[en] Following an introductory section, the subject is discussed under the headings: on the character of research in black hole astrophysics; isolated holes produced by collapse of normal stars; black holes in binary systems; black holes in globular clusters; black holes in quasars and active galactic nuclei; primordial black holes; concluding remarks on the present state of research in black hole astrophysics. (U.K.)

[en] An extended introduction is followed by a section entitled: 'what is the topology of our universe', in which such topics are considered as the underlying manifold, the qualitative behaviour of the light-cones, causal structure, and determinism. In the next section - 'is our universe singular', the famous singularity theorems are discussed. Finally, under 'how noticeably singular is our universe', the issue of cosmic censorship is discussed, i.e. that of whether or not one expects in certain circumstances that surviving observers will be able to detect singular behaviour in spacetime. (U.K.)

[en] The introductory section includes a brief account of the basic mathematical concept of a black hole in a general dynamical context. This is followed by a more detailed examination of the properties of the horizon with particular reference to situations in which the black hole is allowed to tend asymptotically towards a stationary final equilibrium state. A more specialized description is then provided of the properties of the horizon in an exactly stationary state. Quasi-stationary states are next considered. The mass of a black hole is discussed. The final section summarizes the results that provide the justification for the belief that a stationary, asymptotically flat black hole state is fully determined by its mass and angular momentum when external matter and fields are absent, or by its mass, angular momentum, and electric charge if electromagnetic fields are allowed for. (U.K.)

[en] The chapter begins with an elementary account of the Newman-Penrose formalism and proceeds to the study of the basic problems in the theory of the perturbations of the Kerr metric. The main section headings are: the tetrad formalism; the Newman-Penrose formalism; tetrad transformations and related matters; the Kerr metric and the perturbation problem; the solution of Maxwell's equations; gravitational perturbations; the solution of Dirac's equation; the potential barriers surrounding the Kerr black hole and the problem of reflection and transmission; concluding remarks on the properties of the Kerr metric. (U.K.)

[en] Three main approaches to quantizing gravity are summarized: the operator approach, the canonical approach and the path-integral approach. It is shown how the path integral approach can be applied to the quantization of gravity and how it leads to the concepts of black hole temperature and intrinsic quantum mechanical entropy. The discussion is under the following heads: the action in general relativity; complex spacetime; the indefiniteness of the gravitational action; the stationary-phase approximation; zeta function regularization; the background fields (some positive-definite metrics which are solutions of the Einstein equations in vacuum or with a Λ-term); gravitational thermodynamics; beyond one loop; spacetime foam. (U.K.)

[en] The subject is discussed under the following headings: introduction; the quantum ether (other topologies, curved boundaries - acceleration, black holes); the back reaction; the one-loop approximation (formal relations - the role of the Feynman propagator, representations of the function K, the generalized zeta function, regularization and renormalization, conformal invariance and the trace anomaly, conformally flat spacetimes); the full quantum theory (the gauge group, the Feynman functional integral, globally valid gauge conditions - canonical coordinates, factoring out the gauge group, Gaussian averages, the ghost fields, many-particle Green's functions, the Ward-Takahashi identity); conclusion. (U.K.)

[en] An historical summary is given of the development of the theory of general relativity and the various cosmological models that have been proposed. The search for black holes is sketched. Some of the main areas of work in general relativity in the last sixty years or so are introduced under the following headings: the field equations; cosmology; gravitational collapse; quantum gravity; future prospects. (U.K.)

[en] This paper discusses the possibility that a way may be found to describe gravitation, together with suitable matter fields, by an ordinary renormalizable quantum field theory. However, it is chiefly concerned with another possibility, that a quantum field theory which incorporates gravitation may satisfy a generalized version of the condition of renormalizability known as asymptotic safety. The subject is discussed under the headings: renormalizable theories of gravitation; asymptotic safety; physics at ordinary energies; dimensional continuation; gravity in 2 + epsilon dimensions. (U.K.)

[en] The subject is discussed as follows: introduction; basic notions (Hilbert space, the spacetime, field operators, wave equations, commutation relations, Fock bases, Bogoliubov transformations); applications (isotropic (FRW) cosmological models, anisotropic cosmological models, plane waves, black hole spacetimes); conclusion. (U.K.)

[en] Some outstanding problems in connection with the big bang cosmology and relativity theory are reviewed under the headings: enigmas; nostrums and elixirs (the universe as Phoenix (an oscillating universe), the anthropomorphic universe (existence of observers in the present universe), reproducing universes (could a mini big bang bounce, perhaps adding entropy and matter and eventually developing into a suitable home for observers), variable strength of the gravitational interaction and oscillating universes (possible bounce models that have led eventually to the present hospitable environment). (U.K.)