Table of contents

We examine a one-parameter class of actions describing the gravitational interaction between a pair of scalar fields and Einsteinian gravitation. When the parameter is positive the theory corresponds to an axi-dilatonic sector of low-energy string theory. We exploit an SL(2,R) symmetry of the theory to construct a family of electromagnetically neutral solutions with non-zero axion and dilaton charge from solutions of the Brans--Dicke theory. We also comment on solutions to the theory with negative coupling parameter.

We examine the axi-dilatonic sector of low-energy string theory and demonstrate how the gravitational interactions involving the axion and dilaton fields may be derived from a geometrical action principle involving the curvature scalar associated with a non-Riemannian connection. In this geometry the antisymmetric tensor 3-form field determines the torsion of the connection on the frame bundle while the gradient of the metric is determined by the dilaton field. By expressing the theory in terms of the Levi-Civita connection associated with the metric in the `Einstein frame' we confirm that the field equations derived from the non-Riemannian Einstein-Hilbert action coincide with the axi-dilaton sector of the low-energy effective action derived from string theory.

We find a class of (2+1)-dimensional spacetimes admitting Killing spinors appropriate to (2,0) adS-supergravity. The vacuum spacetimes include anti-de Sitter (adS) space and charged extreme black holes, but there are many others, including spacetimes of arbitrarily large negative energy that have only conical singularities, and the spacetimes of fractionally charged point particles. The non-vacuum spacetimes are those of self-gravitating solitons obtained by coupling (2,0) adS supergravity to sigma-model matter. We show, subject to a condition on the matter currents (satisfied by the sigma model), and a conjecture concerning global obstructions to the existence of certain types of spinor fields, that the mass of each supersymmetric spacetime saturates a classical bound, in terms of the angular momentum and charge, on the total energy of arbitrary field configurations with the same boundary conditions, although these bounds may be violated quantum mechanically.

Possible ways of constructing new extended fermionic strings with non-linear N=4 world-sheet supersymmetry are considered. N=4 string theory constraints are required to form a quasi-superconformal algebra, and have conformal dimensions . The most general non-linear N=4 quasi-superconformal algebra is , whose linearization is the so-called `large' N=4 superconformal algebra. The algebra has affine Lie component, and . We construct a quantum BRST charge for this algebra, and show that it is nilpotent only if . This restricts the previously proposed continuous -family of critical N=4 strings to only one theory based on the algebra which is isomorphic to the SO(4)-based Bershadsky--Knizhnik quasi-superconformal algebra. We propose the (non-covariant) Hamiltonian action for the new N=4 string theory. Our results imply the existence of two different critical N=4 fermionic string theories: the `old' one based on the `small' linear N=4 superconformal algebra and having the total ghost central charge , and the new one with the non-linearly realized N=4 supersymmetry, based on the SO(4) quasi-superconformal algebra and having . Both critical N=4 string theories have negative `critical dimensions' and do not admit unitary matter representations.

An interpretation of spacelike singularities in string theory uses target-space duality to relate the collapsing Schwarzschild geometry near the singularity to an inflationary cosmology in dual variables. An appealing picture thus results whereby gravitational collapse seeds the formation of a new universe.

We investigate the Ponzano-Regge and Turaev-Viro topological quantum field theories using spin networks and their q-deformed analogues. We propose a new description of the state space for the Turaev-Viro theory in terms of skein space, to which q-spin networks belong, and give a similar description of the Ponzano-Regge state space using spin networks. We give a definition of the inner product on the skein space and show that this corresponds to the topological inner product, defined as the manifold invariant for the union of two 3-manifolds.

Finally, we look at the relation with the loop representation of quantum general relativity, due to Rovelli and Smolin, and suggest that the above inner product may define an inner product on the loop state space.

A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalization of the Leibniz rules of commutative geometry and uses the bimodule structure of . A special role is played by the extension to the framework of non-commutative geometry of the permutation of two copies of . The construction of the linear connection as well as the definition of torsion and curvature is first proposed in the setting of the derivations based differential calculus of Dubois-Violette and then a general of the Dirac operator based differential calculus of Connes and other differential calculuses is given. The covariant derivative obtained admits an extension to the tensor product of several copies of . These constructions are illustrated with the example of the algebra of matrices.

The hydrostatic equilibrium of a (2+1)-dimensional perfect fluid star in asymptotically anti-de Sitter space is discussed. The interior geometry matches the exterior 2+1 black-hole solution. An upper mass limit is found, analogous to Buchdahl's theorem in 3+1 dimensions, and the possibility of collapse is discussed. The case of a uniform matter density is solved exactly and a new interior solution is presented.

In this paper a full analytic solution for the second post-Newtonian motion of compact binaries with spin is presented. Advantage is taken of the following facts: (i) the second post-Newtonian motion of compact binaries without spin is already solved by a generalized quasi-Keplerian parametrization, (ii) first-order spin--orbit terms cross with Newtonian terms only since spin--orbit terms in compact binary-star systems are numerically of second post-Newtonian order, (iii) in the case of compact binary-star systems spin--spin interaction and quadrupole-deformation contributions are negligible at the second post-Newtonian approximation level. The analytic solution for the quasi-elliptic motion is given in a generalized quasi-Keplerian parametrization. Coordinate time and proper time of one of the bodies are used to parametrize the motion. As a by-product the first post-Newtonian motion of a satellite in the field of a rotating spherical mass is obtained.

Projective collineations in spacetimes, i.e. vector fields generating local groups of geodesic-preserving diffeomorphisms, are studied. The situation for Einstein spaces is resolved completely and some general results are established regarding arbitrary spacetimes. Examples of proper projective collineations are constructed.

We present a graphical way of describing invariants and covariants in (four-dimensional) general relativity. This frees us from the complexity of treating many suffixes. Two new off-shell relations between (mass) invariants are obtained. These are important for 2-loop off-shell calculations in perturbative quantum gravity. We list all independent invariants with dimensions of (mass) and (mass). Furthermore, the six-dimensional Gauss--Bonnet identity is expressed in terms of the independent invariants.

We prove that, under certain conditions, the topology of the event horizon of a four-dimensional asymptotically flat black-hole spacetime must be a 2-sphere. No stationarity assumption is made. However, in order for the theorem to apply, the horizon topology must be unchanging for long enough to admit a certain kind of cross section. We expect this condition is generically satisfied if the topology is unchanging for much longer than the light-crossing time of the black hole.

More precisely, let M be a four-dimensional asymptotically flat spacetime satisfying the averaged null energy condition, and suppose that the domain of outer communication to the future of a cut K of is globally hyperbolic. Suppose further that a Cauchy surface for is a topological 3-manifold with compact boundary in M, and is a compact submanifold of with spherical boundary in (and possibly other boundary components in ). Then we prove that the homology group must be finite. This implies that either consists of a disjoint union of 2-spheres, or is non-orientable and contains a projective plane. Furthermore, , and will be a cross section of the horizon as long as no generator of becomes a generator of . In this case, if is orientable, the horizon cross section must consist of a disjoint union of 2-spheres.

This paper investigates the relationship between the quasilocal energy of Brown and York and certain spinorial expressions for gravitational energy constructed from the Witten--Nester integral. A key feature of the Brown--York method for defining quasilocal energy is that it allows for the freedom to assign the reference point of the energy. When possible, it is perhaps most natural to reference the energy against flat space, i.e. assign flat space the zero value of energy. It is demonstrated that the Witten--Nester integral when evaluated on solution spinors to the Sen--Witten equation (obeying appropriate boundary conditions) is essentially the Brown--York quasilocal energy with a reference point determined by the Sen--Witten spinors. For the case of round spheres in the Schwarzschild geometry, these spinors determine the flat-space reference point. A similar viewpoint is proposed for the Schwarzschild-case quasilocal energy of Dougan and Mason.

The Einstein--Hilbert action with a cosmological term is used to derive an action in 1+1 spacetime dimensions. It is shown that the two-dimensional theory is equivalent to planar symmetry in general relativity. The two-dimensional theory admits black holes and free dilatons, and has a structure similar to two-dimensional string theories. Since by construction these solutions also solve Einstein's equations, such a theory can bring two-dimensional results into the four-dimensional world. In particular the two-dimensional black hole is a black membrane in general relativity.

This paper examines conflicting claims in the literature regarding whether the wave equation satisfies Huygens' principle in a conformally flat spacetime. It is concluded that in general the wave equation does not satisfy Huygens' principle, but there is an exception for the case of the electromagnetic field tensor.

By means of a linear superposition of a couple of tetrads, determining a self-dual type and an anti-self-dual field, the most general non-twisting type N real vacuum solution of the Einstein equation is constructed.

It was noted recently that the ADM diffeomorphism constraint does not generate all observed symmetries for several Bianchi models. We will suggest not using the ADM constraint restricted to homogeneous variables, but some equivalent, which is derived from a restricted action principle. This will generate all homogeneity preserving diffeomorphisms, which will be shown to be automorphism generating vector fields, in class A and class B models. Following Dirac's constraint formalism one will naturally be restricted to the unimodular part of the automorphism group.

The article contains two errors. The first, which results in the loss of a solution, occurs in the statement that the t and r coordinaties can be redefined so that one obtains equation (4.13). This cannot be done in the case of a static solution without introducing t-dependence into the metric functions. For a static spacetime we can go only as far as redefining the t and r coordinates so that the ICKV is given by ξ^a - (t, r, 0, 0). As a result of this correction we obtain an additional static ICKV solution with metric

where c₁, c₂, n are arbitrary constants with 2n²>1. This solution is a transformed version of a special case of solution V of Tolman (1939). If c₁c₂ = 0, the fluid satisfies an equation of state of the form p/µ = constant but, in this case, the ICKV becomes a HV. When n²=1 the solution is conformally flat and is, in fact, the special case of the Schwarzschild interior solution given by equation (A51).

The second error occurs in equation (4.22), which should read

Contrary to the statement following equation (4.22), this spacetime admits only one proper ICKV, namely ; the second claimed ICKV is just the sum of ξ^a and the timelike KV and so is not independent.

Tolman R C 1939 Phys. Rev. 55 364