Table of contents

The motion of relativistic particles around three-dimensional black holes following the Hamilton-Jacobi formalism is studied. It follows that the Hamilton-Jacobi equation can be separated and reduced by quadratures in analogy with the four-dimensional case. It is shown that (i) particles are trapped by the black hole regardless of their energy and angular momentum; and (ii) matter always falls to the centre of the black hole and cannot undertake motion with stable orbits as in four dimensions. For the extreme values of the angular momentum of the black hole, we were able to find exact solutions for the equations of motion and trajectories of a test particle.

It is shown that the surface gravity and temperature of a stationary black hole are invariant under conformal transformations of the metric that are the identity at infinity. More precisely, we find a conformal invariant definition of the surface gravity of a conformal Killing horizon that agrees with the usual definition(s) for a true Killing horizon and is proportional to the temperature as defined by Hawking radiation. This result is reconciled with the intimate relation between the trace anomaly and the Hawking effect, despite the non-invariance of the trace anomaly under conformal transformations.

We examine a recent suggestion by Milburn(1993) that slightly massive electron neutrinos, produced left-handed at the core of the Sun, suffer geodetic precession adequate to render them right-handed (and therefore sterile) in sufficient numbers to solve the solar neutrino problem. In that light, we perform a complete general-relativistic calculation of the geodetic spin precession of an ultrarelativistic particle in the Schwarzschild metric. We conclude that the effect is negligible. In disagreement with Milburn's analysis.

For pt I, see ibid, vol.9, p.801-821 (1992). We further study the universal equations of the supersymmetric (SUSY) modified KdV (MKdV) hierarchy in its generalized form. We show that these equations describe the dynamical quantum equations of the odd series of N=1 minimal (p, q) superconformal field theory coupled to N=1 supergravity, in particular those unitary series with p=2k+3 and q=2k+1. The string susceptibility of these models is gamma _sstr⁽⁰⁾=-2/(2k+1). By constructing the Lax form of the SUSY MKdV hierarchies we give the Virasoro constraints that should be satisfied by the wavefunction of the odd series coupled to N=1 supergravity.

We present a number of new, exact scalar field cosmologies where the potential consists of two or more exponential terms. Such potentials are motivated by supergravity or superstring models formulated in higher-dimensional spacetimes that have been compactified to 3+1 dimensions. We have found solutions in both curved and flat Robertson-Walker spacetimes. These models have a diverse range of properties and often possess several distinct phases, with a smooth transition between ordinary and inflationary expansion. While exponential potentials typically produce power-law inflation, we find models where the inflationary period contains eras of both power-law and exponential growth.

We introduce three nested Lie algebras of infinitesimal 'isometries' of a Galilean spacetime structure which play the role of the algebra of Killing vector fields of a relativistic Lorentz spacetime. Non-trivial extensions of these Lie algebras arise naturally from the consideration of Newton-Cartan-Bargmann automorphisms.

We canonically transform the variables describing plane gravitational waves until the Lagrangian assumes a parametrized free field form and we are able to identify amplitudes for the two polarization states of the gravitational wave. From these amplitudes we construct a complete set of physical observables, defined to be gauge invariant quantities which commute with all the constraints.

Non-local representation of(2+1)-dimensional quantum gravity coupled with the spinor field is considered. We first introduce non-local variables and obtain non-local quantities that are equivalent to the local constraints in some limit. The quantization of these non-local variables is executed in a non-local representation that is similar to the loop representation of pure gravity. The contribution of spinor fields in the state functionals depends not only on the space of loops but also on the space of lines. A state functional which is the solution of all constraint equations is also obtained.

It is shown that if a representation of a *-algebra on a vector space V is an irreducible *-representation with respect to some inner product on V, then under appropriate technical conditions this property determines the inner product uniquely up to a constant factor. Ashtekar(1991) has suggested using the condition that a given representation of the algebra of quantum observables is a *-representation to fix the inner product on the space of physical states. This idea is of particular interest for the quantization of gravity where an obvious prescription for defining an inner product is lacking. The results of this paper show rigorously that Ashtekar's criterion does suffice to determine the inner product in very general circumstances. Two versions of the result are proved: a simpler one which only applies to representations by bounded operators and a more general one which allows for unbounded operators. Some concrete examples are worked out in order to illustrate the meaning and range of applicability of the general theorems.

Using distributional techniques we calculate the energy-momentum tensor of the Schwarzschild geometry. It turns out to be a well defined tensor distribution concentrated on the r=0 region which is usually excluded from spacetime. This provides a physical interpretation for the curvature of this geometry.

Properties of the quasi-Riemannian action and its variations are formally related to laws of thermodynamics. These laws are expected to apply to a broad class or stationary horizon/matter systems in the stationary phase and least action approximations. The example or a system which includes a stationary black hole and either a spin-one gauge field or a generic matter distribution is considered explicitly.

We use the formalism of two-mode squeezed states and the theory of gauge invariant cosmological perturbations to calculate two-point correlation functions and the entropy of density perturbations. In particular we discuss modulation in the power spectrum of density perturbations. We also present a simple picture for entropy in terms of the quantum noise in the state. The relevance of the entropy of density perturbations for large-scale structures in the universe is examined.

An anisotropic cosmological model is investigated by solving Einstein's field equations in (1+3)-dimensional spacetime with topology R¹(X)R¹(X)T² (R¹ is the real line and T² is a two-dimensional torus). Through the process of Kaluze-Klein-type compactification on T² and one-loop quantum corrections to a scalar field, an effective action for (1+1)-dimensional gravity (with time-dependent gravitational constant G_eff as well as a cosmological constant, Lambda _eff) are obtained.

We give a Hamiltonian formulation of relativistic perfect fluids free of any constraints starting from an action principle based on three scalar fields. We show that it can describe not only the barytropic case but also more general perfect fluids with entropy and two or more thermodynamic variables conserved along the fluid flow (i.e. multiconstituent perfect fluids).

The dynamics of the closed universe in a gauge which is determined by an eigenfunction of the 3D Dirac operator is considered. The 3D volume of the universe plays the role of the canonical time parameter in this gauge. The energy of the closed universe is proportional to the corresponding eigenvalue of the 3D Dirac operator. The quantization of GR in this gauge is briefly discussed.

The Einstein equations with pure radiation fields and algebraically special metrics are studied. The symmetry group of these equations is systematically exploited to obtain a large number of new explicit solutions. A family of solutions parametrized by a one function of two variables is obtained in the case when the NUT function M vanishes.

We examine the constraints and the reality conditions that have to be imposed in the canonical theory of four-dimensional gravity formulated in terms of Ashtekar variables. We find that the polynomial reality conditions are consistent with the constraints and make the theory equivalent to Einstein's, as long as the densitized inverse 3-metric is positive definite; in a region of space where it is not, the theory is still consistent, although it does not describe (in general) a real local geometry.

In this paper we consider nets of C* algebras on globally hyperbolic manifolds. The question is considered whether causal information can be incorporated in the net structure. We prove that under quite general conditions net isomorphisms induce isometries of the conformal structures of the underlying spacetimes.

There are two proposals for the classical implementation of signature type change in general relativity. We derive necessary and sufficient junction conditions for both proposals and investigate the extent to which they are equivalent. In the case of discontinuous signature type change there does not exist a non-flat vacuum solution of the Einstein equation. However, we show that in the case of smooth signature type change such solutions exist.

For normal thermodynamic systems superadditivity S, homogeneity H and concavity C of the entropy hold, whereas for (3+1)-dimensional black holes the latter two properties are violated. We show that (1+1)-dimensional black holes exhibit qualitatively new types of thermodynamic behaviour, discussed here for the first time, in which C always holds, H is always violated and S may or may not be violated, depending on the magnitude of the black hole mass. Hence it is now seen that neither superadditivity nor concavity encapsulate the meaning of the second law in all situations.

Recently Brown and York(1993) have devised a new method for defining quasilocal energy in general relativity. Their method employs a Hamilton-Jacobi analysis of an action functional for a spatially bounded spacetime M, and this analysis yields expressions for the quasilocal energy and momentum surface densities associated with the two-boundary B of a spacelike slice of such a spacetime. These expressions are essentially Amowitt-Deser-Misner variables, but with canonical conjugacy defined with respect to the time history ³B of the two-boundary. These boundary ADM variables match previous variables which have arisen directly in the study of black hole thermodynamics. Further, a 'microcanonical' action which features fixed-energy boundary conditions has been introduced in related work concerning the functional-integral representation of the density of quantum states for such a bounded gravitational system. This paper introduces Ashtekar-type variables on the time history ³B of the two-boundary and shows that these variables lead to elegant alternative expressions for the quasilocal surface densities. In addition, it is demonstrated here that both the boundary ADM variables and the boundary Ashtekar-type variables can be incorporated into a larger framework by appealing to the tetrad-dependent Sparling differential forms. Finally, using these results and a tetrad action principle employed by Goldberg(1988), this paper constructs two new tetrad versions of the microcanonical action.

We extend the solution obtained by Lewis(1932) for a stationary cylindrically symmetric vacuum spacetime to include the cosmological constant Lambda .

Through a combination of experiment and numerical modelling, we show that vibration isolators using cantilever springs can be used to achieve vibration levels below 10^-20 m/ square root Hz for frequencies above 100 Hz, and that this performance can probably be achieved for frequencies as low as 25 Hz. Damping or high Q normal modes in all metal isolator structures can be relatively easily achieved by a combination of active and narrow band eddy current damping while the use of a single-stage ultra-low-frequency pre-isolator can eliminate the need for such damping. Utilizing an improved sapphire transducer, new measurements of the isolator's performance are presented. The measurement noise floor reaches 10^-15 m/ square root Hz for vertical displacements and 10^-14 m/ square root Hz for horizontal displacements.

The gravitational interaction between a point mass and a finite, hollow, thick-walled cylinder is calculated, the axial force is derived, and the parametric form of the coupling coefficients k_2p is presented. This theory is applied to the test-masses for the Satellite Test of the Equivalence Principle (STEP) experiment, and an equation is derived for the differential gravitational coupling to these masses which is more than 10⁵ times faster to compute than a Monte-Carlo integration of similar accuracy.

Hawking's method of obtaining black hole evaporation through Bogoljubov transformations between asymptotic modes (in and out) is generalized. The construction is local in that the in-modes (of say positive frequency) are decomposed by Bogoljubov transformations into positive and negative frequency local inertial modes (i.e. those which are solutions of the d'Alembertian in terms of the local normal coordinates). From this follows an interesting re-expression of the local energy-momentum tensor, more particularly of the outgoing energy flux. One finds that even in the local description there exists a partial thermal character parametrized by a local temperature. There exist quantum interference effects as well. These become negligible at large distances from the black hole.

From the Lagrangian R Square Operator Square Operator R one gets an eighth-order theory of gravitation. It has more promising properties than the previously discussed sixth-order ones. The de Sitter solution has the attractor property; we explicitly show how the modes decay. Further, exactly one power law and one pole-like solution exist. Adding the Einstein-Hilbert and other lower order terms with suitably chosen coefficients, we get a theory without tachyons, with the correct Newtonian limit and with cosmological solutions possessing more than one inflationary phase. (Whether double inflation is typical still remains open).