Table of contents

We generalize the dimensionally-reduced Yang–Mills matrix model by adding a d = 1 Chern–Simons term and terms for a bosonic vector. The coefficient, κ, of the Chern–Simons term must be an integer, and hence the level structure. We show at the bottom of the Yang–Mills potential, the low-energy limit, only the linear motion is allowed for D0 particles. Namely, all the particles align themselves on a single straight line subject to κ²/r² repulsive potential from each other. We argue the relevant brane configuration to be D0-branes in a D4 after κ of D8s pass the system.

The theory of stars consisting of noninteracting fermions at zero Kelvin is developed starting from a quantum formulation. Single-particle solutions of the Dirac equation, in a curved spacetime with spherical symmetry, are filled with neutrons in such a way that the mass of the star is a minimum constrained by Einstein's equations of general relativity. Assuming spherical symmetry, this rule leads to quantum ground states with local isotropy (i.e. isotropic pressure) for masses below the Oppenheimer–Volkoff limit, but predicts locally anisotropic stars with mass up to 30% above that limit.

We will present results of a numerical integration of a maximally sliced Schwarzschild black hole using a smooth lattice method. The results show no signs of any instability forming during the evolutions to t = 1000m. The principle features of our method are (i) the use of a lattice to record the geometry, (ii) the use of local Riemann normal coordinates to apply the 1 + 1 ADM equations to the lattice and (iii) the use of the Bianchi identities to assist in the computation of the curvatures. No other special techniques are used. The evolution is unconstrained and the ADM equations are used in their standard form.

We prove local existence and uniqueness of static spherically symmetric solutions of the Einstein–Yang–Mills equations for any action of the rotation group (or SU(2)) by automorphisms of a principal bundle over spacetime whose structure group is a compact semi-simple Lie group G. These actions are characterized by a vector in the Cartan subalgebra of and are called regular if the vector lies in the interior of a Weyl chamber. In the irregular cases (the majority for larger gauge groups) the boundary value problem that results for possible asymptotically flat soliton or black hole solutions is more complicated than in the previously discussed regular cases. In particular, there is no longer a gauge choice possible in general so that the Yang–Mills potential can be given by just real-valued functions. We prove the local existence of regular solutions near the singularities of the system at the centre, the black hole horizon, and at infinity, establish the parameters that characterize these local solutions, and discuss the set of possible actions and the numerical methods necessary to search for global solutions. That some special global solutions exist is easily derived from the fact that (2) is a subalgebra of any compact semi-simple Lie algebra. But the set of less trivial global solutions remains to be explored.

We find considerable evidence supporting the conjecture that four-dimensional quantum Einstein gravity is 'asymptotically safe' in Weinberg's sense. This would mean that the theory is likely to be nonperturbatively renormalizable and thus could be considered a fundamental (rather than merely effective) theory which is mathematically consistent and predictive down to arbitrarily small length scales. For a truncated version of the exact flow equation of the effective average action, we establish the existence of a non-Gaussian renormalization group fixed point which is suitable for the construction of a nonperturbative infinite cut-off limit. The truncation ansatz includes the Einstein–Hilbert action and a higher derivative term.

We investigate the dynamics of a pair of (4 + 1)-dimensional black holes in the moduli approximation and with fixed angular momentum. We find that spinning black holes at small separations are described by the de Alfaro, Fubini and Furlan model. For more than two black holes, we find an explicit expression for the three-body interactions in the moduli metric by associating them with the one-loop three-point amplitude of a four-dimensional ϕ³ theory. We also investigate the dynamics of a three-black-hole system in various approximations.

We describe a new interactive database (GRDB) of geometric objects in the general area of differential geometry. Database objects include, but are not restricted to, exact solutions of Einstein's field equations. GRDB is designed for researchers (and teachers) in applied mathematics, physics and related fields. The flexible search environment allows the database to be useful over a wide spectrum of interests, for example, from practical considerations of neutron star models in astrophysics to abstract space–time classification schemes. The database is built using a modular and object-oriented design and uses several Java technologies (e.g. Applets, Servlets, JDBC). These are platform-independent and well adapted for applications developed for the World Wide Web. GRDB is accompanied by a virtual calculator (GRTensorJ), a graphical user interface to the computer algebra system GRTensorII, used to perform online coordinate, tetrad or basis calculations. The highly interactive nature of GRDB allows systematic internal self-checking and minimization of the required internal records. This new database is now available online at http://grdb.org.

Canonical quantization of an action containing a curvature-squared term requires the introduction of an auxiliary variable. Boulware and coworkers prescribed a technique to choose such a variable, by taking the derivative of the action with respect to the highest derivative of the field variable, present in the action. It has been shown that this technique can even be applied in situations where the introduction of auxiliary variables is not required, leading to the wrong Wheeler–De Witt equation. It has also been pointed out that Boulware's prescription should be taken up only after removing all possible total derivative terms from the action. Once this is done only a unique description of quantum dynamics would emerge. For the curvature-squared term this technique yields, for the first time, a quantum mechanical probability interpretation of quantum cosmology, and an effective potential whose extremization leads to Einstein's equation. We conclude that the Einstein–Hilbert action should essentially be modified by, at least, a curvature-squared term to get a quantum mechanical formulation of quantum cosmology and hence extend our previous work for such an action along with a scalar field.

We consider the evolution of perturbed cosmological spacetime with multiple fluids and fields in Einstein gravity. Equations are presented in gauge-ready forms, and are presented in various forms using the curvature (Φ or φ_χ) and isocurvature (S_(ij) or δϕ_(ij)) perturbation variables in the general background with K and Λ. We clarify the conditions for conserved curvature and isocurvature perturbations in the large-scale limit. Evolutions of curvature perturbations in many different gauge conditions are analysed extensively. In the multi-field system we present a general solution to the linear order in slow-roll parameters.

We study hyperKähler torsion (HKT) structures on nilpotent Lie groups and on associated nilmanifolds. We show three weak HKT structures on ⁸ which are homogeneous with respect to extensions of Heisenberg-type Lie groups. The corresponding hypercomplex structures are of a special kind called Abelian. We prove that on any 2-step nilpotent Lie group all invariant HKT structures arise from Abelian hypercomplex structures. Furthermore, we use a correspondence between Abelian hypercomplex structures and subspaces of p(n) to produce continuous families of compact and noncompact manifolds carrying non-isometric HKT structures. Finally, geometrical properties of invariant HKT structures on 2-step nilpotent Lie groups are obtained.

The asymptotic symmetries of the near-horizon geometry of a lifted (near-extremal) Reissner–Nordstrom black hole, obtained by inverting the Kaluza–Klein reduction, explain the deviation of the Bekenstein–Hawking entropy from extremality. We point out the fact that the extra dimension allows us to justify the use of a Virasoro mode decomposition along the time-like boundary of the near-horizon geometry, AdS₂ × Sⁿ, of the lower-dimensional (Reissner–Nordstrom) spacetime.

The gravitational redshift formula is usually derived in the geometric optics approximation. In this paper we consider an exact formulation of the problem in the Schwarzschild spacetime, with the intention of clarifying under what conditions this redshift law is valid. It is shown that in the case of shocks, the radial component of the Poynting vector can scale according to the redshift formula, under a suitable condition. If that condition is not satisfied, then the effect of backscattering can lead to significant modifications. The results obtained imply that the energy flux of the short wavelength radiation obeys the standard gravitational redshift formula while the energy flux of long waves can scale differently, with redshifts being dependent on the frequency.

We show that the asymptotic dynamics of three-dimensional gravity with a positive cosmological constant is described by Euclidean Liouville theory. This provides an explicit example of a correspondence between de Sitter (dS) gravity and conformal field theories. In the case at hand, this correspondence is established by formulating Einstein gravity with positive cosmological constant in three dimensions as an SL(2, ) Chern–Simons (CS) theory. The de Sitter boundary conditions on the connection are divided into two parts. The first part reduces the CS action to a nonchiral SL(2, ) WZNW model, whereas the second provides the constraints for a further reduction to Liouville theory, which resides on the past boundary of dS₃.

It is often convenient to view Hawking (black-hole) radiation as a process of quantum tunnelling. Within this framework, Kraus and Wilczek (KW) have initiated an analytical treatment of black-hole emission. Notably, their methodology incorporates the effects of a dynamical black-hole geometry. In the current paper, the KW formalism is applied to the case of a charged BTZ black hole. In the context of this interesting model, we are able to demonstrate a non-thermal spectrum, with the usual Hawking result being reproduced at zeroth order in frequency. Considerable attention is then given to the examination of near-extremal thermodynamics.

We show that the zero-frequency limit of the absorption cross section for the minimally-coupled massless scalar field into any stationary black hole with a future horizon with vanishing expansion equals the horizon area if the solution of the scalar field equation with an incident monochromatic plane wave converges to a constant on the horizon, with the error term being at most linear in the frequency.