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Title: Boundary-value problems in quantum gravity and classical solutions
Author: Akbar, M. M.
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 2003
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It is proved that Taub-Bolt infillings are double-valued whereas Taub-Nut and Eguchi-Hanson infillings are unique in arbitrary dimensions. In the case of trivial bundles, there are two or no Schwarzschild infillings. The condition of whether a particular type of infilling exists can be expressed as a limitation on squashing through a functional dependence on dimension in each case. The case of the Eguchi-Hanson metric is solved in arbitrary dimension. The Taub-Nut and the Taub-Bolt are solved in four dimensions and methods for higher dimensions are discussed. For the case of Schwarzschild in arbitrary dimension, thermodynamic properties of the two infilling black-hole solutions are discussed and analytic formulae for their masses are obtained using higher order hypergeometric functions. Convexity of the infilling solutions and isoperimetric inequalities involving the volume of the boundary and the volume of the infilling solutions are investigated. In particular, analogues of Minkowski’s celebrated inequality in flat space are found and discussed. In Chapters 3, the Dirichlet problem is studied for an SU (2) x U(1)-invariant S3 boundary within the class of self-dual Taub-Nut-(anti) de Sitter metrics. Including complex ones there can be a total of three solutions for the infilling although there will be a unique real solution or no real solution depending on the boundary data - the two radii of the S3. Exact solutions of the infilling geometries are obtained making its possible to find their Euclidean actions as analytic functions of the two radii of the S3-boundary. The case of L < 0 is investigated further. For reasonable squashing of the S3, all three infilling solutions have real-valued actions which possess a “cusp catastrophe” structure with a “catastrophe manifold” that shows that the unique real positive-definite solution dominates. The necessary and sufficient condition for the existence of the positive-definite solution is found as a condition on the two radii of the S3. In Chapter 4, the same boundary-value problem is studied for the Taub-Bolt-anti-de Sitter metrics. Such metrics are obtained from the two-parameter Taub-NUT-anti de Sitter family. The condition of regularity results in two bifurcated one-parameter family. It is found that any axially symmetric S3-boundary can be filled in with at least one solution coming from each of these two branches. The infillings appear or disappear catastrophically in pairs as the values of the two radii of S3 are varied; this happens simultaneously for both branches. It is found that the total number in independent infillings is two, six or ten. When the two radii are of the same order and large this number is two. In the isotropic limit, i.e., for round S3 this holds for small radii as well. In Chapter 5, the Dirichlet problem is studied within Euclideanised Schwarzschild-anti de Sitter and anti de Sitter metrics, i.e., for an S1 x Sn boundary. For such boundary data there exist two or no black-holes and always a unique anti de Sitter solution. The black holes have strictly positive and negative specific heats (and hence locally thermodynamically stable and unstable respectively). It is shown that for any radius of the cavity, the larger hole can be globally thermodynamically stable above a critical temperature by demonstrating that a phase transition occurs from hot AdS to Schwarzschild-AdS within the cavity. This gives the Hawking-Page phase transition in the infinite cavity limit. It is found that the case of five dimensions is special in that the masses of the two black holes, and hence other quantities of classical and semi-classical interest, can be obtained exactly as functions of cavity radius and temperature. It is also possible in this case to obtain the minimum temperature (below which no black holes exist) and the critical temperature for phase transition as analytic functions of cavity-radius. In Chapter 6, cosmological and instanton solutions are found for CP1 and CP2 sigma models coupled to gravity with a possible cosmological constant.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available