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.

Supernova and supernova remnant research are two of the most active fields of modern astronomy. SN 1987A has given us a chance to observe a supernova explosion and its aftermath in unprecedented detail, a process that continues to unfold today. Meanwhile, thanks to major advances in optical, radio, and X-ray astronomy, we have gained unprecedented views of the populations and spectrum evolution of supernovae of all kinds. These results have spurred a renaissance in theoretical studies of supernovae. Likewise, samples of well-observed supernovae are becoming large enough that we are closing fast on the goal of using supernovae to determine the cosmic distance scale.

Studies of supernovae and supernova remnants are inextricably linked and we are learning fast about the connections. We now recognize that mass loss from the supernova progenitor star can determine the structure of the circumstellar medium with which the supernova ejecta interact. An outstanding example is the ring around SN1987A. There are many supernovae in which much of the early optical, radio and X-ray emission are due to interaction of the ejecta with circumstellar matter rather than radioactivity within the supernova itself. Just in time for this colloquium, nature provided a particularly spectacular example of such an interacting supernova with SN1993J in M81, one of the brightest supernovae of this century. Moreover, the X-ray spectra of supernova remnants provide a powerful new tool to measure supernova nucleosynthesis yields.

Observational selection effects and the lack of accurate distances for most Galactic SNRs pose problems for studies of the distribution of SNRs in the Galaxy. However, by comparing the observed Galactic longitude distribution of high surface brightness SNRs with that expected from simple models – which avoids some of the problems with selection effects and the lack of distances – a Gaussian scale length of ≈ 7 kpc in Galactocentric radius is obtained for SNRs.

Introduction

The distribution of SNRs in the Galaxy is of interest for many astrophysical studies, particularly in relation to their energy input into the ISM and for comparison with the distributions of possible progenitor populations. Such studies are, however, not straightforward. First, current catalogues of SNRs miss objects due to observational selection effects. Second, there are no reliable distance estimates available for most identified remnants. Here I use a sample of 182 Galactic SNRs from a recently revised catalogue (this proceedings), all but one of which have observed radio flux densities and angular sizes, to derive the distribution of SNRs in the Galaxy by comparing the observed distribution of bright remnants with Galactic longitude with that expected from simple models.

The Problems

The Selection Effects

Although, as discussed by Aschenbach (this proceedings), many new SNRs may soon be identified from the ROSAT X-ray survey, the identification of SNRs in existing catalogues has, generally, been made at radio wavelengths.