Table of contents

A brief account of the life and work of Andrzej Trautman is provided.

The evolution equations of Einstein's theory and of Maxwell's theory - the latter used as a simple model to illustrate the former - are written in gauge-covariant first-order symmetric hyperbolic form with only physically natural characteristic directions and speeds for the dynamical variables. Quantities representing gauge degrees of freedom (the spatial shift vector βⁱ(t, x^j) and the spatial scalar potential ϕ(t, x^j), respectively) are not among the dynamical variables: the gauge and the physical quantities in the evolution equations are effectively decoupled. For example, the gauge quantities could be obtained as functions of (t, x^j) from subsidiary equations that are not part of the evolution equations. Propagation of certain (`radiative') dynamical variables along the physical light cone is gauge invariant while the remaining dynamical variables are dragged along the axes orthogonal to the spacelike time slices by the propagating variables. We obtain these results by (i) taking a further time derivative of the equation of motion of the canonical momentum, and (ii) adding a covariant spatial derivative of the momentum constraints of general relativity (Lagrange multiplier β<sup>i</sup>) or of the Gauss law constraint of electromagnetism (Lagrange multiplier Φ). General relativity also requires a harmonic time-slicing condition or a specific generalization of it that brings in the Hamiltonian constraint when we pass to first-order symmetric form. The dynamically propagating gravity fields straightforwardly determine the `electric' or `tidal' parts of the Riemann tensor.

Amazing effects present in the motion of particles, gyroscopes and charges along a circular light ray in Schwarzschild spacetime could be explained as natural and obvious if one accepts that inertial forces occur only when the actual motion in space deviates from steady motion along a light ray.

We generalize previous work on the classification of (C^∞) symmetries of plane-fronted waves with an impulsive profile. Due to the specific form of the profile it is possible to extend the group of normal-form-preserving diffeomorphisms to include non-smooth transformations. This extension entails a richer structure of the symmetry algebra generated by the (non-smooth) Killing vectors.

It is shown that there is an interesting interplay between self-duality, loop representation and knot invariants in the quantum theory of Maxwell fields in Minkowski spacetime. Specifically, in the loop representation based on self-dual connections, the measure that dictates the inner product can be expressed in terms of the Gauss-linking number of thickened loops.

A new functional calculus, developed recently for a fully non-perturbative treatment of quantum gravity, is used to begin a systematic construction of a quantum theory of geometry. Regulated operators corresponding to areas of 2-surfaces are introduced and shown to be self-adjoint on the underlying (kinematical) Hilbert space of states. It is shown that their spectra are purely discrete, indicating that the underlying quantum geometry is far from what the continuum picture might suggest. Indeed, the fundamental excitations of quantum geometry are one dimensional, rather like polymers, and the three-dimensional continuum geometry emerges only on coarse graining. The full Hilbert space admits an orthonormal decomposition into finite-dimensional subspaces which can be interpreted as the spaces of states of spin systems. Using this property, the complete spectrum of the area operators is evaluated. The general framework constructed here will be used in a subsequent paper to discuss three-dimensional geometric operators, e.g. the ones corresponding to volumes of regions.

We study spacetime Killing vectors in terms of their `lapse and shift' relative to some spacelike slice. We give a necessary and sufficient condition in order for these lapse - shift pairs, which we call Killing initial data (KID), to form a Lie algebra under the bracket operation induced by the Lie commutator of vector fields on spacetime. This result is applied to obtain a theorem on the periodicity of orbits for a class of Killing vector fields in asymptotically flat spacetimes.

We give an elementary proof of the fact that equivalence classes of smooth or differentiable star products on a symplectic manifold M are parametrized by sequences of elements in the second de Rham cohomology space of the manifold. The parametrization is given explicitly in terms of Fedosov's construction which yields a star product when one chooses a symplectic connection and a sequence of closed 2-forms on M. We also show how derivations of a given star product, modulo inner derivations, are parametrized by sequences of elements in the first de Rham cohomology space of M.

The theory obtained as a singular limit of general relativity, if the reciprocal velocity of light is assumed to tend to zero, is known to be not exactly the Newton - Cartan theory, but a slight extension of this theory. It involves not only a Coriolis force field, which is natural in this theory (although not originally Newtonian), but also a scalar field which governs the relation between Newton's time and relativistic proper time. Both fields are or can be reduced to harmonic functions, and must therefore be constants, if suitable global conditions are imposed. We assume this reduction of Newton - Cartan to Newton`s original theory as a starting point and ask for a consistent post-Newtonian extension and for possible differences to the usual post-Minkowskian approximation methods, as developed, for example, by Chandrasekhar. It is shown that both post-Newtonian frameworks are formally equivalent, as far as the field equations and the equations of motion for a hydrodynamical fluid are concerned.

A frame theory encompassing general relativity and Newton - Cartan theory is reviewed. With its help, a definition is given for a one-parameter family of general relativistic spacetimes to have a Newton - Cartan or a Newtonian limit. Several examples of such limits are presented.

Seeking a relativistic quantum infrastructure for gauge physics, we analyse spacetime into three levels of quantum aggregation analogous to atoms, molecules and crystals. Quantum spacetime points with no extension link up to form more complex units with microscopic extension, which link up into networks with macroscopic extension. All non-integrable gauge transport arises from vacuum excitations as in the theory of crystal defects. General covariance is now part of quantum covariance. The particle gauge groups directly indicate the microstructure of the spacetime network. The unit cell of the crystalline vacuum is a fundamental gauge element of the theory. Newton's law of inertia means that the vacuum network is superconducting to some of its excitations, which constitute matter. A simple hypercubical vacuum with off-diagonal long-range order reduces the quantum covariance group to Poincaré, and has bonus internal symmetries like those of the standard model. We cannot make a suitably invariant vacuum or dynamics with the dipole `molecule' proposed earlier. We need at least a quadrupole, as in general relativity.

The null-surface formulation of general relativity - recently introduced - provides novel tools for describing the gravitational field, as well as a fresh physical way of viewing it. The new formulation provides `local' observables corresponding to the coordinates of points - which constitute the spacetime manifold - in a geometrically defined chart, as well as nonlocal observables corresponding to lightcone cuts and lightcones. In the quantum theory, the spacetime point observables become operators and the spacetime manifold itself becomes `quantized', or `fuzzy'. This novel view may shed light on some of the interpretational problems of a quantum theory of gravity. Indeed, as we discuss briefly, the null-surface formulation of general relativity provides (local) geometrical quantities - the spacetime point observables - which are candidates for the long-sought physical operators of the quantum theory.

I describe how the states of a discrete automaton with p sites, each of which may be off or on, can be represented as Majorana spinors associated to a spacetime with signature (p,p). Some ideas about the quantization of such systems are discussed and the relationship to some unconventional formulations and generalizations of quantum mechanics, particularly Jordan's spinorial quantum mechanics, are pointed out. A connection is made to the problem of time and the complex numbers in quantum gravity.

In this paper we prove a statement about the real and pseudoreal (i.e. quaternionic) content of tensor products between real and/or quaternionic representations of compact semisimple Lie algebras (or connected compact semisimple Lie groups) that might be useful in model building when constructing unified theories of fundamental interactions. We also stress the utility of anti-involutions in the description of reality properties.

The propagation of arbitrary information by electromagnetic and gravitational waves in spatially homogeneous and isotropic cosmological models is examined. The test Maxwellian fields and the gravitational perturbations we study depend upon arbitrary functions in the spirit of Trautman's pioneering analysis. We use the covariant and gauge-invariant approach developed by Ellis and Bruni to study cosmological perturbations under this assumption.

We describe how the iterative technique used by Isenberg and Moncrief to verify the existence of large sets of non-constant mean curvature solutions of the Einstein constraints on closed manifolds can be adapted to verify the existence of large sets of asymptotically hyperbolic non-constant mean curvature solutions of the Einstein constraints.

We present a survey of recent results, scattered in a series of papers that have appeared during the past five years, whose common denominator has been the use of cubic relations in various algebraic structures.

Cubic (or ternary) relations can represent different symmetries with respect to the permutation group S₃, or its cyclic subgroup Z₃. Also ordinary or ternary algebras can be divided into different classes with respect to their symmetry properties. We pay special attention to the non-associative ternary algebra of 3-forms (or cubic matrices), and Z₃-graded matrix algebras.

We also discuss the Z₃-graded generalization of Grassmann algebras and their realization in generalized exterior differential forms dξ and d²ξ, with d³ξ=0. A new type of gauge theory based on this differential calculus is presented.

Finally, a ternary generalization of Clifford algebras is introduced, and an analogue of Dirac's equation is discussed, which can be diagonalized only after taking the cube of the Z₃-graded generalization of Dirac's operator. A possibility of using these ideas for the description of quark fields is suggested and discussed in the last section.

The paper follows the lines of earlier papers describing spinors in even-dimensional complex and complexified vector spaces. It discusses the fundamental aspects of pure spinors and questions concerning the notion of a real index in odd-dimensional vector spaces.

Advection-dominated accretion flows could be a unique signature of the presence of black holes in various accreting astrophysical systems such as some quiescent transient x-ray sources and low-luminosity nuclei of galaxies. We present the general framework describing such advection-dominated flows around Kerr black holes and point out several problems that remain to be solved.

Examples in which spacetime might become non-Riemannian appear above Planck energies in string theory or, in the very early Universe, in the inflationary model. The simplest such geometry is metric-affine geometry, in which nonmetricity appears as a field strength, side by side with curvature and torsion. In matter, the shear and dilation currents couple to nonmetricity, and they are its sources. After reviewing the equations of motion and the Noether identities, we study two recent vacuum solutions of the metric-affine gauge theory of gravity. We then use the values of the nonmetricity in these solutions to study the motion of the appropriate test matter. As a Regge-trajectory-like hadronic excitation band, the test matter is endowed with shear degrees of freedom and described by a world spinor.

We consider S² bundles and ' of totally null planes of maximal dimension and opposite self-duality over a four-dimensional manifold equipped with a Weyl or Riemannian geometry. The fibre product ' of and ' is found to be appropriate for the encoding of both the self-dual and the Einstein - Weyl equations for the 4-metric. This encoding is realized in terms of the properties of certain well defined geometrical objects on '. The formulation is suitable for complex-valued metrics and unifies results for all three possible real signatures. In the purely Riemannian positive-definite case it implies the existence of a natural almost Hermitian structure on ' whose integrability conditions correspond to the self-dual Einstein equations of the 4-metric. All Einstein equations for the 4-metric are also encoded in the properties of this almost Hermitian structure on '.

We view the finite rotating universe as a stationary point of the reduced Hamiltonian describing Gödel's symmetric case, casting light - among other things - on the structure of the metric.

The geometry of real and complex light rays (mainly in Minkowski space) is studied using twistor methods. The properties of weak and strong incidence between rays are examined, some apparent anomalies arising for neighbouring rays being explained. Various apparently different interpretations of a complex surface in projective twistor space are given (one coming from the Kerr theorem describing shear-free congruences in Minkowski space). The relations between them are analysed in terms of the caustics of shear-free congruences, null hypersurfaces and a twistor description of spacelike or timelike 2-surfaces. The situation for curved spacetime is also considered. The relation between ray geometry in Minkowski space and a 6-quadric of signature (++++----), with its triality properties, is explored.

We present a simplified model of the collapse of a star to a black hole which can be treated as a dissipative open quantum system giving rise to Hawking radiation.

A completely symmetrical form of the Bel - Robinson (BR) tensor is shown to give rise, in n-dimensional Lorentzian space, to a gravitational density. For n = 4, a physical identification is proposed.

The Yang - Mills - Higgs equations in a spatially bounded subset of the Minkowski space are studied under the assumption of a temporal gauge. It is shown that the Cauchy problem for these equations is uniquely solvable (locally in time) if nonhomogeneous boundary conditions of the metallic type are imposed.