Table of contents

All perfect fluid, spherically symmetric spacetimes admitting a proper inheriting conformal Killing vector (ICKV) which is neither parallel to nor orthogonal to the fluid 4-velocity are found, thereby completing the analysis of a previous work. In particular, the results in this and earlier work are summarized in terms of a general theorem.

The new nonchiral Lagrangian density for the Einstein field equations, which leads to Nester's frames and Hamiltonian formulations in Ashtekar variables, is shown to arise naturally as a canonical 4-form on the second frame bundle, of pure grade 4 in m=4 Clifford algebra. This insight allows such Lagrangians to be written as easily for higher dimensional theories.

The authors present the finite-harmonic solution of the constraint equations of the spinor representation of the relativistic string. Choosing a gauge, they make a harmonic decomposition in the form of a product representation. This finite-harmonic approach is then compared with that of Hughston and Shaw (1988). They describe a recursive method for relating series and product parameters, and comment briefly on the question of a generalization for the infinite harmonic case and on the quantization of such systems.

The coupling of a N=4 affine gauge theory to N=1 supergravity is investigated. For the unbroken Kac-Moody theory the known sum rule for finite Lie multiplets is extended to cover the infinite dimensional Kac-Moody multiplet. For the broken Kac-Moody theory, the infinite Kac-Moody multiplet splits into an infinite set of finite multiplets representing a finite Lie symmetry * the U(1) generated by L₀; in this case there is a separate sum rule for each Lie multiplet. The gravitationally coupled theory is more restricted but at the level of this investigation is not qualitatively different from the corresponding global theory.

The authors propose a minisuperspace construction of the ground-state (no boundary) wavefunction for a radiation-dominated universe. As in the case of a free massless conformally invariant scalar field discussed by Hartle and Hawking (1983), the functional integrations over the gravitational and gauge fields can be performed independently. The gauge part of the wavefunction is semiclassically computed with the help of the (anti-)selfdual solutions of the Euclideanized SO(4)-symmetric Einstein-Yang-Mills (EYM) systems. Implications of their construction are discussed.

A mathematical treatment of perturbed Friedmann models is given. Local existence, convergence and analytic linearization stability for the constraints for k=0, k+or-1 are shown. To prove this, special function spaces are introduced in order to invert certain elliptic differential operators. These function spaces can also be employed to formulate precisely some decomposition theorems used for the study of linearized perturbations (see Stewart and Durrer) where elliptic operators also have to be inverted. The results for the constraints cannot easily be generalized to the evolution equations. In particular one-parameter families which depend analytically on a parameter do not exist in general. This is a generalization of a result of Rendall (1990).

Dust-filled Friedmann-Robertson-Walker universes are canonically quantized in a 'slow-time' gauge t=p_R, where p_R is the canonical momentum conjugate to the scale factor R. In the cases of flat or hyperbolic 3-space the Hamiltonian operator turns out to be time dependent and self-adjoint for all t. Both models are shown to be singular, thus disproving a conjecture put forward by Gotay and Demaret (1983) to the effect that self-adjoint quantum dynamics in a slow-time gauge is always non-singular.

The classical theory of a topological model which is constructed by setting the kinetic term of a (one-dimensional) sigma model at the classical level equal to zero is examined. The constraints and the geometry of the phase space of the supersymmetric extensions of the topological theory are studied. The rigid symmetries of a (supersymmetric) sigma model and its corresponding topological theory are gauged. Finally, the author examines the quantum theory of the one-dimensional sigma model in the limit where its kinetic term is set equal to zero and compare it with the quantum theory of the corresponding topological model.

On the basis of the link class solutions for quantum general relativity found by Rovelli and Smolin in the loop representation, the author describes a possible procedure for assigning a Hilbert space structure to this set of solutions. On this Hilbert space operators can be defined by going back to the connection representation; this is possible consistently since the 'gauge choice' one makes by picking out specific loops in the link classes only corresponds to general coordinate transformations on the observables connected to the operators. In this way the loop transform of Rovelli and Smolin is restricted to the physical states of the theory. However, the results still contain products of delta functions at one point, and thus the description is not complete until a satisfying regularization procedure has been found. Nevertheless this formalism shows how physical observables of general relativity can be defined in a natural way as operators on the space of smooth link classes.

For pt.I see ibid, vol.6, p.1599 (1989). Continuing from earlier work, the structure of the naked singularity in Vaidya-Papapetrou models for the gravitational collapse of null dust is further analysed here. The families of nonspacelike geodesics terminating at the singularity are classified and it is shown that the singularity exhibits a directional behaviour in terms of curvature growth along the singular geodesics.

The investigation of the Dirac equation by the WKB-Maslov method (complex sprout method) in Minkowski-Cartan space with an arbitrary external electromagnetic field is carried out. The Ehrenfest-type complete set of quasi-classical trajectory-coherent states (TCS) for the Dirac equation up to the order of O(h(cross)³2/) at to 0 is constructed. These states have the form of wavepackets moving along the worldline of the charged particle. From the Dirac equation a classical relativistic spin equation is derived by the use of the TCS which coincides with the generalization of the well known Bargmann-Michel-Telegdi equation in the case of an arbitrary external torsion field.

The problem of determining apparent horizons in (3+1) numerical relativity is considered. The method proposed by Nakamura and co-workers (1984) is applied to initial data for several time-symmetric Schwarzschild black holes to determine the quantitative numerical properties of the algorithm. Several refinements are suggested to the method which lead to improved stability and convergence. Some circumstances in which it is not appropriate to use the method are found.

Resting upon the Ehlers-Pirani-Schild axiomatic approach to spacetime theory, the author considers Weyl manifolds as reasonable spacetime models. He defines and discuss several concepts associated with a timelike vector field (=observer field=reference frame) on such a spacetime model. In particular, he derives propagation equations for rotation, expansion and shear of V, prove some kinematical vorticity theorems and present a new method how to measure length curvature operationally. As important mathematical tools, he uses the orthogonal decomposition of the Lie derivative and of the covariant derivative; the latter is the Weylian generalization of the Fermi-Walker derivative.

Properties of totally consistent equations of motion of interacting gauge fields of all spins s=0, 1/2, 1, . . ., infinity in 3+1 dimensions are discussed in some detail. The equations under consideration are explicitly general coordinate invariant, possess all necessary higher-spin gauge symmetries, reduce to the standard free massless equations in the linearized approximation and contain Einstein equations with the cosmological term in the spin-2 sector.