On the Dynamics of Noncircular Accretion Discs

The classical picture of an accretion disc is of a geometrically thin, nearly axisymmetric, fluid flow undergoing supersonic circular motion and slowly accreting due to angular momentum transport by the disc turbulence. There are, however, strong theoretical and observational grounds for considering non-circular motion in accretion discs. In this thesis I study the dynamics of such non-circular accretion discs using a mix of analytical and semi-analytical methods. One such example of a non-circular accretion disc is an eccentric disc, where the dominant fluid motion consists of slowly evolving, nested, confocal, Keplerian ellipses. I focus on the dynamics of these eccentric discs, along with the dynamical non-axisymmetric vertical structure that they set up. In the first part of this thesis I consider eccentric waves in ideal fluid discs. I present a secular Hamiltonian theory describing the evolution of the disc orbits due to pressure gradients and show that it can be used to calculate the eccentric standing wave patterns in the disc (the eccentric modes). I derive a ``short wavelength'' theory for nearly circular orbits that nevertheless can have substantially non-axisymmetric surface density/pressure due to nonlinear eccentricity gradients and disc twist. I use this to show that the pressure profile and precessional forces (such as general relativistic apsidal precession) can focus eccentric waves, causing them to become highly nonlinear in the inner disc. In the second part of this thesis I consider the action of non-ideal terms on the eccentric disc, such as excitation and damping by viscosity. I derive the ordinary differential equations describing simple horizontally invariant ``laminar flows'' in a local model of an eccentric disc, which can be used to study the disc's dynamical vertical structure. I also move beyond the purely hydrodynamic models of the preceding chapters and consider the action of magnetic fields in a local model. Finally I apply the theory presented in this thesis to the highly eccentric discs expected from the tidal disruption of a star by a supermassive black hole. It is currently an open question how, and indeed if, the discs in tidal disruption events circularise. As a step towards understanding the evolution of the disc orbits, I calculate the dynamical vertical structure of highly eccentric discs, emphasising the role of radiation pressure and thermal stability, and showing that magnetic fields may be important where the disc is highly compressed near the periapsis

Set-Theoretic Geology

A ground of the universe V is a transitive proper class W subset V, such that W is a model of ZFC and V is obtained by set forcing over W, so that V = W[G] for some W-generic filter G subset P in W . The model V satisfies the ground axiom GA if there are no such W properly contained in V . The model W is a bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle of V is the intersection of all grounds of V . The generic mantle of V is the intersection of all grounds of all set-forcing extensions of V . The generic HOD, written gHOD, is the intersection of all HODs of all set-forcing extensions. The generic HOD is always a model of ZFC, and the generic mantle is always a model of ZF. Every model of ZFC is the mantle and generic mantle of another model of ZFC. We prove this theorem while also controlling the HOD of the final model, as well as the generic HOD. Iteratively taking the mantle penetrates down through the inner mantles to what we call the outer core, what remains when all outer layers of forcing have been stripped away. Many fundamental questions remain open.Comment: 44 pages; commentary concerning this article can be made at http://jdh.hamkins.org/set-theoreticgeology

Local tomography and the Jordan structure of quantum theory

Using a result of H. Hanche-Olsen, we show that (subject to fairly natural constraints on what constitutes a system, and on what constitutes a composite system), orthodox finite-dimensional complex quantum mechanics with superselection rules is the only non-signaling probabilistic theory in which (i) individual systems are Jordan algebras (equivalently, their cones of unnormalized states are homogeneous and self-dual), (ii) composites are locally tomographic (meaning that states are determined by the joint probabilities they assign to measurement outcomes on the component systems) and (iii) at least one system has the structure of a qubit. Using this result, we also characterize finite dimensional quantum theory among probabilistic theories having the structure of a dagger-monoidal category

Minisuperspaces: Observables and Quantization

A canonical transformation is performed on the phase space of a number of homogeneous cosmologies to simplify the form of the scalar (or, Hamiltonian) constraint. Using the new canonical coordinates, it is then easy to obtain explicit expressions of Dirac observables, i.e.\ phase space functions which commute weakly with the constraint. This, in turn, enables us to carry out a general quantization program to completion. We are also able to address the issue of time through ``deparametrization'' and discuss physical questions such as the fate of initial singularities in the quantum theory. We find that they persist in the quantum theory {\it inspite of the fact that the evolution is implemented by a 1-parameter family of unitary transformations}. Finally, certain of these models admit conditional symmetries which are explicit already prior to the canonical transformation. These can be used to pass to quantum theory following an independent avenue. The two quantum theories --based, respectively, on Dirac observables in the new canonical variables and conditional symmetries in the original ADM variables-- are compared and shown to be equivalent.Comment: 34 page