We consider supersymmetric AdS/CFT gravity dual pairs and their deformations. First, we consider supersymmetric and integrable deformations: the beta deformation of N = 4 SYM and the gamma deformation, a three-parameter generalization. Then, we consider the eta and lambda deformations of the string worldsheet in AdS5 × S5. Then, the Yang–Baxter deformation, and the generalized supergravity equations.

For extremal black holes, we have the attractor mechanism, originally defined in the context of N = 2 supergravity. This is then interpreted and described in the Sen’s entropy function formalism. The attractor mechanism exists also in five-dimensional gauged supergravity, and by embedding it in string theory, we can relate it to holography and the AdS/CFT correspondence.

After an introduction to general relativity and supersymmetry, the formalism of supergravity is defined, on-shell, off-shell, and in superspace, using coset theory and local superspace. Higher dimensions, extended susy, and KK reduction are also defined. Then, various applications are described: dualities and solution-generating techniques, solutions and their susy algebra, gravity duals and deformations, supergravity on the string worldsheet and superembeddings, cosmological inflation, no-go theorems and Witten’s positive energy theorem, compactification of low-energy string theory and toward embedding the Standard Model using supergravity, susy breaking and minimal supergravity.

We examine cosmological inflation in supergravity. We start with N = 1 supergravity with a single chiral superfield, then consider D-term inflation, with the example of the FI model. We examine possible field redefinitions. Supergravity models with slow-roll conditions satisfied are found. A special embedding of any inflationary model into supergravity is defined. The “alpha attractors” defined by Kallosh and Linde in N = 1 supergravity are defined.

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.