Does the world really need a new textbook on general relativity? I feel that my first duty in presenting this book should be to provide a convincing affirmative answer to this question.

There already exists a vast array of available books. I will not attempt here to make an exhaustive list, but I will mention three of my favourites. For its unsurpassed pedagogical presentation of the elementary aspects of general relativity, I like Schutz's A first course in general relativity. For its unsurpassed completeness, I like Gravitation by Misner, Thorne, and Wheeler. And for its unsurpassed elegance and rigour, I like Wald's General Relativity. In my view, a serious student could do no better than start with Schutz for an outstanding introductory course, then move on to Misner, Thorne, and Wheeler to get a broad coverage of many different topics and techniques, and then finish off with Wald to gain access to the more modern topics and the mathematical standard that Wald has since imposed on this field. This is a long route, but with this book I hope to help the student along. I see my place as being somewhere between Schutz and Wald – more advanced than Schutz but less sophisticated than Wald – and I cover some of the few topics that are not handled by Misner, Thorne, and Wheeler.

In the winter of 1998 I was given the responsibility of creating an advanced course in general relativity. The course was intended for graduate students working in the Gravitation Group of the Guelph-Waterloo Physics Institute, a joint graduate programme in Physics shared by the Universities of Guelph and Waterloo.

Our purpose in this chapter is to develop the mathematical techniques required in the description of congruences, the term designating an entire system of nonintersecting geodesics. We will consider separately the cases of timelike geodesics and null geodesics. (The case of spacelike geodesics does not require a separate treatment, as it is virtually identical to the timelike case; it is also less interesting from a physical point of view.) We will introduce the expansion scalar, as well as the shear and rotation tensors, as a means of describing the congruence's behaviour. We will derive a useful evolution equation for the expansion, known as Raychaudhuri's equation. On the basis of this equation we will show that gravity tends to focus geodesics, in the sense that an initially diverging congruence (geodesics flying apart) will be found to diverge less rapidly in the future, and that an initially converging congruence (geodesics coming together) will converge more rapidly in the future. And we will present Frobenius' theorem, which states that a congruence is hypersurface orthogonal – the geodesics are everywhere orthogonal to a family of hypersurfaces – if and only if its rotation tensor vanishes.

The chapter begins (in Section 2.1) with a review of the standard energy conditions of general relativity, because some of these are required in the proof of the focusing theorem. It continues (in Section 2.2) with a pedagogical introduction to the expansion scalar, shear tensor, and rotation tensor, based on the kinematics of a deformable medium. Congruences of timelike geodesics are then presented in Section 2.3, and the case of null geodesics is treated in Section 2.4.

This chapter covers three main topics that can all be grouped under the rubric of hypersurfaces, the term designating a three-dimensional submanifold in a fourdimensional spacetime.

The first part of the chapter (Sections 3.1 to 3.3) is concerned with the intrinsic geometry of a hypersurface, and it examines the following questions: Given that the spacetime is endowed with a metric tensor gαβ, how does one define an induced, three-dimensional metric hab on a specified hypersurface? And once this three-metric has been introduced, how does one define a vectorial surface element that allows vector fields to be integrated over the hypersurface? While these questions admit straightforward answers when the hypersurface is either timelike or spacelike, we will see that the null case requires special care.

The second part of the chapter (Sections 3.4 to 3.6) is concerned with the extrinsic geometry of a hypersurface, or how the hypersurface is embedded in the enveloping spacetime manifold. We will see how the spacetime curvature tensor can be decomposed into a purely intrinsic part – the curvature tensor of the hypersurface – and an extrinsic part that measures the bending of the hypersurface in spacetime; this bending is described by a three-dimensional tensor Kab known as the extrinsic curvature. We will see what constraints the Einstein field equations place on the induced metric and extrinsic curvature of a hypersurface.

The third part of the chapter (Sections 3.7 to 3.11) is concerned with possible discontinuities of the metric and its derivatives across a hypersurface.

This first chapter is devoted to a brisk review of the fundamentals of differential geometry. The collection of topics presented here is fairly standard, and most of these topics should have been encountered in a previous introductory course on general relativity. Some, however, may be new, or may be treated here from a different point of view, or with an increased degree of completeness.

We begin in Section 1.1 by providing definitions for tensors on a differentiable manifold. The point of view adopted here, and throughout the text, is entirely unsophisticated: We do without the abstract formulation of differential geometry and define tensors in the old-fashioned way, in terms of how their components transform under a coordinate transformation. While the abstract formulation (in which tensors are defined as multilinear mappings of vectors and dual vectors into real numbers) is decidedly more elegant and beautiful, and should be an integral part of an education in general relativity, the old approach has the advantage of economy, and this motivated its adoption here. Also, the old-fashioned way of defining tensors produces an immediate distinction between tensor fields in spacetime (four-tensors) and tensor fields on a hypersurface (three-tensors); this distinction will be important in later chapters of this book.

Covariant differentiation is reviewed in Section 1.2, Lie differentiation in Section 1.4, and Killing vectors are introduced in Section 1.5. In Section 1.3 we develop the mathematical theory of geodesics. The theory is based on a variational principle and employs an arbitrary parameterization of the curve.

Long after Newton's equations of motion were well known in the physics and mathematics communities, there were a series of formal developments that helped to give a deeper understanding of the significance of these equations. These developments also provided a mathematical framework for describing and understanding new theories as they arose. This framework is useful not only for systems of point particles, but also for classical field theories, such as the Max well theory, and even for quantum field theories and string theories.

In this appendix, we survey some of the basic concepts and tools that have been developed. When all is said and done, the main message is that the central object for describing a theory, be it a classical or a quantum theory, is the action S. In classical theories the equations of motion are obtained by making the action stationary. In quantum theories one computes a probability amplitude by integrating the phase exp (i S/ħ) h over all possible trajectories.

Classical mechanics

Each dynamical system can be characterized by a set of independent variables that specify its degrees of freedom. For example, in the case of a theory of a collection of point particles, a possible choice is the positions of the particles. Let us refer to these coordinates as qi(t). The superscript i is mean to label both the directions of space and the particles. So if there are N particles in three dimensions, it takes 3N values.

A minimal action principle is a basic ingredient of any field theory. With it (with an action) we can systematically find conserved currents and charges, canonically conjugate momenta, and a Hamiltonian (which is necessary for canonical quantization), etc. On the other hand, it is easier to deal with actions than with equations of motion; it is easier to include new fields and couplings in the action respecting certain symmetries than to invent new consistent equations of motion for them and modifications of the equations of motion of the old fields.

In this chapter we are going to study in detail several action principles for GR and for more general theories we will be concerned with later on. First, we will study the standard second-order Einstein–Hilbert action that we found as the result of imposing self-consistency on the Fierz–Pauli theory coupled to matter. We will derive the Einstein equations from it and we will find the right boundary term that will allow us to impose boundary conditions on the variations of the metric δgμν only, not on its derivatives. We will do the same for theories including a scalar and in a conformal frame that is not Einstein's. In these theories, an extra scalar factor K (which could be e–2φ in the string effective action) appears multiplying the Ricci scalar and obtaining the gravitational equations becomes more involved.

We are also going to study the behavior of the Einstein–Hilbert action under GCTs and we will obtain the Bianchi (gauge) identity and Noether current associated with them and see how they are modified by the addition of boundary terms to the action.

In the previous chapter we obtained and studied the Schwarzschild solution of the vacuum Einstein equations and arrived at the BH concept. However, many of the general features of BHs that we discussed, such as the no-hair conjecture, make reference to BHs in the presence of matter fields. In this chapter we are going to initiate the study and construction of BH solutions of the Einstein equations in the presence of matter fields, starting with the simplest ones: massless scalar and vector fields.

The (unsuccessful) search for BH solutions of gravity coupled to a scalar field will allow us to deepen our understanding of the no-hair conjecture.

The (successful) search for BH solutions of gravity coupled to a vector field will allow us to find the simplest BH solution different from Schwarzschild's: the Reissner–Nordström (RN) solution. Simple as it is, it has very interesting features, in particular, the existence of an extreme limit with a regular horizon and zero Hawking temperature that will be approached with positive specific heat, as in standard thermodynamical systems. Later on we will relate some of these properties to the unbroken supersymmetry of the extreme RN (ERN) solution, which will allow us to reinterpret it as a self-gravitating supersymmetric soliton interpolating between two vacua of the theory.

The ERN BH is the archetype of the more complicated self-gravitating supersymmetric solitons that we are going to encounter later on in the context of superstring low-energy effective actions (actually, one of our goals will be to recover it as a superstring solution) and many of its properties will be shared by them. Furthermore, the four-dimensional Einstein–Maxwell system exhibits electric–magnetic duality in its simplest form.