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In this thesis, the implications of the existence of Killing spinors in a spacetime are investigated. In particular, it is shown that in vacuum and electrovacuum spacetimes a Killing spinor, along with some assumptions on the associated Killing vector in an asymptotic region, guarantees that the spacetime is locally isometric to a member of the Kerr or Kerr-Newman family. It is shown that the characterisation of these spacetimes in terms of Killing spinors is an alternative expression of characterisation results of Mars (Kerr) and Wong (Kerr-Newman) involving restrictions on the Weyl curvature and matter content. In the next section, the construction of a geometric invariant characterising initial data for the Kerr-Newman spacetime is described. This geometric invariant vanishes if and only if the initial data set corresponds to exact Kerr-Newman initial data, and so characterises this type of data. First, the characterisation of the Kerr-Newman spacetime in terms of Killing spinors is illustrated. The space spinor formalism is then used to obtain a set of four independent conditions on an initial Cauchy hypersurface that guarantee the existence of a Killing spinor on the development of the initial data. Following a similar analysis in the vacuum case, the properties of solutions to the approximate Killing spinor equation are studied, and used to construct the geometric invariant. Finally, the problem of Killing spinor initial data in the characteristic problem is investigated. It is shown that data need only be speci ed on the bifurcation surface of the two intersecting null hypersurfaces in order to guarantee the existence of a Killing spinor in a neighbourhood of the bifurcation surface. This characterises the class of spacetimes known as distorted black holes, which include but is strictly larger than the Kerr family of spacetimes.

I study linear waves on higher dimensional Schwarzschild black holes and Schwarzschild de Sitter spacetimes. In the first part of this thesis two decay results are proven for general finite energy solutions to the linear wave equation on higher dimensional Schwarzschild black holes. I establish uniform energy decay and improved interior first order energy decay in all dimensions with rates in accordance with the 3 + 1-dimensional case. The method of proof departs from earlier work on this problem. I apply and extend the new physical space approach to decay of Dafermos and Rodnianski. An integrated local energy decay estimate for the wave equation on higher dimensional Schwarzschild black holes is proven. In the second part of this thesis the global study of solutions to the linear wave equation on expanding de Sitter and Schwarzschild de Sitter spacetimes is initiated. I show that finite energy solutions to the initial value problem are globally bounded and have a limit on the future boundary that can be viewed as a function on the standard cylinder. Both problems are related to the Cauchy problem in General Relativity.

The existence and -if exists- the nature of the solutions of the scalar wave equation in spacetimes with closed timelike curves are investigated. The general properties of the solutions on some class of spacetimes are obtained. Global monochromatic solutions of the scalar wave equation are obtained in flat wormholes of dimensions 2+1 and 3+1. The solutions are in the form of infinite series involving cylindirical and spherical wave functions and they are elucidated by the multiple scattering method. Explicit solutions for some limiting cases are illustrated as well. The results of 2+1 dimensions are verified by using numerical methods.

Cylindrically symmetric shell sources are studied and a general form of induced energy-momentum tensor of a collapsing shell is presented. The actual movement of the shell is numerically solved for a case when dynamics of the system is determined by energy conservation per unit length. It is then shown that additional assumptions can prevent the collapse, e. g. induced energy-momentum tensor as perfect fluid or null dust of counter-rotating particles. Collapse can be prevented also by introducing Thorne's C-energy into the system.

<p>In this thesis we study multipole moments of axisymmetric spacetimes. Using the recursive definition of the multipole moments of Geroch and Hansen we develop a method for computing all multipole moments of a stationary axisymmetric spacetime without the use of a recursion. This is a generalisation of a method developed by Herberthson for the static case.</p><p>Using Herberthson’s method we also develop a method for finding a static axisymmetric spacetime with arbitrary prescribed multipole moments, subject to a specified convergence criteria. This method has, in general, a step where one has to find an explicit expression for an implicitly defined function. However, if the number of multipole moments are finite we give an explicit expression in terms of power series.</p> / Note: The two articles are also available in the pdf-file. Report code: LiU-TEK-LIC-2006:4.

Charged or rotating black holes possess an inner horizon beyond which determinism is lost. However, the strong cosmic censorship conjecture claims that even small perturbations will turn the horizon into a singularity beyond which the spacetime is inextendible, preventing the loss of determinism. Motivated by this conjecture, this dissertation studies free scalar quantum fields on various black-hole spacetimes to test whether quantum effects can lead to the formation of a singularity at the inner horizon in cases where classical perturbations cannot. The starting point is the investigation of the behaviour of real-scalar-field observables near the inner horizon of Reissner-Nordström-de Sitter spacetimes. Using semi-analytical methods, we find that quantum effects can indeed uphold the censorship conjecture. Subsequently, we consider charged scalar fields on the same spacetime and observe that a first-principle calculation is essential to accurately describe the quantum effects at the inner horizon. As a first step towards an extension of these results to rotating black holes, we rigorously construct the Unruh state for the real scalar field on slowly rotating Kerr-de Sitter spacetimes. We show that it is a well-defined Hadamard state and can therefore be used to compute expectation values of the stressenergy tensor and other non-linear observables.:1 Introduction 7 2 An introduction to quantum fields and black holes 13 2.1 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 A brief introduction to AQFT . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 An introduction to microlocal analysis . . . . . . . . . . . . . . . . . . . 24 2.4 An introduction to black-hole spacetimes . . . . . . . . . . . . . . . . . 28 2.4.1 The Reissner-Nordström-de Sitter spacetime . . . . . . . . . . . 28 2.4.2 The Kerr-de Sitter spacetime . . . . . . . . . . . . . . . . . . . . 32 2.5 Free scalar fields in black-hole spacetimes . . . . . . . . . . . . . . . . . 37 3 Computing the energy flux of the real scalar field 43 3.1 Strong cosmic censorship on RNdS . . . . . . . . . . . . . . . . . . . . 43 3.2 The Klein-Gordon equation on RNdS . . . . . . . . . . . . . . . . . . . 45 3.3 Extension to the charged scalar field on RNdS . . . . . . . . . . . . . . . 52 3.4 The energy flux at the Cauchy horizon . . . . . . . . . . . . . . . . . . . 53 4 The charged scalar field in Reissner-Nordström-de Sitter 63 4.1 The Unruh state for the charged scalar field . . . . . . . . . . . . . . . . 65 4.2 The renormalized current . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3 The current in the Unruh state - numerical results . . . . . . . . . . . . . 80 4.4 The charged scalar field at the inner horizon . . . . . . . . . . . . . . . . 86 5 The Unruh state on Kerr-de Sitter 97 5.1 Null geodesics in the Kerr-de Sitter spacetime . . . . . . . . . . . . . . . 98 5.2 The Unruh state on Kerr-de Sitter . . . . . . . . . . . . . . . . . . . . . . 107 5.3 The Hadamard property of the Unruh state . . . . . . . . . . . . . . . . . 120 5.3.1 The Hadamard condition in O . . . . . . . . . . . . . . . . . . . 123 5.3.2 The Hadamard condition on M\O . . . . . . . . . . . . . . . . . 128 6 Summary and discussion 139 A Bibliography 143

In this thesis we study multipole moments of axisymmetric spacetimes. Using the recursive definition of the multipole moments of Geroch and Hansen we develop a method for computing all multipole moments of a stationary axisymmetric spacetime without the use of a recursion. This is a generalisation of a method developed by Herberthson for the static case. Using Herberthson’s method we also develop a method for finding a static axisymmetric spacetime with arbitrary prescribed multipole moments, subject to a specified convergence criteria. This method has, in general, a step where one has to find an explicit expression for an implicitly defined function. However, if the number of multipole moments are finite we give an explicit expression in terms of power series. / <p>Note: The two articles are also available in the pdf-file. Report code: LiU-TEK-LIC-2006:4.</p>

Esta tese tem por base três artigos publicados pelo autor e colaboradores. O primeiro artigo trata do problema da quantização de modelos pseudoclássicos de partículas escalares em campos de fundo não-abelianos, cujo foco é a dedução desses modelos pseudo-clássicos usando métodos de integral de trajetória. O segundo artigo investiga a possibilidade de realizar modelos de gravitação dilatônica em variedades não-comutativas em duas dimensões. Para tanto, vale-se de um método de análise de vínculos e simetrias especialmente desenvolvido para gravitação não-comutativa em duas dimensões. O terceiro artigo discute modelos renormalizáveis em espaços-tempo não-comutativos com parâmetro de não-comutatividade bifermiônico em quatro dimensões. / This thesis is based on three published papers by the author and co-authors. The rst article treats the quantization problem of pseudoclassical models of scalar particles in non-Abelian backgrounds, which aims at deriving these models using path-integral methods. The second article examines the possibility of realizing dilaton gravity models in noncommutative two-dimensional manifolds. It relies upon a method of analysis of constraints and symmetries especially developed for non-commutative dilaton gravities in two dimensions. The third article discusses renormalizable models in noncommutative spacetime with bifermionic noncommutative parameter in four dimensions.

Submitted by RODRIGO DE CASTRO LIMA null (castro.lima.rodrigo@gmail.com) on 2017-03-21T14:21:46Z No. of bitstreams: 1 Dissertação Rodrigo.pdf: 1498013 bytes, checksum: 43cddf583f4f6c01f01e4ada8c8e38bd (MD5) / Approved for entry into archive by Luiz Galeffi (luizgaleffi@gmail.com) on 2017-03-22T14:43:07Z (GMT) No. of bitstreams: 1 lima_rc_me_guara.pdf: 1498013 bytes, checksum: 43cddf583f4f6c01f01e4ada8c8e38bd (MD5) / Made available in DSpace on 2017-03-22T14:43:07Z (GMT). No. of bitstreams: 1 lima_rc_me_guara.pdf: 1498013 bytes, checksum: 43cddf583f4f6c01f01e4ada8c8e38bd (MD5) Previous issue date: 2017-02-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O campo espinorial ELKO associado às partículas de matéria de spin 1/2 e dimensão de massa 1, construídos em um conjunto completo de autoespinores de helicidade dual do operador conjugação de carga, é um canditado a descrever a matéria escura. Devido a sua natureza, possui diversas aplicações cosmológicas. Neste trabalho, o estudo de tais espinores é realizado sob universo de Friedmann-Robertson-Walker plano, homogênio e isotrópico. À luz da Teoria Quântica de Campos em Espaços-Tempo Curvos, com as transformações de Bogoliubov, obteve-se a densidade de partículas criadas em uma métrica modelo de regime assintóticamente plano no passado e no futuro, permitindo-se comparar com soluções exatas presentes na literatura para produção de partículas escalares e férmions de Dirac. Discute- se, também, a compatibilidade do fenômeno de criação de partículas provenientes da Teoria Quântica de Campos em Espaço Curvos com os estudos de Prigogine para descrição deste fenômeno à escala cosmológica, considerando-se um universo termodinamicamente aberto. / The ELKO spinor field associated with spin 1/2 particles and mass dimension 1, con- structed on a complete set of dual helicity eigenspinors of the charge conjugation operator, is a candidate for describing dark matter. Due to its nature, it has several cosmological applications. In this work, the study of such spinors is performed under a homogenous and isotropic Friedmann-Robertson-Walker universe. In the light of the Quantum Field Theory in Curved Space-Time, with the Bogoliubov transformations, we obtained the density of particles created in a model of asymptotically plane metric in the past and in the future, allowing to compare with exact solutions present in the literature for the production of scalar particles and Dirac fermions. It is also discussed the compatibility of the phenomenon of particle creation from the Quantum Field Theory in Curved Space with the studies of Prigogine to describe this phenomenon at the cosmological scale, considering a thermodynamically open universe.