In this thesis we look at mobile agent solutions to black hole search and related problems. Mobile agents are computational entities that are autonomous, mobile, and can interact with their environment and each other. The black hole search problem is for a team of these agents to work together to map or explore a graph-like network environment where some elements of the network are dangerous to the agents. Most research into black hole search has focussed on finding a single dangerous node: a black hole. We look at the problem of finding multiple black holes and, in the case of dangerous graph exploration, multiple black links as well. We look at the dangerous graph exploration problem in the network model. The network model is based on a normal static computer network modelled as a simple graph. We give an optimal solution to the dangerous graph exploration problem using agents that start scattered on nodes throughout the network. We then make the problem more difficult by allowing an adversary to delete links during the execution of the algorithm and provide a solution using scattered agents. In the last decade or two, types of networks have emerged, such as ad hoc wireless networks, that are by their nature dynamic. These networks change quickly over time and can make distributed computations difficult. We look at black hole search in one type of dynamic network described by the subway model, which we base on urban subway systems. The model allows us to look at the cost of opportunistic movement by requiring the agents to move using carriers that follow routes among the network's sites, some of which are black holes. We show that there are basic limitations on any solution to black hole search in the subway model and prove lower bounds on any solution's complexity. We then provide two optimal solutions that differ in the agents' starting locations and how they communicate with one another. Our results provide a small window into the cost of deterministic distributed computing in networks that have dynamic elements, but which are not fully random.

The semi-classical black hole tunnelling method is a useful technique to calculate black hole temperature and understand black hole thermodynamics. I will investigate the black hole tunnelling method in detail. I will compare two different approaches used to calculate black hole tunnelling. The tunnelling method can be applied to a broad range of spacetimes and I will show this explicitly in order to demonstrate the robustness of the tunnelling technique. In particular, I will apply the tunnelling method to spacetimes including: Rindler (the method can recover the Unruh temperature), and more general spacetimes (such as Kerr-Newman and Taub-NUT). I will also discuss the 5d Kerr-Gödel spacetimes in detail (while showing a previous unobserved property of these spaces). Once the parameter space of Kerr-Gödel is understood in detail, I will show how the tunnelling method can also be successfully applied to the Kerr-Gödel black hole. Finally, the key result of my thesis involves extending the tunnelling method to model fermion emission. The previous tunnelling calculations all involved the emission of scalar particles. I will model the emission of spin-1/2 fermions from various spacetimes including the Rindler spacetime and general non-rotating black holes. I will also model the emission of charged spin-1/2 fermions from the Kerr-Newman spacetime to show that the method is also applicable to rotating spacetimes. In all these cases I show that the correct Hawking temperature (Unruh temperature in the case of Rindler) is recovered for spin-1/2 fermion emission. Although this final result is not surprising, it is an important result because it confirms that Dirac particles will radiate from the black hole at the same temperature as scalar particles. It has always been assumed that this is the case but there is very little literature involving fermion radiation of black holes. So the results of my calculations are twofold, I demonstrate that Dirac particles are emitted at the same temperature as scalar particles from a black hole and it shows how robust the semi-classical tunnelling technique is.

I modify the quasilocal energy formalism of Brown and York into a purely Hamiltonian form. As part of the reformulation, I remove their restriction that the time evolution of the boundary of the spacetime be orthogonal to the leaves of the time foliation. Thus the new formulation allows an arbitrary evolution of the boundary which physically corresponds to allowing general motions of the set of observers making up that boundary. I calculate the rate of change of the quasilocal energy in such situations, show how it transforms with respect to boosts of the boundaries, and use the Lanczos-Israel thin shell formalism to reformulate it from an operational point of view. These steps are performed both for pure gravity and gravity with attendant matter fields. I then apply the formalism to characterize naked black holes and study their properties, investigate gravitational tidal heating, and combine it with the path integral formulation of quantum gravity to analyze the creation of pairs of charged and rotating black holes. I show that one must use complex instantons to study this process though the probabilities of creation remain real and consistent with the view that the entropy of a black hole is the logarithm of the number of its quantum states.

The semi-classical black hole tunnelling method is a useful technique to calculate black hole temperature and understand black hole thermodynamics. I will investigate the black hole tunnelling method in detail. I will compare two different approaches used to calculate black hole tunnelling. The tunnelling method can be applied to a broad range of spacetimes and I will show this explicitly in order to demonstrate the robustness of the tunnelling technique. In particular, I will apply the tunnelling method to spacetimes including: Rindler (the method can recover the Unruh temperature), and more general spacetimes (such as Kerr-Newman and Taub-NUT). I will also discuss the 5d Kerr-Gödel spacetimes in detail (while showing a previous unobserved property of these spaces). Once the parameter space of Kerr-Gödel is understood in detail, I will show how the tunnelling method can also be successfully applied to the Kerr-Gödel black hole. Finally, the key result of my thesis involves extending the tunnelling method to model fermion emission. The previous tunnelling calculations all involved the emission of scalar particles. I will model the emission of spin-1/2 fermions from various spacetimes including the Rindler spacetime and general non-rotating black holes. I will also model the emission of charged spin-1/2 fermions from the Kerr-Newman spacetime to show that the method is also applicable to rotating spacetimes. In all these cases I show that the correct Hawking temperature (Unruh temperature in the case of Rindler) is recovered for spin-1/2 fermion emission. Although this final result is not surprising, it is an important result because it confirms that Dirac particles will radiate from the black hole at the same temperature as scalar particles. It has always been assumed that this is the case but there is very little literature involving fermion radiation of black holes. So the results of my calculations are twofold, I demonstrate that Dirac particles are emitted at the same temperature as scalar particles from a black hole and it shows how robust the semi-classical tunnelling technique is.

Nous analysons des bulles d'espace-temps d'épaisseur finie en relativité générale. Les conditions d'énergie sont utilisées afin d'obtenir un ensemble de critères permettant de restreindre la structure du bord de la bulle. Dans le cas des bulles statiques et à symétrie sphérique, nous obtenons quatre inégalités différentielles équivalentes aux trois conditions d'énergie les plus communes. Nous montrons qu'elles sont équivalentes à un ensemble de deux inégalités différentielles simples lorsque le potentiel gravitationnel effectif a une forme particulière. Nous paramétrons alors l'espace-temps de manière à rendre la vérification de ces inégalités plus simple lorsqu'il sera question de bulles d'espace-temps. Nous traitons en particulier quatre formes de bulles, toutes caractérisées par un extérieur de type Schwarzschild de Sitter. Nous montrons que notre méthode donne les bons résultats lorsque la limite où l'épaisseur de la bulle tend vers zéro est prise. Nous terminons par un traitement succinct du problème d'une onde gravitationnelle se propageant dans un nuage de bulles d'espace-temps. / We analyze space-time bubbles of finite thickness in general relativity. We use the energy conditions to restrict their structures. In the case of static, spherically symmetric bubbles, we get a set of four differential inequalities. If the effective gravitational potential is taken of a particular form, we show that they can be further reduced to a set of two differential inequalities. We then parameterize the bubble's wall in a particular way, simplifying the inequalities, and easing the application of boundary conditions on our solutions. We then treat four different cases of bubbles that all have a Schwarzschild de Sitter exterior. We show that in the limit where the thickness of the bubble's wall goes to zero, we recover the standard results. Lastly, we treat gravitational waves propagating in a dilute gas of non-interacting space-time bubbles.

I construct anti-de Sitter boson stars in Einstein-Gauss-Bonnet gravity coupled to a (D-1)/(2)-tuplet of complex massless scalar field both perturbativelyand numerically in D=5,7,9,11 dimensions. Due to the choice of scalar fields, these solutions possess just a single helical Killing symmetry. For each choice of the Gauss-Bonnet parameter &alpha;&#8800;&alpha;_cr, the central energy density at the center of the boson star, q_0 completely characterizes the one parameter family of solutions. These solutions obey the first law of thermodynamics, in the case of the numerics, to within 1 part in 10^6. I describe the dependence of the boson star mass, angular momentum and angular velocity on &alpha; and on the dimensionality. For &alpha;<&alpha;_cr and D>5, these quantities exhibit damped oscillations about finite central values and the central energy density tends to infinity. The Kretschmann invariant at the center of the boson star diverges in the limit of diverging central energy. This contrasts the D=5 case, where the Kretschmann invariant diverges at a finite value of the central energy density. Solutions where &alpha;<&alpha;_cr, correspond to negative mass boson stars, and the for all dimensions the boson star mass and angular momentum decrease exponentially as the central energy density tends toward infinity with the Kretschmann invariant diverging only when in the limit the central energy density diverges. I also briefly discuss the difficulties of numerically obtaining single Killing vector hairy black hole solutions and present the explicit boundary conditions for both Einstein gravity and Einstein-Gauss-Bonnet gravity.

An investigation of information loss in black hole spacetimes is performed. We demon- strate that the definition of particles as energy levels of the Harmonic oscillator will not have physical significance in general and is thus not a good instrument to study the ra- diation of black holes. This is due to the ambiguity of the choice of coordinates on the phase space of the quantum field. We demonstrate how to identify quantum states in the functional Schr ̈dinger picture. o We demonstrate that information is truly lost in the case of a Vaidya black hole (a black hole formed from null dust) if we neglect back reaction. This is done by quantizing the constrained classical system of a Klein-Gordon field in a Vaidya background. The interaction picture of quantum mechanics can be applied to this system. We find a physically well motivated vacuum state for a spherically symmetric space- time with an extra conformal Killing vector. We also demonstrate how to calculate the response of a particle detector in the a LeMaitre-Tolman-Bondi spacetime with a self- similarity. Finally, some of the claims and confusion surrounding Unruh radiation, Hawking radiation and the equivalence principle are investigated and shown to be false.

An investigation of information loss in black hole spacetimes is performed. We demon- strate that the definition of particles as energy levels of the Harmonic oscillator will not have physical significance in general and is thus not a good instrument to study the ra- diation of black holes. This is due to the ambiguity of the choice of coordinates on the phase space of the quantum field. We demonstrate how to identify quantum states in the functional Schr ̈dinger picture. o We demonstrate that information is truly lost in the case of a Vaidya black hole (a black hole formed from null dust) if we neglect back reaction. This is done by quantizing the constrained classical system of a Klein-Gordon field in a Vaidya background. The interaction picture of quantum mechanics can be applied to this system. We find a physically well motivated vacuum state for a spherically symmetric space- time with an extra conformal Killing vector. We also demonstrate how to calculate the response of a particle detector in the a LeMaitre-Tolman-Bondi spacetime with a self- similarity. Finally, some of the claims and confusion surrounding Unruh radiation, Hawking radiation and the equivalence principle are investigated and shown to be false.

In this thesis we look at mobile agent solutions to black hole search and related problems. Mobile agents are computational entities that are autonomous, mobile, and can interact with their environment and each other. The black hole search problem is for a team of these agents to work together to map or explore a graph-like network environment where some elements of the network are dangerous to the agents. Most research into black hole search has focussed on finding a single dangerous node: a black hole. We look at the problem of finding multiple black holes and, in the case of dangerous graph exploration, multiple black links as well. We look at the dangerous graph exploration problem in the network model. The network model is based on a normal static computer network modelled as a simple graph. We give an optimal solution to the dangerous graph exploration problem using agents that start scattered on nodes throughout the network. We then make the problem more difficult by allowing an adversary to delete links during the execution of the algorithm and provide a solution using scattered agents. In the last decade or two, types of networks have emerged, such as ad hoc wireless networks, that are by their nature dynamic. These networks change quickly over time and can make distributed computations difficult. We look at black hole search in one type of dynamic network described by the subway model, which we base on urban subway systems. The model allows us to look at the cost of opportunistic movement by requiring the agents to move using carriers that follow routes among the network's sites, some of which are black holes. We show that there are basic limitations on any solution to black hole search in the subway model and prove lower bounds on any solution's complexity. We then provide two optimal solutions that differ in the agents' starting locations and how they communicate with one another. Our results provide a small window into the cost of deterministic distributed computing in networks that have dynamic elements, but which are not fully random.

We will consider the scenario of the co-rotation of a fluid star (in specific, a neutron star) and a black hole. The neutron star (or primary)is assumed to have constant angular velocity. The tidal effects on the primary are investigated. First, the centrally condensed approximation is applied, where both bodies are considered as point sources. In the second treatment, the primary is treated as an incompressible and homogeneous fluid mass, which in addition to its own gravity is subject to centrifugal and Coriolis forces, derived from fluid motions. The black hole (or secondary) is treated as a rigid sphere and can be regarded as a point mass. The equilibrium figure is derived. The problem is then adapted to include vorticity and a pseudo-Newtonian potential. The coalescence of neutron star - black hole binaries and their importance to gravitational wave detection is also discussed. / Mathematical Sciences / M. Sc. (Applied Mathematics)