Table of contents

Classical and Quantum Gravity (CQG) welcomes articles on experimental gravitation. This includes articles on instrumentation that is mostly of interest for gravitational experiments, as long as the relationship to gravitation is clearly explained in the paper. Experimental authors should also take into account that the readership of CQG is broad, consisting of theorists and experimentalists. It is therefore beneficial to include in the introduction a summary that places the findings in a meaningful context for such a readership.

The Editorial Board

We revise the problem of quantizing the relativistic particle. We present the modified canonical scheme, which allows us to construct the consistent relativistic quantum mechanics as well as to incorporate arbitrary backgrounds. This quantum mechanics literally reproduces the one-particle sector of the corresponding quantum field theory. Moreover, this construction gives a possible solution for the well known, old problem of how to construct a consistent quantum mechanics on the basis of a relativistic wave equation.

It has been suggested that if the Universe satisfies a flat, multiply connected, perturbed Friedmann-Lemaître model, then cosmic microwave background data from the COBE satellite implies that the minimum size of the injectivity diameter (shortest closed spatial geodesic) must be larger than about (2/5) of the horizon diameter. To show that this claim is misleading, a simple T²×R universe model of injectivity diameter a quarter of this size, i.e. a tenth of the horizon diameter, is shown to be consistent with COBE four-year observational maps of the cosmic microwave background. This is done using the identified circles principle.

We show that the recently formulated equivalence principle (EP) implies a basic cocycle condition both in Euclidean and Minkowski spaces, which holds in any dimension. This condition, that in one dimension is sufficient to fix the Schwarzian equation, implies a fundamental higher-dimensional Möbius invariance which, in turn, unequivocally fixes the quantum version of the Hamilton-Jacobi equation. This also holds in the relativistic case, so that we obtain both the time-dependent Schrödinger equation and the Klein-Gordon equation in any dimension. We then show that the EP implies that masses are related by maps induced by the coordinate transformations connecting different physical systems. Furthermore, we show that the minimal coupling prescription, and therefore gauge invariance, arises quite naturally in implementing the EP. Finally, we show that there is an antisymmetric 2-tensor which underlies quantum mechanics and sheds new light on the nature of the quantum Hamilton-Jacobi equation.

A TQFT in terms of general gauge fixing functions is discussed. In a covariant gauge it yields the Donaldson-Witten TQFT. The theory is formulated on a generalized phase space where a simplectic structure is introduced. The Hamiltonian is expressed as the anticommutator of off-shell nilpotent BRST and anti-BRST charges. Following the original ideas of Witten a time reversal operation and the corresponding inner product are defined. We present a non-covariant gauge fixing that gives rise to a manifestly time reversal invariant Lagrangian and a positive definite Hamiltonian, with the previously introduced inner product. As a consequence, the indefiniteness problem of some of the kinetic terms of the Witten's action is resolved. The construction allows then a consistent interpretation of Floer groups in terms of the cohomology of the BRST charge which is explicitly independent of the background metric. The relation between the BRST cohomology and the ground states of the Hamiltonian is then completely established. The topological theories arising from the covariant, Donaldson-Witten, and non-covariant gauge fixing are shown to be quantum equivalent by using the operatorial approach.

A homothetic, static, spherically symmetric solution to the massless Einstein-Klein-Gordon equations is described. There is a curvature singularity which is central, null, bifurcate, massless and marginally trapped. The spacetime is therefore extreme in the sense of lying at the threshold between black holes and naked singularities, just avoiding both. A linear perturbation analysis reveals two types of dominant mode. One breaks the continuous self-similarity by periodic terms reminiscent of discrete self-similarity, with an echoing period within a few per cent of the value observed numerically in near-critical gravitational collapse. The other dominant mode explicitly produces a black hole, white hole, eternally naked singularity or regular dispersal, the latter indicating that the background is critical. The black hole is not static but has constant area, the corresponding mass being linear in the perturbation amplitudes, explicitly determining a unit critical exponent. It is argued that a central null singularity may be a feature of critical gravitational collapse.

We consider the non-relativistic effective field theory of `extreme black holes' in the Einstein-Maxwell-dilaton theory with an arbitrary dilaton coupling. We investigate the finite-temperature behaviour of a gas of `extreme black holes' using the effective theory. The total energy of the classical many-body system is also derived.

We consider the spacetimes corresponding to static global monopoles with interior boundaries corresponding to a black hole horizon and analyse the behaviour of the appropriate ADM mass as a function of the horizon radius r_H. We find that for small enough r_H, this mass is negative as in the case of the regular global monopoles, but that for large enough r_H the mass becomes positive, encountering an intermediate value for which we have a black hole with zero ADM mass.

We find an exact solution to the charged two-body problem in (1+1)-dimensional lineal gravity which provides the first example of a relativistic system that generalizes the Majumdar-Papapetrou condition for static balance.

We apply Cartan's method of equivalence to construct invariants of a given null hypersurface in a Lorentzian spacetime. This enables us to fully classify the internal geometry of such surfaces and hence solve the local equivalence problem for null hypersurface structures in four-dimensional Lorentzian spacetimes.

We analyse the classical limit of kinematic loop quantum gravity in which the diffeomorphism and Hamiltonian constraints are ignored. We show that there are no quantum states in which the primary variables of the loop approach, namely the SU(2) holonomies along all possible loops, approximate their classical counterparts. At most a countable number of loops must be specified. To preserve spatial covariance, we choose this set of loops to be based on physical lattices specified by the quasiclassical states themselves. We construct `macroscopic' operators based on such lattices and propose that these operators be used to analyse the classical limit. Thus, our aim is to approximate classical data using states in which appropriate macroscopic operators have low quantum fluctuations.

Although, in principle, the holonomies of `large' loops on these lattices could be used to analyse the classical limit, we argue that it may be simpler to base the analysis on an alternate set of `flux'-based operators. We explicitly construct candidate quasiclassical states in two spatial dimensions and indicate how these constructions may generalize to three dimensions. We discuss the less robust aspects of our proposal with a view towards possible modifications. Finally, we show that our proposal also applies to the diffeomorphism-invariant Rovelli model which couples a matter reference system to the Hussain-Kucha{r} model.

We show that in the Maxwell-Chern-Simons theory of topologically massive electrodynamics the Dirac string of a monopole becomes a cone in anti-de Sitter space with the opening angle of the cone determined by the topological mass, which in turn is related to the square root of the cosmological constant. This proves to be an example of a physical system, a priori completely unrelated to gravity, which nevertheless requires curved spacetime for its very existence. We extend this result to topologically massive gravity coupled to topologically massive electrodynamics within the framework of the theory of Deser, Jackiw and Templeton. The two-component spinor formalism, which is a Newman-Penrose type approach for three dimensions, is extended to include both the electrodynamical and gravitational topologically massive field equations. Using this formalism exact solutions of the coupled Deser-Jackiw-Templeton and Maxwell-Chern-Simons field equations for a topologically massive monopole are presented. These are homogeneous spaces with conical deficit. Pure Einstein gravity coupled to the Maxwell-Chern-Simons field does not admit such a monopole solution.

A tensor description of perturbative Einsteinian gravity about an arbitrary background spacetime is developed. By analogy with the covariant laws of electromagnetism in spacetime, gravito-electromagnetic potentials and fields are defined to emulate electromagnetic gauge transformations under substitutions belonging to the gauge symmetry group of perturbative gravitation. These definitions have the advantage that on a flat background, with the aid of a covariantly constant timelike vector field, a subset of the linearized gravitational field equations can be written in a form that is fully analogous to Maxwell's equations (without awkward factors of four and extraneous tensor fields). It is shown how the remaining equations in the perturbed gravitational system restrict the time dependence of solutions to these equations and thereby prohibit the existence of propagating vector fields. The induced gravito-electromagnetic Lorentz force on a test particle is evaluated in terms of these fields together with the torque on a small gyroscope. It is concluded that the analogy of perturbative gravity to Maxwell's description of electromagnetism can be valuable for (quasi-)stationary gravitational phenomena but that the analogy has its limitations.

Equations (50)-(51) should read

ds²=r^-4mdz² +r^4m+2 dφ²+ (1-8 ε)^-1 e^8m(1+4m) r^4m(1+2m) (dr²-dt²) (50)

E=1/8-(1/8-ε)e^-8m(1+4m) r^-8m2 (51)

The author thanks Bill Bonnor for pointing out the problem with the original form of the metric.