In 1931 Chandrasekhar established an upper bound for the mass of a cold self–gravitating star in thermal equilibrium (Chandrasekhar 1931a, 1931b). This leads one to consider the ultimate fate of a star which, having radiated all its thermo–nuclear energy, still has a mass beyond the critical limit (a few solar masses). Once the nickel and iron core has been formed, there exists no possibility for any further nuclear reactions; the core must therefore undergo gravitational collapse. The collapse may cease by the time the core has reached nuclear densities, which leads to the formation of a neutron star, provided that the mass of the collapsing part lies below the critical value. If this is not the case, then nothing can prevent total gravitational collapse (Chandrasekhar 1939, Oppenheimer and Snyder 1939, Oppenheimer and Volkoff 1939), resulting in the formation of a black hole (Wheeler 1968; see Israel 1987 for a historical review).

Birkhoff's theorem (Birkhoff 1923), which states that a spherically symmetric spacetime is locally isometric to a part of the Schwarzschild–Kruskal metric (Kruskal 1960), yields a significant simplification in the discussion of the spherically symmetric collapse scenario (Harrison et al. 1965). However, in order to treat more general situations, one has to find the generic features of gravitational collapse in general relativity. This was achieved by Geroch, Hawking, Penrose and others in the late sixties and early seventies (Hawking and Penrose 1970; see also Hawking and Ellis 1973, Clarke 1975, 1993): The singularity theorems show that - in contrast to Newtonian gravity - deviations from spherical symmetry, internal pressure or rotation do not prevent the formation of a singularity.

Einstein's field equations form a set of nonlinear, coupled partial differential equations. In spite of this, it is still sometimes possible to find exact solutions in a systematic way by considering space-times with symmetries. Since the laws of general relativity are covariant with respect to diffeomorphisms, the corresponding reduction of the field equations must be performed in a coordinate–independent way. This is achieved by using the concept of Killing vector fields. The existence of Killing fields reflects the symmetries of a spacetime in a coordinate–invariant manner.

A spacetime (M, g) admitting a Killing field gives rise to an invariantly defined 3–manifold Σ. However, Σ is only a hypersurface of (M, g) if it is orthogonal to the Killing trajectories. In general, Σ must be considered to be a quotient space M/G rather than a subspace of M. (Here G is the 1–dimensional group generated by the Killing field.) The projection formalism for M/G was developed by Geroch (1971, 1972a), based on earlier work by Ehlers (see also Kramer et al. 1980). The invariant quantities which play a leading role are the twist and the norm of the Killing field.

In the first section of this chapter we compile some basic properties of Killing fields. The twist, the norm and the Ricci 1–form assigned to a Killing field are introduced in the second section. Using these quantities, we then give the complete set of reduction formulae for the Ricci tensor.

In the third section we apply these formulae to vacuum space-times. In particular, we introduce the vacuum Ernst potential and derive the entire set of field equations from a variational principle.

The area theorem is probably one of the most important results in classical black hole physics. It asserts that (under certain conditions which we specify below) the area of the event horizon of a predictable black hole spacetime cannot decrease. This result bears a resemblance to the second law of thermodynamics. The analogy is reinforced by the similarity of the mass variation formula to the first law of ordinary thermodynamics. Within the classical framework the analogy is basically of a formal, mathematical nature. There exists, for instance, no physical relationship between the surface gravity, κ, and the classical temperature of a black hole, which must be assigned the value of absolute zero. Nevertheless, on account of the Hawking effect, the relationship between the laws of black hole physics and thermodynamics gains a deep physical significance: The temperature of the black–body spectrum of particles created by a black hole is κ/2π. This also sheds light on the analogy between the entropy and the area of a black hole.

The Killing property of a stationary event horizon implies that its surface gravity is constant. If the Killing fields are integrable (that is, in static or circular spacetimes), the zeroth law of black hole physics is a purely geometrical property of Killing horizons. Otherwise, it is a consequence of Einstein's equations and the dominant energy condition.

The Komar expression for the mass of a stationary spacetime provides a formula giving the mass in terms of the total angular momentum, the angular velocity, the surface gravity and the area of the horizon.

Quantum mechanics, as for example in the case of a non-relativistic particle, can be treated in either of two ways. One can work with the differential-equation form of the theory, by studying the Schrödinger equation. Alternatively, one can study the Feynman path integral, which gives the integral form of the Schrödinger differential approach. The Feynman path integral has the advantage of incorporating the boundary conditions on the particle, for example that the particle is at spatial position xa at an initial time ta, and at position xb at final time tb. The path integral leads naturally to a semi-classical expansion of the quantum amplitude, valid asymptotically as the action of the classical solution of the equations of motion becomes large compared to Planck's constant ħ.

One moves from quantum mechanics to quantum gravity by replacing the spatial argument x of the wave function by the three-dimensional spatial geometry hij(x). A typical quantum amplitude is then the amplitude to go from an initial three-geometry hijI to a final geometry hijF, specified (say) on identical three-surfaces ΣI, ΣF. To complete the description in the asymptotically flat case, one needs to specify asymptotic parameters such as the time T between the two surfaces, measured at spatial infinity. To make the classical boundary-value problem elliptic and (one hopes) well-posed, one rotates to imaginary time –iT. The Feynman path integral would again give a semi-classical expansion of the quantum amplitude, were it not for the infinities present in the loop amplitudes.

Introduction

Before embarking on the full theory of N = 1 supergravity in the following chapters, it is necessary to review some of what is known about quantum cosmology based on general relativity, possibly coupled to spin-0 or spin-1/2 (non-supersymmetric) matter. The ideas presented in this chapter, based to a considerable extent but not exclusively on Hamiltonian methods, will recur throughout the book. Perhaps the main underlying idea is that there is an analogy between the classical dynamics of a point particle with position x and that of a three-geometry hij(x). The theory of point-particle dynamics, when written in parametrized form [Kuchař 1981] and cast into Hamiltonian form, and the theory of general relativity, again in Hamiltonian form, bear a strong resemblance. In the Hamiltonian form of general relativity, hij(x) can be taken to be the ‘coordinate’ variable, corresponding to x in particle dynamics. In section 2.2, for parametrized particle dynamics, it is shown following [Kuchař 1981] how a constraint arises classically in the Hamiltonian theory, which, when quantized, gives the appropriate Schrödinger or wave equation for the quantum wave function ψ(x, t). As described in subsequent sections, the quantization of the analogous constraint in general relativity gives the Wheeler–DeWitt equation [DeWitt 1967, Wheeler 1968], a second-order functional differential equation for the wave function Ψ[hij(x)], which contains all the information in quantum gravity, if only one could solve and interpret it.

The Hamiltonian form of general relativity is derived from the Einstein–Hilbert Lagrangian in section 2.3.

The application of canonical methods to gravity has a long history [De-Witt 1967]. In [Dirac 1950] a general Hamiltonian approach was presented, which allowed for the presence of constraints in a theory, due to the momenta not being independent functions of the velocities. In particular, this occurs in general relativity, because of the underlying coordinate invariance of gravity. The general approach above was applied to general relativity in [Dirac 1958a,b, 1959] and further described in [Dirac 1965]. It was seen that there are four constraints, usually written ℋi(i = 1,2,3) and ℋ⊥, associated with the freedom to make coordinate transformations in the spatial and normal directions relative to a hypersurface t = const. in the Hamiltonian decomposition. Classically, these four constraints must vanish for allowed initial data. In the quantum theory, as will be seen in chapter 2, these constraints become operators on physically allowed states Ψ, which must obey ℋiΨ = 0, ℋ⊥Ψ = 0. Here, in the simplest representation, Ψ is a functional of the spatial metric hij(x). It was shown in [Higgs 1958, 1959] that the constraints ℋiΨ = 0 precisely describe the invariance of the wave function under spatial coordinate transformations. The Hamiltonian formulation of gravity was also studied by [Arnowitt et al. 1962], who provided the standard definition of the mass or energy M of a spacetime, as measured at spatial infinity.

Quasars, which can be a thousand times brighter than an ordinary galaxy, are the most distant objects observable in the Universe. How quasars produce the luminosity of 1013 suns in a volume the size of the solar system continues to be a major question in astronomy. Distant quasars are very rare objects whose study has been blocked by their scarcity. Recent technical advances, however, have opened new paths for their discovery. Forty quasars with redshifts greater than 4 have been found since 1986. Redshift 4 corresponds to a light travel time of more than 10 billion years. As a result, we are now able to probe the epoch shortly after the Big Bang when quasars may have first formed and to study the universe when it was less than a tenth its present age.

Quasars were one of the main discoveries thirty years ago that revolutionized astronomy. While they and the black holes thought to occur in their centers have become household words today, quasars are as enigmatic in many ways as they were when first discovered. Whatever their nature, they offer us views of the Universe never before seen, especially at distances far beyond what astronomers of the previous generation expected to see. In this chapter I wish to review briefly their history, how extraordinary their properties are, and how they serve as probes of the Universe to nearly as far as the visible horizon.