter 1, we focused on quantum field theories of free fermions. In order to construct renormalizable interacting quantum field theories, we must introduce additional fields. The requirement of renormalizability imposes two constraints. First, the couplings in the interaction Lagrangian must have nonnegative mass dimension.

In this chapter, we devise a set of Feynman rules to describe matrix elements of processes involving spin-1/2 fermions. The rules are developed for two-component fermions and are then applied to tree-level decay and scattering processes and the fermion self-energy functions in the one-loop approximation.

Despite the successes of the Standard Model, it seems very likely that new physics is present at the TeV scale.