In this dissertation, we prove a number of results regarding the conformal method of finding solutions to the Einstein constraint equations. These results include necessary and sufficient conditions for the Lichnerowicz equation to have solutions, global supersolutions which guarantee solutions to the conformal constraint equations for near-constant-mean-curvature (near-CMC) data as well as for far-from-CMC data, a proof of the limit equation criterion in the near-CMC case, as well as a model problem on the relationship between the asymptotic constants of solutions and the ADM mass. We also prove a characterization of the Yamabe classes on asymptotically Euclidean manifolds and resolve the (conformally) prescribed scalar curvature problem on asymptotically Euclidean manifolds for the case of nonpositive scalar curvatures. This dissertation includes previously published coauthored material.

This thesis considers various aspects of general relativity in more than four spacetime dimensions. Firstly, I review the generalization to higher dimensions of the algebraic classification of the Weyl tensor and the Newman-Penrose formalism. In four dimensions, these techniques have proved useful for studying many aspects of general relativity, and it is hoped that their higher dimensional generalizations will prove equally useful in the future. Unfortunately, many calculations using the Newman-Penrose formalism can be unnecessarily complicated. To address this, I describe new work introducing a higher-dimensional generalization of the so-called Geroch-Held-Penrose formalism, which allows for a partially covariant reformulation of general relativity. This approach provides great simplifications for many calculations involving spacetimes which admit one or two preferred null directions. The next chapter describes the proof of an important result regarding algebraic classification in higher dimensions. The classification is based upon the existence of a particular null direction that is aligned with the Weyl tensor of the geometry in some appropriate sense. In four dimensions, it is known that a null vector field is such a multiple Weyl aligned null direction (WAND) if and only if it is tangent to a shearfree null geodesic congruence. This is not the case in higher dimensions. However, I have formulated and proved a partial generalization of the result to arbitrary dimension, namely that a spacetime admits a multiple WAND if and only if it admits a geodesic multiple WAND.Moving onto more physical applications, I describe how the formalism that we have developed can be applied to study certain aspects of the stability of extremal black holes in arbitrary dimension. The final chapter of the thesis has a rather different flavour. I give a detailed analysis of the properties of a particular solution to the Einstein equations in five dimensions: the Pomeransky-Sen'kov doubly spinning black ring. I study geodesic motion around this black ring and demonstrate the separability of the Hamilton-Jacobi equation for null, zero energy geodesics. I show that this unexpected separability can be understood in terms of a symmetry described by a conformal Killing tensor on a four dimensional spacetime obtained by a Kaluza-Klein reduction of the original black ring spacetime.

We develop new variational techniques, acting on classes of Lagrangians with the same functional dependence but arbitrary functional form, for the derivation of general, strongly conserved quantities, supplementing the usual procedure for deriving weak conservation laws via Noether's theorem. Using these new techniques we generate and generalize virtually all energy-momentum complexes currently known. In the process we discover and understand the reason for the difficulties associated with energy-momentum complexes in general relativity. We study a Palatini variation of a novel Lagrangian due to Nissani. We find that Nissani's principal claim, that his Lagrangian specifies Riemannian geometry in the presence of a generalized matter tensor, is not in fact justifiable, and prove that his Lagrangian is not unique. We speculate on the possibility of deriving a general-relativistic analog of Maxwell's current equation, a matter current equation, yielding an entirely new approach to the idea of energy-momentum in general relativity. We develop the SL(2,C) x U(1) spinor formalism naturally combining the gravitational and electromagnetic potentials in a single object--the spinor connection. Variably charged matter is rigourously introduced, through the use of spin densities, in the unified potential theories we develop. We generate both the Einstein-Maxwell equations and new equations. The latter generalize both the Maxwell equation and the Einstein equation which includes a new "gravitational stress-energy tensor". This new tensor exactly mimicks the electromagnetic stress-energy tensor with Riemann tensor contractions replacing Maxwell tensor contractions. We briefly consider the introduction of matter. A Lagrangian generalizing the two spinor Dirac equations has no gravitational currents and the electromagnetic currents must be on the light cone. A Lagrangian generalizing the Pauli equations has both gravitational and electromagnetic currents. The equations of both Lagrangians demonstrate beautifully how the divergence of the total stress-energy tensor vanishes in this formalism. In the theory of the generalized Einstein-Maxwell and Pauli equations we succeed in deriving an equation describing a generalized matter-charge current density. / Graduate

Apresentamos neste trabalho uma generalização da teoria de perturbações cosmológicas para o caso de universos homogêneos e anisotrópicos, caracterizados por um espaço-tempo do tipo Bianchi I. Como aplicação da teoria, investigamos as conseqüências de uma fase inflacionária e anisotrópica do universo dos pontos de vista clássico e quântico. Após uma discussão da evolução do espaço-tempo de fundo nós quantizamos os modos perturbativos para, em seguida, construir o espectro de potências das perturbações de curvatura e de ondas gravitacionais do fim da inflação. Nossos resultados mostram que as principais características de uma fase anisotrópica primordial do universo são: (1) dependência direcional dos espectros de potências, (2) acoplamento entre as perturbações de curvatura e as ondas gravitacionais e (3) espectros distintos para as diferentes polarizações das ondas gravitacionais em grandes escalas cosmológicas. Todos esses efeitos são importantes apenas em grandes escalas cosmológicas e, localmente, recuperamos a teoria isotrópica de perturbações cosmológicas. Nossos resultados dependem de uma escala característica que pode, embora não seja estritamente necessário, ser ajustada a alguma escala observável. / In this work we generalize the standard theory of cosmological perturbations to the case of homogeneous and anisotropic universes described by a Bianchi I spacetime metric. As an application of this theory we investigate the predictions of an inflationary anisotropic phase, both at the classical and quantum level. After discussing the evolution of the background spacetime, we solve and quantize the perturbation equations in order to predict the power spectra of the curvature perturbations and gravity waves at the end of inflation. Our results show that the main features of an early anisotropic phase are: (1) a dependence of the spectra on the direction of the modes, (2) a coupling between curvature perturbations and gravity waves, and (3) the fact that the two gravity waves polarisations do not share the same spectrum on large scales. All these effects are significant only on large scales and die out on small scales where isotropy is recovered. Finally, our results depend on a characteristic scale that can, but a priori does not have to, be tuned to some observable scale.

In this thesis we study near-horizon geometries of extreme black holes. We first consider stationary extreme black hole solutions to the Einstein-Yang-Mills theory with a compact semi-simple gauge group in four dimensions, allowing for a negative cosmological constant. We prove that any axisymmetric black hole of this kind possesses a near-horizon AdS2 symmetry and deduce its near-horizon geometry must be that of the abelian embedded extreme Kerr-Newman (AdS) black hole. We show that the near-horizon geometry of any static black hole is a direct product of AdS2 and a constant curvature space. We then consider near-horizon geometry in Einstein gravity coupled to a Maxwell field and a massive complex scalar field, with a cosmological constant. We prove that assuming non-zero coupling between the Maxwell and the scalar fields, there exists no solution with a compact horizon in any dimensions where the massive scalar is non-trivial. This result generalises to any scalar potential which is a monotonically increasing function of the modulus of the complex scalar. Next we determine the most general three-dimensional vacuum spacetime with a negative cosmological constant containing a non-singular Killing horizon. We show that the general solution with a spatially compact horizon possesses a second commuting Killing field and deduce that it must be related to the BTZ black hole (or its near-horizon geometry) by a diffeomorphism. We show there is a general class of asymptotically AdS3 extreme black holes with arbitrary charges with respect to one of the asymptotic-symmetry Virasoro algebras and vanishing charges with respect to the other. We interpret these as descendants of the extreme BTZ black hole. However descendants of the non-extreme BTZ black hole are absent from our general solution with a non-degenerate horizon. We then show that the first order deformation along transverse null geodesics about any near-horizon geometry with compact cross-sections always admits a finite-parameter family of solutions as the most general solution. As an application, we consider the first order expansion from the near-horizon geometry of the extreme Kerr black hole. We uncover a local uniqueness theorem by demonstrating that the only possible black hole solutions which admit a U(1) symmetry are gauge equivalent to the first order expansion of the extreme Kerr solution itself. We then investigate the first order expansion from the near-horizon geometry of the extreme self-dual Myers-Perry black hole in 5D. The only solutions which inherit the enhanced SU(2) X U(1) symmetry and are compatible with black holes correspond to the first order expansion of the extreme self-dual Myers-Perry black hole itself and the extreme J = 0 Kaluza-Klein black hole. These are the only known black holes to possess this near-horizon geometry. If only U(1) X U(1) symmetry is assumed in first order, we find that the most general solution is a three-parameter family which is more general than the two known black hole solutions. This hints the possibility of the existence of new black holes.

Philosophiae Doctor - PhD / The matter power spectrum is key to understanding the growth of large-scale structure in the Universe. Upcoming surveys of galaxies in the optical and HI will probe increasingly large scales, approaching and even exceeding the Hubble scale at the survey redshifts. On these cosmological scales, surveys can in principle provide the best constraints on dark energy (DE) and modified gravity models and will be able to test general relativity itself. However, in order to realise the potential of these surveys, we need to ensure that we are using a correct analysis, i.e. a general relativistic analysis, on cosmological scales. There are two fundamental issues underlying the general relativistic (GR) analysis. Firstly, we need to correctly identify the galaxy overdensity that is observed on the past light cone. Secondly, we need to account for all the distortions arising from observing on the past light cone, including redshift distortions (with all general relativistic effects included) and volume distortions. These general elativistic effects appear in the angular power spectra of matter in redshift space. We compute these quantities, taking into account all general relativistic large-scale effects, and including the important contributions from redshift space distortions and lensing convergence. This is done for self-consistent models of DE, known as ‘quintessence’, which have only been very recently treated in the GR approach. Particularly, we focus mainly on computing the predictions (i.e. the power spectra) that need to be confronted with future data. Hence we compute the GR angular power spectra, correcting the 3D Newtonian calculation for several quintessence models. We also compute the observed 3D power spectra for interacting DE (which until now have not previously been studied in the GR approach) – in which dark matter and DE exchange energy and momentum. Interaction in the dark sector can lead to large-scale deviations in the power spectrum, similar to GR effects or modified gravity. For the quintessence case, we found that the DE perturbations make only a small contribution on the largest scales, and a negligible contribution on smaller scales. Ironically, the DE perturbations remove the false boost of large-scale power that arises if we impose the (unphysical) assumption that the DE perturbations vanish. However, for the interacting DE (IDE) case, we found that if relativistic effects are ignored, i.e. if they are not subtracted in order to isolate the IDE effects, the imprint of IDE will be incorrectly identified – which could lead to a bias in constraints on IDE, on horizon scales. Moreover, we found that on super-Hubble scales, GR corrections in the observed galaxy power spectrum are able to distinguish a homogeneous DE (being one whose density perturbation in comoving gauge vanishes) from the concordance model (and from a clustering DE) – at low redshifts and for high magnification bias. Whereas the matter power spectrum is incapable of distinguishing a homogeneous DE from the concordance model. We also found that GR effects become enhanced with decreasing magnification bias, and with increasing redshift.

Techniques for numerically constructing initial data in the 3+1 formalism of general relativity (GR) are studied, using the theoretical framework described in Bowen and York (1980), Physical Review D 21(8), 2047-2056. The two main assumptions made are maximal slicing and 3-conformal flatness of the generated spaces. For ease of numerical solution, axisymmetry is also assumed, but all the results should extend without difficulty to the non-axisymmetric case. The numerical code described in this thesis may be used to construct vacuum spaces containing arbitrary numbers of black holes, each with freely specifyable (subject to the axisymmetry assumption) position, mass, linear momentum, and angular momentum. It should be emphasised that the time evolution of these spaces has not yet been attempted. There are two significant innovations in this work: the use of a new boundary condition for the surfaces of the black holes, and the use of multiple coordinate patches in the numerical solution. The new boundary condition studied herein requires the inner boundary of the numerical grid to be a marginally trapped surface. This is in contrast to the approach used in much previous work on this problem area, which requires the constructed spaces to be conformally isometric under a "reflection mapping" which interchanges the interior of a specified black hole with the remainder of the space. The new boundary condition is found to be easy to implement, even for multiple black holes. It may also prove useful in time evolution problems. The coordinate choice scheme introduced in this thesis uses multiple coordinate patches in the numerical solution, each with a coordinate system suited to the local physical symmetries of the region of space it covers. Because each patch need only cover part of the space, the metrics on the individual patches can be kept simple, while the overall patch system still covers a complicated topology. The patches are linked together by interpolation across the interpatch boundaries. Bilinear interpolation suffices to give accuracy comparable with that of common second order difference schemes used in numerical GR. This use of multiple coordinate patches is found to work very well in both one and two black hole models, and should generalise to a wide variety of other numerical GR problems. Patches are also found to be a useful (if somewhat over-general) way of introducing spatially varying grid sizes into the numerical code. However, problems may arise when trying to use multiple patches in time evolution problems, in that the interpatch boundaries must not become spurious generators or reflectors of gravitational radiation, due to the interpolation errors. These problems have not yet been studied. The code described in this thesis is tested against Schwarzschild models and against previously published work using the Bowen and York formalism, reproducing the latter within the limits of error of the codes involved. A number of new spaces containing one and two black holes with linear or angular momentum are also constructed to demonstrate the code, although little analysis of these spaces has yet been done. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

This study is an investigation of analytical and numerical predictions of the relativistic perihelion shift of the planets in the Solar System. These shifts are a consequence of the general theory of relativity and the theoretical expression for the perihelion shift can be derived using different approximations. Two of these approximations are presented in this report. Due to these approximations, it is of interest to investigate when the analytical expression provides accurate values of perihelion shifts. This is performed by calculating the perihelion shifts for the planets in the Solar System numerically and comparing the results. To summarize the results, the numerical method slightly outperforms the analyt- ical method for almost all planets in the Solar System but the results are overall similar. Furthermore, a parameter study is conducted to investigate how the numerical and an- alytical perihelion shift predictions are dependent on these parameters. The parameter study shows that the methods diverge for some cases and based on this predictions are made for when numerical methods could be of use when predicting perihelion shifts.

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