In Chapter 8 we learned that Lorentz boosts and rotations are part of a group called the Lorentz group. Adding translations in space and time, one gets an even larger group, called the Poincaré group. An understanding of this group is very helpful in the construction and interpretation of relativistic theories. Since a relativistic theory has the same basic equations in any inertial frame, these equations should be invariant – or transform among themselves – under all Poincaré group transformations. This places a severe restriction on the structure of allowed theories.

It is natural to ask whether theories could be restricted further by requiring that they have additional symmetries beyond those described by the Poincaré group. The answer, of course, is that there is a variety of possibilities for other types of symmetry besides those that are contained in the Poincaré group. In considering additional symmetries, we can make the following distinction: either the added symmetries act independently from the Poincaré symmetries or they do not. In the first case the additional symmetries form a group G by themselves, and the total symmetry of the theory is a product of the Poincaré symmetries and the G symmetries. In particular, this implies that the G symmetries commute with rotations, and therefore the generators of G must be rotationally invariant. In the second case – which is the one that is realized in supersymmetrical theories – there are new symmetry generators that are not rotationally invariant.

In our study of several solutions in the previous chapters we have mentioned that some special properties that arise for special values of the parameters (mass, charges) are related to supersymmetry; more precisely, to the existence of (unbroken) supersymmetry. Those statements were a bit surprising because we were dealing with solutions of purely bosonic theories (Einstein–Maxwell, Kaluza–Klein …).

The goal of this chapter is to explain the concept and implications of unbroken supersymmetry and how it can be applied in purely bosonic contexts, including pure GR. Supersymmetry will be shown to have a very deep meaning, underlying more familiar symmetries that can be constructed as squares of supersymmetries. At the very least, supersymmetry can be considered as an extremely useful tool that simplifies many calculations and demonstrations of very important results in GR that are related directly or indirectly to the positivity of energy (a manifest property of supersymmetric theories).

As a further reason to devote a full chapter to this topic, unbroken supersymmetry is a crucial ingredient in the stringy calculation of the BH entropy by the counting of microstates. It ensures the stability of the solution and the calculation under classical and quantum perturbations.

To place this subject in a wider context, we will start by giving in Section 13.1 a general definition of residual (unbroken) symmetry and we will relate it to the definition of a vacuum. Vacua are characterized by their symmetries, which determine the conserved charges of point-particles moving in them and, ultimately, the spectra of quantum-field theories (QFTs) defined on them.

Suppose that a subset H of the elements of a group G also satisfies the axioms for a group using the same multiplication rule as in the definition of G. In this case H is called a subgroup of G. Every group has two trivial subgroups. One is the entire group G itself, and the other is the one element set {e} consisting of only the identity element. Any other subgroups are called proper.

Whenever one has a group G with a subgroup H, one can define certain sets called cosets. Specifically, consider the set of group elements formed by multiplying each element of H from the left by a specific group element a ∈ G. This set has the same number of elements as H, but it is not identical to H unless a happens to belong to H. It is conventional to denote this coset by a H. More specifically, since we have used left multiplication, this is a left coset. Right cosets H a are defined in an analogous manner. Note that a coset contains the identity element e only if the subgroup H contains the element a−1. Also, since H satisfies the axioms of a group, this is equivalent to the statement that H contains a. Thus we learn that the coset is either equal to H or it does not contain the identity element. Thus all cosets, other than H itself, are not subgroups of G.

String theory has lived for the past few years during a golden era in which a tremendous upsurge of new ideas, techniques, and results has proliferated. In what form they will contribute to our collective enterprise (theoretical physics) only time can tell, but it is clear that many of them have started to have an impact on closely related areas of physics and mathematics and, even if string theory does not reach its ultimate goal of becoming a theory of everything, it will have played a crucial, inspiring role.

There are many interesting things that have been learned and achieved in this field that we feel can (and perhaps should) be taught to graduate students. However, we have found that this is impossible without the introduction of many ideas, techniques, and results that are not normally taught together in standard courses on general relativity, field theory or string theory, but which have become everyday tools for researchers in this field: black holes, strings, membranes, solitons, instantons, unbroken supersymmetry, Hawking radiation . … They can, of course, be found in various textbooks and research papers, presented from various viewpoints, but not in a single reference with a consistent organization of the ideas (not to mention a consistent notation).

These are the main reasons for the existence of this book, which tries to fill this gap by covering a wide range of topics related, in one way or another, to what we may call semiclassical string gravity.

Hands-on exercise: the stress tensor of a tub of water

To complete this exercise you will need a lab assistant and the following supplies:

• A 2 or 3 liter tub filled with water.

• A square piece of stiff cardboard not more than a third as wide as the tub on a side.

• A marking pen with waterproof ink.

Fill the tub with water. Let it sit at rest on a table or counter. In your mind's eye, imagine Euclidean coordinate system axes for the tub, with the z axis in the vertical direction, and the x and y axes in the horizontal directions in a right-handed orientation relative to the z axis. These coordinate axes will be referred to as the tub frame.

Using the waterproof pen, draw a large round mark on one side of the cardboard. This will represent the direction normal to the surface of the cardboard. On the same side of the cardboard, draw two arrows, each of which is parallel to one side of the square piece of cardboard. These are the two directions tangential to the surface of the cardboard.

Leaving the tub of water at rest, insert the cardboard into the water so that the normal to the surface points in the positive x direction according to the tub frame. One of the arrows on the surface of the cardboard should be pointing in the positive y direction in the tub frame.

Let no one ignorant of Mathematics enter here.

Inscription above the doorway of Plato's Academy

The standard approach to general relativity (GR) is purely geometrical: spacetime is curved by its energy content according to Einstein's equation and test particles move along geodesics. This point of view is what makes GR a theory completely different from the theories that describe all the other known interactions that are special-relativistic field theories (SRFTs) that, after quantization, explain the interaction between two charged bodies as the interchange of quanta of the field.

The enormous success of relativistic quantum field theories with a gauge principle made it unavoidable to try to find a theory of that kind to describe gravitational interactions at a classical and quantum level. This path was followed by many people and it was found that such a theory, whose starting point is the linear perturbation theory of GR (the Fierz–Pauli theory for a free, massless spin-2 particle), would be self-consistent only after the introduction of an infinite number of non-linear terms whose summation should be equivalent to the full non-linear GR theory. Thus, this approach may lead to a different justification of Einstein's theory and provides an alternative interpretation of it that is worth studying. Some of the predictions of GR can be obtained at leading or next to leading order in this approach. Since this is not the standard approach, there are only a few complete treatments in the literature: the book [386], based on Feynman's lectures on gravitation, that also contains many references, some of which we will follow in Section 3.2; and also Deser's lectures on the gravitational field [300]. Reference [30] is also an excellent review with many references.

In the previous chapter, we introduced increasingly complex theories of gravity, starting from GR, to accommodate fermions and we saw that the generalizations of GR that we had to use could be thought of as gauge theories of the symmetries of flat spacetime.

A very important development of the last few decades has been the discovery of supersymmetry and its application to the theory of fundamental particles and interactions. This symmetry relating bosons and fermions can be understood as the generalization of the Poincaré or AdS groups which are the symmetries of our background spacetime to the super-Poincaré or super-AdS (super-)groups which are the symmetries of our background superspacetime, a generalization of standard spacetime that has fermionic coordinates.

It is natural to construct generalizations of the standard gravity theories that can be understood as gauge theories of the (super-)symmetries of the background (vacuum) superspacetime. These generalizations are the supergravity (SUGRA) theories. Given that the kind of fermions that one can have depends critically on the spacetime dimension, the SUGRA theories that one can construct also depend critically on the spacetime dimension. Furthermore, one can extend the standard bosonic spacetime in different ways by including more than one (N) set of fermionic coordinates. This gives rise to additional supersymmetries relating them and, therefore, to supersymmetric field theories and SUGRA theories with N supersymmetries. The latter are also known as extended SUGRAs (SUEGRAs).

In the previous chapter we have seen how scalar fields coupled to gravity arise naturally in KK compactification. In Part III we are also going to see that scalar fields are also present, even before compactification, in some higher-dimensional supergravity theories that are the low-energy effective-field theories of certain superstring theories. In all these examples the scalar fields couple in a characteristic way to vector (or p-form in higher dimensions) field strengths. In this chapter we are going to study first, in Section 12.1, a simple model that synthesizes the main features of those theories.

The a-model describes a real scalar coupled to gravity and to a vector-field strength. The coupling is exponential and depends on a parameter a (hence the name “a-model” that we are giving it here). Since the scalar can be identified in some cases with the string dilaton (or with the KK scalar, which is called also the dilaton sometimes), these models are also generically referred to as dilaton gravity. We will be able to obtain BH-type solutions for general values of a and in any dimension d ≥ 4; however, only a handful of values of a actually occur in the theories of interest, although they occur in many different ways (embeddings [620]).

After studying the main properties of these dilaton BHs, we are going to study in Section 20.1 a more complex (four-dimensional) model that involves several scalar and vector fields. We are going to obtain extreme BH solutions that can be understood as composite BHs.

The asymptotically flat, static, spherically symmetric Schwarzschild and RN BH solutions that we have studied in the two previous chapters were the only solutions of the Einstein and Einstein–Maxwell equations with those properties. To find more solutions, we have to relax these conditions or couple to gravity more general types of matter, as we will do later on. If we stay with the Einstein(–Maxwell) theory, one possibility is to look for static, axially symmetric solutions and another possibility is to relax the condition of staticity and only ask that the solution be stationary, which implies that we have to relax the condition of spherical symmetry as well and look for stationary, axisymmetric spacetimes. In the first case one finds solutions like those in Weyl's family [949, 950] which can be interpreted as describing the gravitational fields of axisymmetric sources with arbitrary multipole momenta or Melvin's solution [692] (which has cylindrical symmetry and was constructed earlier by Bonnor [165] via a Harrison transformation [499] of the vacuum), among many others. In the second case, we find the Kerr–Newman BHs [617, 723] with angular momentum and electric or magnetic charge and also the Taub–Newman–Unti–Tambourino (Taub–NUT) solution [724, 879], which may but need not include charges. The Taub–NUT metric does not describe a BH because it is not asymptotically flat. In fact, the only stationary axially symmetric BHs of the Einstein–Maxwell theory belong to the Kerr–Newman family of solutions (see e.g. [532, 533]).

The Taub–NUT solution has a number of features that are particularly interesting for us, which we are going to discuss in this chapter.