Table of contents

An example is given of a spacetime with an isometry group having spherical orbits but where the spacetime is not the direct product of a typical orbit with an orthogonal 2-space. It is shown that, if a general spacetime is not such a product, then it has a covering space which is.

The action for the classical model of the electron exhibiting Zitterbewegung is generalised to curved space by introducing a spin connection. The dynamical equations and the symplectic structure are given for several different choices of the variables. In particular, the authors obtain the equation of motion for spin and compare it with the Papapetrou equation (1951).

An exact spatially inhomogeneous solution of Einstein's equations is obtained where the energy stress tensor is that due to a viscous fluid and the false vacuum. The solution undergoes an exponential expansion with homogenisation, isotropisation and the 3-space section becomes flat in the timescale ( Lambda /3)^-1/2 as in the case of homogeneous models studied by Wald (1983).

The geometrical theory of finite incidence structures is used to classify the possible configurations of the bosonic field, characteristic of 11-dimensional supergravity, as well as to solve the corresponding field equations, in the case of Abelian anisotropic spatially homogenous cosmological models. Solutions with different splittings of the ten spatial dimensions are discussed in connection with the time behaviour of both the spatial volume associated with the directions along which the bosonic field lies and the complementary volume. This behaviour depends clearly on the type of configuration of the bosonic field as well as on the amount of anisotropy of the models.

The authors clarify the structure of gauged N=2 supergravity with a cosmological term of arbitrary sign in the context of conformal supergravity. They find that there exists a larger variety of these theories than has been anticipated. They explain the presence of negative-metric states, and exhibit a theory with zero cosmological constant in which the photon couples to a nilpotent charge.

The authors relate the anomalies of string theories to the mathematical obstructions of Dirac's canonical quantisation. The cohomological features of the deformation approach to the quantisation procedure determine the topological nature of such anomalies.

It is shown that the existence of a global gravitational anomaly in string theory on a ten-dimensional spacetime M under a coordinate transformation pi is related to an invariant of the smooth structure of an 11-manifold associated to M and pi . The cancellation of global gravitational anomalies in ten-dimensional flat space follows from the fact that the 11-sphere admits exactly 992 distinct smooth structures. The possibility of an anomaly occurring on a general spacetime can be discussed in terms of smoothing theory.

Hyper-Kahler non-linear sigma models in two spacetime dimensions with (4, 0) supersymmetry are ultraviolet finite due to a conspiracy of the supersymmetry and Lorentz invariances. The authors show this by performing the harmonic superspace formalism to develop (4, 0) supersymmetrically covariant Feynman rules. In this framework, it is manifest that every graph is ultraviolet finite.

The connection between sigma models on supergroup manifolds and superstrings is studied. The degeneracy of the Cartan metric and the necessity of using infinite-dimensional representations are the main ingredients of the present approach. The example of the bosonic string is discussed in some detail and both heterotic and Green-Schwarz covariant actions are rederived.

The sigma model interpretation of superstrings is studied. It is shown that additional local fermionic symmetries in these models are a consequence of the degeneracy of the Cartan metric and the fact that the structure constants of the supersymmetry algebra form a Clifford algebra.

Gives a new method for deriving transverse inverse propagators in string field theory from the knowledge of null secondary vectors in (super) conformal field theory in two dimensions. The author proves that the locality of such operators is equivalent to the vanishing of the singular subset of the null secondary vectors at particular eigenvalues of the Hamiltonian H=L₀. Furthermore, the author gives a new method of deriving the 'minimal' inverse propagators introduced originally by Friedan (1985).

A new formalism is presented for the derivation of index theorems from the supersymmetric quantum mechanics of the Dirac operator, based on a discrete approximation to the path integral. Operator ordering in H=(i gamma ^mu D_mu )² dictates the form of the action (after N insertions of intermediate coherent states), and the N to infinity limit yields the correct (normalised) form of the index theorem for the U(1) anomaly. It is established that internal degrees of freedom may be represented by fermions and/or bosons. In the purely gravitational case, the bosonic formulation yields a generating function for the contribution to the anomaly for spinor fields carrying arbitrary irreps (1/2A, 1/2B) of the local SO(4) group.

The authors give the supersymmetric generalisation of the duality between topologically massive and self-dual vector fields in three dimensions. In particular, this shows that the topologically massive theory with two interacting sets of gauge vectors admits couplings to matter.

The authors discuss the possibility of obtaining the non-zero one-loop topological Casimir effect in supergravity theories and evaluate it for a number of supergravity models in the flat homogeneous Clifford-Klein spacetimes with topologies M_q=(S¹)^q*R^4-q, q=1, 2, 3. These models are pure N=1 supergravity, N=1 supergravity coupled to matter and gauge fields, N=2 supergravity coupled to matter fields and N=8 supergravity. The results of calculations permit us to estimate the probability of the quantum birth of the Universe, possessing the effective M_q topology and filled with the fields of any of the above theories in the semiclassical approximation. In all cases it turns out that the birth of the Universe with more isotropic topology is more plausible. They also outline the perspectives of an extension of the results obtained.

The authors study the effect of quantum fluctuations on the roll-down rate of the inflation field in a semiclassical approximation; this is done by treating the inflation field as a classical random field. The quantum fluctuations are simulated by a noise term in the equation of motion. They consider two different inflationary scenarios (new and chaotic inflation) and find that the roll-down rate of the median value of the inflation field is increased by the quantum fluctuations. Non-linear effects may become important in the later stages of the inflationary regime.

Studies Teitelboim's propagation amplitude (1982, 1983) of quantum gravity in spatially homogeneous spacetimes of class A. In the case of Bianchi types IX and VIII, the amplitude is given by averaging a Green function of the superHamiltonian operator over spatial coordinate transformations. For the remaining class A types, the amplitude depends on the choice of spatial coordinates. The author evaluates the amplitude analytically at the isotropic limit of type IX. The physical interpretation is discussed.

It is shown that spherical Robinson-Trautman spacetimes filled with a homogeneous time-varying radiation field approach the Vaidya metric as the retarded time goes to infinity. This is a generalisation of an earlier result on the vacuum Robinson-Trautman spacetimes.

The authors consider spacetimes of general relativity admitting a preferred null direction l^mu and a two-dimensional Abelian group of isometries G₂. A null tetrad formulation of the Killing equations is given, as well as a classification of G₂ according to the orientation of l^mu with respect to the group transitivity surfaces. Two theorems concerning the action of the isometry group on l^mu are presented: the first one deals with spacetimes admitting at least one Killing vector, while the second one deals with spacetimes admitting a G₂. The Einstein field equations with a shear-free and diverging null dust source are integrated under the assumptions: (i) the spacetime admits an Abelian G₂ whose transitivity surfaces are non-orthogonal to and do not contain the null dust propagation vector; (ii) there exists a Killing vector k^mu whose magnitude is almost everywhere bounded at the endpoints of the null dust rays. These spacetimes are of Petrov type II, non-asymptotically flat, and the G₂ is non-orthogonally transitive. By switching off the null dust the authors have explicitly obtained the underlying vacuum metrics which generalise some of the diverging Petrov type D vacuum metrics found by Kinnersley (1969).

Spin coefficient techniques based on the horizon of a spacetime are used to obtain an intrinsic, coordinate-free, characterisation of the final shape of the black hole.

Although it is considered physically evident, a rigorous proof that a static stellar model is spherically symmetric is still lacking. The author discusses the relevance of the positive-mass theorem in proving the spherical symmetry of a static stellar model. As an example the author proves the uniqueness of a particular static perfect fluid solution due to Wyman (1930) using the positive-mass theorem.

The Novikov coordinates for the Kruskal-Schwarzschild spacetime are derived from the Tolman model (1934) and generalised. It is then shown that models with identical two-sheets-and-a-neck topologies can have a non-zero density and that other unusual topologies are possible for Tolman models. The introduction of matter into such a model is found to split the horizons in the two sheets, and to reduce communication between them. The topology of the neck always requires paired white and black holes, which in these models are the big bang and big crunch singularities. It is concluded that worm holes between universes can also exist in cosmological (i.e. dynamic, non-vacuum) models.

A new approach to the Regge calculus, developed in a previous paper, (ibid., vol.4, p.391, 1987), is used in conjunction with the velocity potential version of relativistic fluid dynamics due to Schutz (1970) to calculate relativistic model stars. The results are compared with those obtained when the Tolman-Oppenheimer-Volkov equations are solved by other numerical methods. The agreement is found to be excellent.

The unconstrained light-front action of higher-order (third-order bosonic and second-order fermionic) topologically massive supergravity is given. The generator of the light-front time evolution is determined as a functional of the two unique physical variables the system has: one (real) for the bosonic massive spin-2 excitation and other (Grassmannian, real) for the massive spin-3/2 degree of freedom. The analysis is performed in the natural torsion-sensitive supersymmetric generalisation of the recently introduced dreibein light-front gauge. It shows in a particular transparent fashion how the dynamics of the fermionic sector, whose first layer consists of the non-(locally) conformal invariant Rarita-Schwinger action is driven by the (local) conformal invariance of the supersymmetric Chern-Simons layer of the full action.

A geometric approach based on the chronoinvariance group introduced by Zelmanof (1956) and Cattaneo (1958) is used in order to describe the behaviour of superconductors in arbitrary external gravitational and electromagnetic fields. In this manner the physical interpretation of the equations obtained with respect to a chosen reference frame is automatically achieved. The starting point is a model of a superconductor which is manifestly covariant and gauge invariant within a generalisation of the London (1950) and Ginzburg-Landau formulation. In the weak-field approximation and for a stationary gravitational field the London-DeWitt and Schiff-Barnhill effects (1966) are recovered.

The motion of spinless test particles falling freely in the plane-wave solution of simple supergravity is studied. The polarisation of this wave is discussed in relation to general relativity. In a restricted sense, the plane wave of supergravity is proven to be equivalent to the corresponding relativistic solution. (This statement is, however, not in contradiction with the Aichelburg-Urbantke proof (1978) of non-triviality of the supergravity plane wave).

The authors generalise the result derived for the Einstein theory with the cosmological term (the general asymptotic solution containing four arbitrary functions of three coordinates) to fourth-order gravity. For scale-invariant fourth-order gravity the expanding generalised de Sitter solution is found to be an attractor, i.e. for t to infinity an open neighbourhood of solutions approach this one. For (R+R² gravity), one obtains an intermediate de Sitter stage with a high probability that is followed by a power-law Friedman stage. Finally, they argue that the inclusion of ordinary inhomogeneously distributed matter does not alter these results.

It is suggested that the existence of gravitomagnetic monopoles may imply not only the quantisation of mass-energy but also the quantisation of frequency (and other rate-like quantities). It is possible that the magnitude of the poles may be determined by cosmological considerations and whilst it is unlikely that they can exist except at large distances from the Earth, their detection is, in principle, possible using gyroscopic rings. It is shown that these detection processes may involve an effect in which gyro-particles are created or destroyed.

The theory of a free scalar field with conformal coupling in curved spacetime with some special metrics is considered. The integral representations for the Green function G approximately _in(0 mod T phi (x) phi (x') mod 0)_in in the form of integrals with Schwinger-De Witt kernel over contours in the complex plane of proper time are obtained. It is shown how the transitions from a unique Green function in Euclidean space to different Green functions in Minkowski space and vice versa can be carried out.

Discusses the quantisation of the scalar and vector fields with non-zero masses inside the event horizon of de Sitter spacetime by using the static coordinate system. The commutators for creation and annihilation operators are derived from the local canonical commutation relations. The author then calculates the response of these fields to monochromatic multipole sources in the vacuum defined in the coordinate system which describes the expanding half of de Sitter spacetime. It is found, as expected, that it is the same as that in thermal equilibrium in the static coordinate system with the well known Hawking temperature.

The general asymptotically flat solution of the five-dimensional vacuum equations R_mu nu =0 with a light-like symmetry and a space-like axial symmetry is investigated. It is shown that this solution is reduced to a special pp wave which has been recently investigated by the author, (ibid., vol.18, p.899, 1986). The consequences of this result for the author's concept of extended massless particles are discussed.

The authors show that the plane-fronted electromagnetic waves in our physical spacetime can be regarded as geometrical ripples in a six-dimensional spacetime with torsion. The extra dimensions are curled up into a compact S².

For pt.I see ibid., vol 2, p.77, (1985). The problem of wire singularities in the Robinson-Trautman solutions is investigated in the case of axially symmetric line elements. A special class of metrics is studied in detail and its time evolution is found, analogous to that of a heat wave.

Considers the generalisation of the Bach-Lanczos Lagrangian in matrix relativity where it is no longer a topological invariant, and find that for certain structures of the matrix affine connection a Yang-Mills type Lagrangian is obtained. Thus the author contemplates the possibility of interpreting non-Abelian gauge fields as arising from an otherwise topological invariant.

This comment is concerned with simple progressing solutions of the wave equation Square Operator phi =0. In flat spacetime, these were all found by Friedlander in 1946. It is shown that all of the cases he enumerated are special cases of a single progressing-wave solution in a plane-wave gravitational background. Finally, progressing waves are used to obtain a generalisation, to plane-wave spacetime, of Whittaker's formula (1903) for solutions of the wave equations in flat space.