Table of contents

A string in the Einstein universe is defined by making the polar angle have period beta . The Green function of a massless scalar field is derived and the vacuum average of the energy-momentum tensor is calculated.

Kaluza-Klein cosmological models are investigated in the vicinity of a spacelike singularity. A new parametrisation of the Kasner exponents is given for any spacetime dimension, which reduces the mixmaster dynamics to a combination of a translation and an isometry or a dilating inversion. Using this parametrisation, chaos is proven to hold for spacetime dimensions n<or=10. For n>or=11, the chaotic behaviour is shown to become unstable and to be replaced by monotonic Kasner asymptotics. These results explicitly establish conjectures formulated in previous work.

The authors show how the consistency problem of the Kaluza-Klein ansatz can be resolved when the extra dimension space is identified as a coset space.

The unconstrained harmonic potential giving rise to the general (8q-3)-parameter q-instanton solution of the SU(2) Yang-Mills equations is found. The original problem of solving the non-linear equations of self-duality is reduced to a linear differential equation on S². The explicit solution is obtained in the case of a 't Hooft type ansatz.

Vacuum to vacuum and source-free Einstein-Maxwell to source-free Einstein-Maxwell generalised Kerr-Schild metrics are studied. It turns out that such spacetimes admit a geodesic null vector and are such that the vacuum and Einstein-Maxwell first-order perturbation field equations are respectively satisfied.

The equations of motion for the gravitational scattering of two point masses obtained in a post-linear approximation to general relativity are integrated by iteration starting with uniform straight-line motion. The ensuing conservation laws for energy and linear momentum disprove Rosenblum's recent claim of an energy loss in a post-linear approximation scheme.

The author argues that the version of the pilot wave interpretation of quantum mechanics which uses a non-local non-Schrodinger force is inconsistent when applied to distributions with small numbers of particles. Thus, no version of the pilot wave interpretation (sometimes called the de Broglie-Bohm, or causal, interpretation) can be applied to the wavefunction of quantum cosmology because in any version of this interpretation, there is only one particle, the universe.

The Lagrangian for the N=4 supersymmetric extension of the chiral boson is obtained from a truncation of the N=4 spinning string. This Lagrangian completes the list given by W. Siegel (1984) of Lagrangians for the chiral boson and its supersymmetric extensions, which can be obtained from truncation of (spinning) strings.

For pt. IV see ibid., vol.1, p.189-92, 1987. Partly in the interest of clarifying previous work, massive spin-2 tensor equations, compatible in an arbitrary V₄, are constructed by transcription of corresponding known compatible equations for spinors xi ^{mu nu rho sigma} , eta ^{mu lambda nu rho} . These equations are, in fact, such that they cannot govern the history of a real field. The deficiencies implicit in an ansatz made previously are examined. One incidental section contains a timely discussion of the semantics of the terms 'inconsistency' and 'incompatibility'; in another it is shown that certain spin-2 equations due to Tauber (1980) are in fact incompatible.

The standard notions of asymptotically flat spacetimes have been challenged by the examples found by Isaacson, Welling and Winicour (1984), which show logarithmic asymptotic behaviour. Here, a geometrical definition of general future asymptotically flat (GeFAF) spacetimes is introduced, and its implications are systematically studied throughout the paper. Conformal techniques are used, but no reference to field equations is made. In particular logarithmic behaviour is studied. For the general GeFAF case it is deduced that the radiation fields always behave like 1/r asymptotically.

The Palatini variational principle for the Hilbert-type gravitational theories is generalised to the case of manifolds with boundaries. The surface terms, which should be added to the volume action integral are obtained for the general metric-affine theory and for the Einstein-Cartan theory with torsion.

Various methods of obtaining an exact solution of Einstein field equations representing shear-free motion of a spherically symmetric perfect fluid have been discussed. The methods due to McVittie (1967) and Nariai (1967), where one assumes the metric in a convenient form form one group, and the methods due to Kustaanheimo and Qvist (1948), Chakravarty et al. (1976) and Wyman (1976) where one chooses suitably an arbitrary function of integration form the other group.

Using York's method, the author discusses techniques for numerically constructing GR initial data on a Cauchy surface representing spacetimes containing arbitrary numbers of black holes, each with arbitrary mass, position, (linear) momentum and spin (angular momentum). The initial data slices discussed are maximal and 3-conformally flat. The author restricts the discussion to axisymmetric asymptotically flat vacuum (outside the black holes) slices, although these restrictions are not essential.

The coupled gravitational and electromagnetic perturbations of the Reissner-Nordstrom solution are obtained by means of complex potentials. The complete metric and vector potential perturbations are given in terms of the first derivatives of a set of potentials which obeys a system of four first-order linear partial differential equations that can be solved by separation of variables.

The equations of motion for the Green-Schwarz superstring are solved in the ON-gauge (in principle, also outside the ON-gauge), the object being to obtain a precise understanding of the fermionic gauge symmetry. In three and four dimensions, the solution is the most general one. This treatment is not satisfactory in six or ten dimensions, but the author conjectures that the gauge symmetry 'is'-in a well defined sense-a quaternion in D=6, and an octonion in D=10.

The authors use the Euclidean functional integral prescription to construct the ground-state wavefunction of the closed bosonic string including ghost contributions. They work in 26 dimensions where the sole contribution to the wavefunction comes from the string and ghost classical actions as well as the tachyonic centre of mass motion. They show that the ghost action must include a surface term to be consistent with the ghost boundary conditions. The result obtained represents the usual string plus ghost ground state and is BRST invariant.

The N=1 locally supersymmetric non-linear sigma -model with the generalised Wess-Zumino-Witten term in two spacetime dimensions is constructed by the Noether procedure. The consideration is generalised to the N=2 locally supersymmetric sigma -model with torsion, using superconformal twisted chiral multiplets in the framework of the N=2, d=2 superconformal tensor calculus. The two-loop superconformal anomalies for the above theories are calculated and the critical dimensions for various N-extended Fermi string theories in the same approximation are derived.

The authors explicitly construct the vacua of the bosonic and fermionic string and corresponding ghost systems as holomorphic vector bundles over Diff(S¹)/S¹. They give a simple method for calculating curvatures of these bundles. Possible application to string field theory is briefly discussed.

An open string in Minkowski spacetime corresponds to a quasi-periodic null curve. Such null curves have a natural twistor description when viewed as classical objects. The author treats the first quantisation of loops in real four-dimensional twistor space. Such loops correspond to open strings in three-dimensional spacetime. The geometry and reality structures pertaining to twistors in three dimensions are reviewed and the twistor description of null geodesics is presented as a prototype for the discussion of null curves.

The traditional Abelian and non-Abelian Kaluza-Klein ansatze are analysed from a geometrical point of view using a Bochner-Yano type theorem for geodesic Killing vector fields. It is shown how the presence of Killing vectors in the ansatz gives rise to constraints on the Yang-Mills fields and the curvature. The conserved Komar quantities associated with these Killing vectors are computed and compared with the corresponding ADM charges. A crucial difference between them in the presence of external fields is pointed out.

The authors give a complete description of 'standard' N=2 Poincare supergravity in harmonic superspace. The main constraint they impose is preservation of the analytic subspace by supergravity transformations, so as to allow the coupling of supergravity to analytic matter superfields. Supplemented by 'conventional' and conformal breaking constraints, this requirement is shown to define the whole theory in terms of an unconstrained analytic prepotential.

The authors develop the off-shell unconstrained formulation of N=2 Einstein supergravity in harmonic superspace. The elements of differential geometry are constructed in terms of the prepotentials (the analytic vielbeins of the harmonic covariant derivative D⁺⁺). Thus, the authors find the solution to the constraints of the first off-shell version of the theory. A number of useful 'building blocks' with simple transformation properties are constructed from the prepotentials. They are used to obtain a density for the full supervolume of harmonic superspace. One can then write down the invariant action for this version of the theory. The proof of the invariance relies on the existence of a new 'hybrid' basis in harmonic superspace in which semi-chirality together with analyticity are manifest.

This paper concludes the formulation of N=2 supergravity in harmonic superspace. The authors generalize the approach to include conformal supergravity. The superconformal group is then compensated by coupling to a Maxwell and various matter multiplets. All the previously known versions of N=2 Einstein supergravity are reproducible in this way. The authors give one example (with the non-linear multiplet as a compensator). Their main result is a new version of N=2 Einstein supergravity which involves an off-shell complex hypermultiplet with its infinitely many auxiliary fields. They believe this version to be fundamental.

It has been shown that the most general type of gauge invariance for a single massless spin-2 field is either 'normal' spin-2 gauge invariance or general covariance. The authors extend that analysis to the case of a collection of spin-2 fields. They obtain the general solution of the integrability condition which determines whether a given candidate infinitesimal symmetry arises from an exact symmetry. This general solution is a new type of gauge invariance involving associative commutative algebras in a manner analogous to the way gauge symmetries of collections of spin-1 fields are based on Lie algebras.

For pt. I see ibid., vol.4, p.1267-78, 1987. A geometrical interpretation is provided for the new gauge symmetries which were found in I to be applicable to a collection of massless spin-2 fields. On a real manifold, M, the author introduces the notion of tensor fields valued in an associative commutative algebra, A, and shows that in the case where A has as identity element, most of the standard constructions and results of ordinary real-valued differential geometry concerning derivative operators, metrics and curvature can be directly carried over to the algebra-valued case.

The authors consider the spontaneous compactification induced by a scalar sector in the form of a non-linear sigma model. A very general class of solutions is given by Riemannian submersions from the extra dimensional space onto the space in which the scalar fields take values. An explicit example is constructed taking for the extra dimensional space a generalised Hopf manifold. A massless gauge field is associated with a vertical Killing vector of the Lee type.

The change in the phase of the wavefunction of a test particle in a (4+n)-dimensional Kaluza-Klein theory is computed. It is found to contain the gravitational and non-Abelian phase shifts separately and simultaneously.

In the light-front formulation of particle dynamics the authors introduce transverse creation and annihilation operators. Using these they formulate a free-field theory containing all massless bosonic representations of the Poincare group. They then derive a cubic vertex as a non-linear realisation in four dimensions. This vertex reproduces previously known cubic interaction terms for arbitrary integer helicity as well as interactions between different helicities. The authors also give a complete list of all possible cubic couplings between massless Bose fields.

The structure of a complex manifold with a bilinear (complex-valued) metric is examined. It is shown to admit spacetime-like submanifolds, the choice of which corresponds to a gauge freedom. Particles which follow complex geodesics of the manifold have quantum-like behaviour.

Z-function regularisation of the zero point energy is used to calculate the vacuum energy density for differential forms representing massive fields in certain spacetimes of the form R^M*S^N. Explicit results are given for massive vectors and 2-forms in the static Einstein universe. The authors discuss the massless limit of the results and mention the relationship to the curved spacetime renormalisation group.

The density matrix rho ( alpha , phi ; alpha ', phi ') on the mini-superspace corresponding to a Friedmann universe with scale factor alpha containing a (homogeneous) massive scalar field phi is investigated on the basis of the linear coupling model of higher multipoles as proposed by Halliwell and Hawking (1985). The wavefunctions for all multipoles of the matter field and for the tensor modes of the metric are calculated explicitly. They can be considered as continuously measuring alpha and phi . It turns out that in this way the 'intrinsic time variable' alpha is rendered a perfectly classical quantity in not allowing the observation of superpositions of different alpha , while phi can be regarded to be defined within the width of a narrow Gaussian.

For a scalar quantum field coupled to curved spacetime the author formulates the moment hierarchy associated with the Einstein-Vlasov equations with first-order quantum corrections and studies possible solutions for minimal as well as for conformal coupling.

The authors consider the response of accelerated detectors which couple to the components of the stress-energy tensor of the field linearly and locally. They show that these detectors become excited if put on an accelerated trajectory and evaluate the rate of excitation for the simple case when the coupling is to the trace of Tⁱ_k. These results arise from the fact that detectors respond to the power spectrum of the two-point function (O mod T_ik(x)T_lm(y) mod O) rather than to (O mod T_ik mod O)_reg. The latter quantity vanishes in the accelerated frame but the former does not. The consequences of the result are discussed.

Israel-Wilson metrics are solutions to the Einstein-Maxwell equations in the Lorentzian regime. The author proves theorems showing that these metrics can consistently be analytically continued into the Euclidean sector where they can be made to satisfy asymptotically flat or asymptotically Euclidean boundary conditions. He shows that these spacetimes can have complicated topologies and suggests that they provide a possible model for the gravitational vacuum. The author notes that these metrics admit a gauge covariant constant supergravity transformation.

The evolution of spatially closed cosmological models including vacuum polarisation effects is discussed. An intermediate inflationary stage and the later particle production and thermalisation lead to the transition of the initially Planck sized vacuum universe to the standard radiation dominated Friedman universe of reasonable size. The non-linearity of the equations causes the essential dependence of the global behaviour of solutions on the initial conditions and the existence of oscillating vacuum solutions for discrete initial values.

The author investigates the decay of one massive conformally coupled scalar particle into two photons and the time-reversed decay according to the pi ⁰-2 gamma -model interaction in a spatially flat Robertson-Walker universe. Besides a few conceptual remarks concerning decay processes in pair-creating external fields, special attention is drawn to the problem of CPT non-invariance. It is found that both decay processes are not CPT invariant if (i) the expansion of the universe provides a T-asymmetric background and (ii) the conditions for free-particle creation are fulfilled for at least one of the quantum fields participating in the interaction. The results are illustrated for particular scale factors of the universe.

Starting from locally rotationally symmetric (LRS) perfect-fluid metrics and by using a Kerr-Schild method the author obtains new exact perfect-fluid metrics of the same type. The method is used to give a large class of new LRS cosmological models explicitly starting from the known ones.

This paper is in two parts. In the first part, the author makes a conjecture connected with Penrose's hypothesis of zero Weyl tensor at initial singularities and presents some evidence for believing in it. In the second, he calculates the quasi-local mass of a comoving cylinder in some LRS cosmological models and investigates its behaviour near a singularity.

A solution to Einstein's equations is given for a one-parameter family of perfect fluids satisfying the conditions -1/7<p/ mu <0, mu >0, p<0, mu +p>0. The metric is static and plane symmetric (x, y), and of Petrov type D. The only singularity occurs at z=0 where mu =+ infinity , p=- infinity while mu , p to 0 as z to + infinity . The semi-infinite layer of fluid remains at rest in the x, y, z frame under the pressure gradient, having zero shear, zero spin and zero expansion.

A quantum interference effect due to the scalar field in the five-dimensional Kaluza-Klein theory is derived.