The electroweak unification appears mainly in the neutral current processes. The transition probabilities of all of them are predicted in terms of the weak mixing angle. Measuring the weak mixing angle.

Theory predicts the existence of three vector bosons, W+, W− and Z0. It does not predict their masses, but precisely states how they are related with two measured quantities: the Fermi constant and the weak mixing angle. The UA1 experiment and the discovery of the vector bosons.

The precision tests of the electroweak theory performed at the LEP electron–positron collider and at the Tevatron proton–antiproton collider.

The last missing element of the SM, the Higgs boson. The spontaneous symmetry breaking and the boson. The searches at LEP and at the Tevatron. The Large Hadron Collider and the ATLAS and CMS experiments. The discovery of 2012. Checking Higgs physics, measuring its mass and width, its spin and parity, its couplings to the bosons and to the fermions. All agree with the predictions of the Standard Model, so completing the experimental verification of its basic building blocks.

The Bianchi classification of 3-dimensional Lie algebras is introduced by the Schucking method: mapping the structure constants of the algebras into the set of 3×3 matrices, and then considering all the inequivalent combinations of eigenvalues and eigenvectors. A general 4-dimensional metric with a symmetry algebra of Bianchi type is derived. The general metric of a spatially homogeneous and isotropic (= Robertson–Walker, R–W) spacetime is derived. The possible Bianchi types of R–W spacetimes are demonstrated.

It is shown that the Riemann tensor can be calculated in a simpler way when the metric is represented by a basis of differential forms. The formulae for the basis components of the Christoffel symbols (called Ricci rotation coefficients) and of the Riemann tensor are derived. A still-easier way to calculate the Riemann tensor, by using algebraic computer programs, is briefly advertised.

This is a very brief listing of topics that belong to the general area of relativity but were not included in this book.

Solutions of the Einstein and Einstein–Maxwell equations for spherically symmetric metrics (those of Schwarzschild and Reissner–Nordstr\“{o}m) are derived and discussed in detail. The equations of orbits of planets and of bending of light rays in a weak field are derived and discussed. Two methods to measure the bending of rays are presented. Properties of gravitational lenses are described. The proof (by Kruskal) that the singularity of the Schwarzschild metric at r = 2m is spurious is given. The relation of the r = 2m surface to black holes is discussed. Embedding of the Schwarzschild spacetime in a 6-dimensional flat Riemann space is presented. The maximal extension of the Reissner–Nordstr\“{o}m metric (by the method of Brill, Graves and Carter) is derived. Motion of charged and uncharged particles in the Reissner–Nordstr\“{o}m spacetime is described.

The metric tensor and the (pseudo-)Riemannian manifolds are defined. The results of the earlier chapters are specialised to this case, in particular the affine connection coefficients are shown to reduce to the Christoffel symbols. The signature of a metric, the timelike, null and spacelike vectors are defined and the notion of a light cone is introduced. It is shown that in two dimensions the notion of curvature agrees with intuition. It is also shown that geodesic lines extremise the interval (i.e. the ‘distance’). Mappings between Riemann spaces are discussed. Conformal curvature (= the Weyl tensor) is defined and it is shown that zero conformal curvature on a manifold of dimension >=4 implies that the metric is proportional to the flat one. Conformal flatness in three dimensions and the Cotton–York tensor are discussed. Embeddings of Riemannian manifolds in Riemannian manifolds of higher dimension are discussed and the Gauss–Codazzi equations derived. The Petrov classification of conformal curvature tensors in four dimensions with signature (+ - - -) is introduced at an elementary level.

Spinors are defined, their basic properties and relation to tensors are derived. The spinor image of the Weyl tensor is derived and it is shown that it is symmetric in all four of its spinor indices. From this, the classification of Weyl tensors equivalent to Petrov’s (by the Penrose method) is derived. The equivalence of these two approaches is proved. The third (Debever’s) method of classification of Weyl tensors is derived, and its equivalence to those of Petrov and Penrose is demonstrated. Extended hints for verifying the calculations (moved to the exercises section) are provided.

Parallel transport of vectors and tensor densities along curves is defined using the covariant derivative. A geodesic is defined as such a curve, along which the tangent vector, when parallely transported, is collinear with the tangent vector defined at the endpoint. Affine parametrisation is introduced.

The curvature tensor is defined via the commutators of second covariant derivatives acting on tensor densities. It is shown that curvature is responsible for the path-dependence of parallel transport. Algebraic and differential identities obeyed by the curvature tensor are derived. The geodesic deviation is defined, and the equation governing it is derived.

Maxwell’s equations in curved spacetime are presented, and Einstein’s equations with electromagnetic field included in the sources are derived. The attempt to unify electromagnetism with gravitation in the Kaluza–Klein theory is presented.

The derivation of the Einstein equations is presented following Einstein’s method. Hilbert’s derivation (from a variational principle) is also presented. The Newtonian limit of Einstein’s theory is discussed. A Bianchi type I solution of Einstein’s equations with a dust source is derived. A brief review of other theories of gravitation (Brans–Dicke, Bergmann–Wagoner, Einstein–Cartan and Rosen) is presented. The matching conditions for different metrics are derived. The weak-field approximation to general relativity is presented.

The Robertson–Walker metrics are presented as the simplest candidates for the models of our observed Universe. The Friedmann solutions of the Einstein equations (which follow when a R–W metric is taken as an ansatz), with and without the cosmological constant, are derived and discussed in detail. The Milne–McCrea Newtonian analogues of the Friedmann models are derived. Horizons in the R–W models are discussed following the classical Rindler paper. The conceptual basis of the inflationary models is critically reviewed.

This chapter contains detailed hints on how to solve the more difficult exercises and verify some other complicated calculations.