Table of contents

Using recent observational constraints on cosmological density parameters, together with recent mathematical results concerning small volume hyperbolic manifolds, we argue that, by employing pattern repetitions, the topology of nearly flat small hyperbolic universes can be observationally undetectable. This is important in view of the fact that quantum cosmology may favour hyperbolic universes with small volumes, and from the expectation, coming from inflationary scenarios, that Ω₀ is likely to be very close to one.

We suggest a new solution of the initial spacetime singularity. In this approach the initial singularity of spacetime corresponds to a zero-size singular gravitational instanton characterized by a Riemannian metric configuration ( + + + + ) in dimension D = 4. Connected with some unexpected topological data corresponding to the zero scale of spacetime, the initial singularity is thus not considered in terms of divergences of physical fields but can be resolved within the framework of topological field theory. Then it is suggested that the `zero-scale singularity' can be understood in terms of topological invariants (in particular, the first Donaldson invariant ∑<sub>i</sub>(-1)<sup>n<sub>i</sub></sup>). With this perspective, here we introduce a new topological index, connected with zero scale, of the form {Z}<sub>β = 0</sub> = Tr (-1)<sup>s</sup>, which we call the `singularity invariant'. Interestingly, this invariant also corresponds to the invariant topological current yield by the hyperfinite II_∞ von Neumann algebra describing the zero scale of spacetime. Then we suggest that the (pre-)spacetime is in thermodynamical equilibrium at the Planck-scale and is therefore subject to the KMS condition. This might correspond to a unification phase between the `physical state' (Planck scale) and the `topological state' (zero scale). Then we conjecture that the transition from the topological phase of the spacetime (around the zero scale) to the physical phase observed beyond the Planck scale should be deeply connected to the supersymmetry breaking of the N = 2 supergravity.

We present a study of Dirac quantum fields in a four-dimensional de Sitter spacetime. The theory is based on the requirement of precise analyticity properties of the waves and the correlation functions in the complexification of the de Sitter manifold. Holomorphic de Sitter spinorial plane waves are introduced in this way and used to construct the two-point functions, whose properties are fully characterized. The physical interpretation of the analyticity properties of Wightman's functions in terms of a KMS-type thermal condition is also given.

The Proca equation with negative mass-square is studied in a refractive and absorptive spacetime. The generation of superluminal radiation fields by subluminal currents is discussed. The possibility of time-symmetric wave propagation is analysed in the context of the Wheeler-Feynman absorber theory; it is shown how advanced modes of the Proca field can be turned into retarded ones in a permeable spacetime capable of producing an absorber field. A microscopic oscillator model for the permeability is suggested. Tachyonic Liénard-Wiechert potentials are studied and strictly causal retarded wave solutions are obtained. Energy transfer by superluminal radiation is discussed, and explicit formulae for the spectral energy density and intensity are derived. Superluminal radiation fields generated by classical damped oscillators carrying tachyonic charge are investigated, including the tachyonic analogue to Thomson and Rayleigh cross sections. The Maxwell equations for negative mass-square are derived, their non-local generalization to frequency-dependent permeabilities, as well as the Poynting theorem for superluminal radiation in an absorptive spacetime.

A flow invariant is a quantity depending only on the ultraviolet (UV) and infrared (IR) conformal fixed points and not on the flow connecting them. Typically, its value is related to the central charges a and c. In classically conformal field theories, scale invariance is broken by quantum effects and the flow invariant a_UV-a_IR is measured by the area of the graph of the beta function between the fixed points. There exists a theoretical explanation of this non-trivial fact. On the other hand, when scale invariance is broken at the classical level, it is known empirically that the flow invariant equals c_UV-c_IR in massive free-field theories, but a theoretical argument explaining why it is so is still missing. A number of related open questions are answered here. A general formula of the flow invariant is found, which also holds when the stress tensor has improvement terms. The conditions under which the flow invariant equals c_UV-c_IR are identified. Several non-unitary theories are used as a laboratory, but the conclusions are general and an application to the Standard Model is addressed. The analysis of the results suggests some new minimum principles, which might point towards a new understanding of quantum field theory.

We argue that 11-dimensional supergravity can be described by a nonlinear realization based on the group E₁₁. This requires a formulation of 11-dimensional supergravity in which the gravitational degrees of freedom are described by two fields which are related by duality. We show the existence of such a description of gravity.

Recent observations suggest that the ratio of the total density to the critical density of the universe, Ω₀, is likely to be very close to one, with a significant proportion of this energy being in the form of a dark component with negative pressure. Motivated by this result, we study the question of observational detection of possible non-trivial topologies in universes with Ω₀~1, which include a cosmological constant. Using a number of indicators we find that as Ω₀→1, increasing families of possible manifolds (topologies) become either undetectable or can be excluded observationally. Furthermore, given a non-zero lower bound on |Ω₀-1|, we can rule out families of topologies (manifolds) as possible candidates for the shape of our universe. We demonstrate these findings concretely by considering families of manifolds and employing bounds on cosmological parameters from recent observations. We find that given the present bounds on cosmological parameters, there are families of both hyperbolic and spherical manifolds that remain undetectable and families that can be excluded as the shape of our universe. These results are of importance in future search strategies for the detection of the shape of our universe, given that there are an infinite number of theoretically possible topologies and that the future observations are expected to put a non-zero lower bound on |Ω₀-1| which is more accurate and closer to zero.

Superembeddings which have bosonic codimension zero are studied in three, four and six dimensions. The worldvolume multiplets of these branes are off-shell vector multiplets in these dimensions, and their self-interactions include a Born-Infeld term. It is shown how they can be written in terms of standard vector multiplets in flat superspace by working in the static gauge. The action formula is used to determine both Green-Schwarz-type actions and superfield actions.

We consider the problem of the Hamiltonian reduction of Einstein's equations on a (3 + 1)-vacuum spacetime that admits a foliation by constant mean curvature compact spacelike hypersurfaces M of Yamabe type -1. After a conformal reduction process, we find that the reduced Einstein flow is described by a time-dependent non-local dimensionless reduced Hamiltonian H_reduced which is strictly monotonically decreasing along any non-constant integral curve of the reduced Einstein system. We discuss relationships between H_reduced, the σ-constant of M, the Gromov norm |M|| and the hyperbolic σ-conjecture. As examples, we consider Bianchi models that spatially compactify to manifolds of Yamabe type -1. For these models we show that under the reduced Einstein flow, H_reduced asymptotically approaches either the σ-constant or in the hyperbolizable case, the conjectured σ-constant, as suggested by our general theory. In the non-hyperbolizable cases, the conformal metric of the reduced Einstein flow volume-collapses M along either circular fibres, embedded tori, or collapses the entire manifold to a point, and in each case, the collapse occurs with bounded curvature. We consider applications of these results to future all-time small-data existence theorems for spatially compact spacetimes.

We study the occurrence, visibility, and curvature strength of singularities in dust-containing Szekeres spacetimes (which possess no Killing vectors) with a positive cosmological constant. We find that such singularities can be locally naked, Tipler strong and develop from a non-zero-measure set of regular initial data. When examined along timelike geodesics, the singularity's curvature strength is found to be independent of the initial data.

The issue of the local visibility of the shell-focusing singularity in marginally bound spherical dust collapse is considered from the point of view of the existence of future-directed null geodesics with angular momentum which emanate from the singularity. The initial data (i.e. the initial density profile) at the onset of collapse is taken to be of class C³. Simple necessary and sufficient conditions for the existence of a naked singularity are derived in terms of the data. It is shown that there exist future-directed non-radial null geodesics emanating from the singularity if and only if there exist future-directed radial null geodesics emanating from the singularity. This result can be interpreted as indicating the robustness of previous results on radial geodesics, with respect to the presence of angular momentum.

A stability criterion is derived for self-similar solutions with perfect fluids which obey the equation of state P = kρ in general relativity. A wide class of self-similar solutions turn out to be unstable against the so-called kink mode. The criterion is directly related to the classification of sonic points. The criterion gives a sufficient condition for instability of the solution. For a transonic point in collapse, all primary-direction nodal-point solutions are unstable, while all secondary-direction nodal-point solutions and saddle-point ones are stable against the kink mode. The situation is reversed in expansion. The applications are the following: the expanding flat Friedmann solution for 1/3 ≤ k < 1 and the collapsing one for 0 < k ≤ 1/3 are unstable; the static self-similar solution is unstable; nonanalytic self-similar collapse solutions are unstable; the Larson–Penston (attractor) solution is stable for this mode for 0 < k ≲ 0.036, while it is unstable for 0.036 ≲ k; the Evans–Coleman (critical) solution is stable for this mode for 0 < k ≲ 0.89, while it is unstable for 0.89 ≲ k. The last application suggests that the Evans–Coleman solution for 0.89 ≲ k is not critical because it has at least two unstable modes.

The local and global properties of the Levi-Civita (LC) solutions coupled with an electromagnetic field are studied and some limits to the vacuum LC solutions are given, from which the physical and geometrical interpretations of the free parameters involved in the solutions are made clear. Sources for both the LC vacuum solutions and the LC solutions coupled with an electromagnetic field are studied, and in particular it is found that all the LC vacuum solutions with σ ≥ 0 can be produced by cylindrically symmetric thin shells that satisfy all the energy conditions, weak, dominant and strong. When the electromagnetic field is present, the situation changes dramatically. In the case of a purely magnetic field, all the solutions can be produced by physically acceptable cylindrical thin shells, while in the case of a purely electric field, no such shells are found for any choice of the free parameters involved in the solutions.

We prove the global unique existence of classical solutions to the Einstein equations coupled with the nonlinear Klein-Gordon equations for small initial data under spherical symmetry. We also obtain the decay estimates of the solutions. For its proof we reduce the system to a single first-order integro-differential equation, and use the contraction mapping theorem in the appropriate function spaces.

Convexity and convex functions play an important role in theoretical physics. To initiate a study of the possible uses of convex functions in general relativity, we discuss the consequences of a spacetime (M, g_μν) or an initial data set (Σ, h_ij, K_ij) admitting a suitably defined convex function. We show how the existence of a convex function on a spacetime places restrictions on the properties of the spacetime geometry.

In the preceding paper of this series of articles we established peakedness properties of a family of coherent states that were introduced by Hall for any compact gauge group and were later generalized to gauge field theory by Ashtekar, Lewandowski, Marolf, Mourão and Thiemann.

In this paper we establish the `Ehrenfest property' of these states which are labelled by a point (A,E), a connection and an electric field, in the classical phase space. By this we mean that the expectation values of the elementary operators (and of their commutators divided by iℏ, respectively) in a coherent state labelled by the (A,E) are, to zeroth order in ℏ, given by the values of the corresponding elementary functions (and of their Poisson brackets, respectively) at the point (A,E).

These results can be extended to all polynomials of elementary operators and to a certain non-polynomial function of the elementary operators associated with the volume operator of quantum general relativity. These findings are another step towards establishing that the infinitesimal quantum dynamics of quantum general relativity might, to lowest order in ℏ, indeed be given by classical general relativity.

The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in four dimensions. A relativistic spin network is a graph labelled by unitary irreducible representations of the Lorentz group appearing in the direct integral decomposition of the space of L² functions on three-dimensional hyperbolic space. To `evaluate' such a spin network we must perform an integral; if this integral converges we say that the spin network is `integrable'. Here we show that a large class of relativistic spin networks are integrable, including any whose underlying graph is the 4-simplex (the complete graph on five vertices). This proves a conjecture of Barrett and Crane, whose validity is required for the convergence of their state sum model.

With the help of two special functions c(;η) and s(;η), we propose the five-dimensional axisymmetric solutions relating to harmonics with the nonzero parameter η. In a special case, we obtain a complete five-dimensional metric with one harmonic. We also discuss the flatness of the metric signature.

We describe the potential functions for = 4B supersymmetric quantum mechanics with D(2,1;α) symmetry.

We show that the black hole perturbations of the Hayward static solution to the massless Einstein-Klein-Gordon equations are actually gauge artifacts resulting from the linearization of a coordinate transformation.