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.

Future Higgs factories can reach impressive precision on Higgs property measurements. In this paper, instead of conventional focus of Higgs precision in certain interaction bases, we explore its sensitivity to new physics models at the electron-positron colliders. In particular, we study two categories of new physics models, Standard Model (SM) with a real scalar singlet extension, and Two Higgs Double Model (2HDM) as examples of weakly-interacting models, Minimal Composite Higgs Model (MCHM) and three typical patterns of the more general operator counting for strong interacting models as examples of strong dynamics. We perform a global fit to various Higgs search channels to obtain the 95% C.L. constraints on the model parameter space. In the SM with a singlet extension, we obtain the limits on the singlet-doublet mixing angle sin theta, as well as the more general Wilson coefficients of the induced higher dimensional operators. In the 2HDM, we analyze tree level effects in tan beta vs. cos(beta-alpha) plane, as well as the one-loop contributions from the heavy Higgs bosons in the alignment limit to obtain the constraints on heavy Higgs masses for different types of 2HDM. In strong dynamics models, we obtain lower limits on the strong dynamics scale. In addition, once deviations of Higgs couplings are observed, they can be used to distinguish different models. We also compare the sensitivity of various future Higgs factories, namely Circular Electron Positron Collider (CEPC), Future Circular Collider (FCC)-ee and International Linear Collider (ILC).

6d (2, 0) theory on M5 branes is investigated by considering its KK modes on a 2d space. Selecting KK modes on different 2d spaces amounts to choosing different set of selfdual strings as the perturbative degrees of freedom thus will give the 6d theories related to each other by U-duality. The 4d effective theory for the KK modes is studied via the M5-D3 duality. Except for the (p, q) open strings, which is the KK mode arising from the selfdual strings, the 3-string junction should also be added since it is the bound state of the (p, q) open strings. The quantization of the 3-string junctions gives the fields, which, when lifted to 6d, may account for the conformal anomaly of the 6d (2, 0) theory. The interaction between the open strings and the 3-string junctions is also considered. The Lagrangian and the corresponding N=4 supersymmetry transformation is obtained up to some additional terms to be added. Although the original 6d (2, 0) theory is not constructed directly, the 4d effective theory for the KK modes gives an equivalent description, from which the 6d S-matrix can be calculated.

The study of general relativity in higher dimensions has proven to be a fruitful avenue of research, revealing new applications of the theory, for instance in understanding strongly coupled quantum field theories through the holographic principle, and proposing an explanation of the hierarchy problem through TeV gravity scenarios. To understand the non-linear regime of higher dimensional general relativity, such as that involved in the merger of black holes, we use numerical relativity to solve the Einstein equations. In this thesis we develop and demonstrate several diagnostic tools and new initial data for use in numerical relativity simulations of higher dimensional spacetimes, and use these to investigate binary black hole systems. Firstly, we present a formalism for calculating the gravitational waves in a numerical simulation of a higher dimensional spacetime, and apply this formalism to the example of the head on merger of two equal mass black holes. In doing so, we simulate the merger of black holes in up to 10 spacetime dimensions for the first time, and investigate the dependence of the energy radiated away in gravitational waves on the number of dimensions. We also apply this formalism to the example of head on unequal mass black hole collisions, investigating the dependence of radiated energy and momentum on the number of dimensions and the mass ratio. This study complements and sheds further light on previous work on the merger of point particles with black holes in higher dimensions, and presents evidence for a link between the regime studied, and the large $D$ regime of general relativity where $D$ is the number of spacetime dimensions. We also present initial data that enables us to study black holes with initial momentum and angular momentum, putting in place the framework needed to study problems such as the scattering cross section of black holes in higher dimensions, and the nature of black hole orbits in higher dimensions. Finally, we present, and demonstrate the use of, an apparent horizon finder for higher dimensional spacetimes. This allows us to calculate a black hole's mass and spin, which characterise the black hole.

General relativity, one of the pillars of our understanding of the universe, has been a remarkably successful theory. It has stood the test of time for more than 100 years and has passed all experimental tests so far. Most recently, the LIGO collaboration made the first-ever direct detection of gravitational waves, confirming a long-standing prediction of general relativity. Despite this, several fundamental mathematical questions remain unanswered, many of which relate to the global existence and the stability of solutions to Einstein's equations. This thesis presents our efforts to use numerical relativity to investigate some of these questions. We present a complete picture of the end points of black ring instabilities in five dimensions. Fat rings collapse to Myers-Perry black holes. For intermediate rings, we discover a previously unknown instability that stretches the ring without changing its thickness and causes it to collapse to a Myers-Perry black hole. Most importantly, however, we find that for very thin rings, the Gregory-Laflamme instability dominates and causes the ring to break. This provides the first concrete evidence that in higher dimensions, the weak cosmic censorship conjecture may be violated even in asymptotically flat spacetimes. For Myers-Perry black holes, we investigate instabilities in five and six dimensions. In six dimensions, we demonstrate that both axisymmetric and non-axisymmetric instabilities can cause the black hole to pinch off, and we study the approach to the naked singularity in detail. Another question that has attracted intense interest recently is the instability of anti-de Sitter space. In this thesis, we explore how breaking spherical symmetry in gravitational collapse in anti-de Sitter space affects black hole formation. These findings were made possible by our new open source general relativity code, GRChombo, whose adaptive mesh capabilities allow accurate simulations of phenomena in which new length scales are produced dynamically. In this thesis, we describe GRChombo in detail, and analyse its performance on the latest supercomputers. Furthermore, we outline numerical advances that were necessary for simulating higher dimensional black holes stably and efficiently.

This thesis contains two main themes. The first is Einstein's theory of general relativity in higher dimensions, while the second is M-theory. The first part of the thesis concerns the use of classification techniques based on the Weyl curvature in an attempt to systematically study higher dimensional general relativity and its solutions. After a review of the various classification schemes, the application of these schemes to the study of higher dimensional solutions is explained. The first application of the tensor approach that is discussed is the systematic classification of higher dimensional axisymmetric solutions. A complete classification of all algebraically special axisymmetric solutions to the vacuum Einstein equation in higher dimensions is presented. Next, the study of perturbations of higher dimensional solutions within this framework and the possibility of decoupling equations for black hole solutions of interest, as has been successfully done in four dimensions, is considered. In the case where such a decoupling of the perturbations is possible, a map for constructing solutions of the perturbation equation is presented and is applied to the Kerr/CFT correspondence. Also, the property of gravitational radiation emitted from an isolated source in higher dimensions is considered and the tensor classification scheme is used to derive the peeling property of the Weyl tensor in higher dimensions. This is shown to be different to that which occurs in four dimensions. Finally, after an in-depth exposition of the spinor classification scheme and its relation to the tensor approach, solutions belonging to the most special type in the spinor classification are classified. In addition, the classification of the black ring in this scheme is discussed. The second part of the thesis explores the use of generalised geometry as a tool for better understanding M-theory. After briefly reviewing the curious phenomenon of M-theory dualities, it is explained how generalised geometry can be used to show that these symmetries are not exclusive to compactifications of the theory, but can be made manifest without recourse to compactification. Finally, results regarding the local symmetries of M-theory in the generalised geometry framework for a particular symmetry group are presented.

We present a method for solving the Einstein-Maxwell equations in a five dimensional, asymptotically flat, black hole spacetime with three commuting Killing vector fields. In particular, we show that by reducing the dimension of the Einstein-Maxwell equations in a Kaluza-Klein like manner we can determine the components of the metric and vector potential which lie in the direction of the Killing vector fields. These components are determined by nine scalar fields each of which satisfy a partial differential equation in two variables. These equations take the form of an elliptic operator set equal to a nonlinear source. We find evidence that particular combinations of these fields satisfy Dirichlet boundary conditions, and are well suited to numerical solution using Green functions. Using this method we generate numerical solutions to the 4+1 Einstein-Maxwell equations corresponding to charged generalizations of the Myers-Perry solution. We also discover symmetry relations among the scalar equations which constrain their functional forms and posit the existence of two rigidity-theorem-like relations for electrovac spacetimes and sketch how their use generalizes our method to N+1 dimensions.

We present a method of quantum renormalization group, which makes it possible to construct a bulk theory for a general conformal field theory in the context of anti-de Sitter/conformal field theory correspondence. We demonstrate that within this method it is possible to construct scalar field theory in anti-de Sitter space. We also demonstrate that from a conformal field theory possessing global symmetry, it is possible to construct non-abelian gauge theory in anti-de Sitter space. / Thesis / Master of Science (MSc)

Made available in DSpace on 2016-12-12T20:15:53Z (GMT). No. of bitstreams: 1 A-inicio.pdf: 106467 bytes, checksum: 8ecfda3cfa81a009fae7a31b2dea68e8 (MD5) Previous issue date: 2008-10-31 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this Dissertation we study the scattering of massless monochromatic scalar waves by 5-dimensional (5D) Schwarzschild-Tangherlini black holes. We compute the scattering phase shifts using Prüfer method for high frequencies and high-values of the angular multipole index l. The phase shifts are then used to analyze the glory effect, in comparison with the respective 4D values. In the case of low frequencies th total absorption is completely absent in some cases. The oscillatory behavior of the phase shift for large values of l, typical in the 4D case, now becomes absent. For high frequencies, the total absorption starts at much lower values of l, compared to the 4D case. / Nesta Dissertação estudamos o espalhamento de ondas escalares monocromáticas sem massa por buracos negros de Schwarzschild-Tangherlini em 5 dimensões (5D). Computamos os desvios de fase usando o método de Prüfer para altas freqüências e altos valores do índice de multipolo l. Os desvios de fase são para análise do efeito glória, comparando com os respectivos valores resultantes do caso do espalhamento em 4D. Para baixas freqüências a absorção total em 5D é, em alguns casos, totalmente ausente para l + 0. O comportamento oscilatório do desvio de fase para grandes valores de l, típico em 4D, agora desaparece. Para altas freqüências a absorção total em 5D começa para valores de l muito menores que aqueles respectivos ao caso 4D.

In this work we study properties of the higher-dimensional generally rotating black hole space-time so-called Kerr-NUT-(A)dS and the related spaces with the same explicit and hidden symetries as the Kerr-NUT-(A)dS spacetime. First, we search commuta- tivity conditions for classical (charged) observables and their operator analogues, then we investigate a fulfilment of these conditions in the metioned spaces. We calculate the curvature of these spaces and solve the charged Hamilton-Jacobi and Klein-Gordon equations by the separation of the variables for an electromagnetic field, which pre- serves integrability of motion of a charged particle and mutual commutativity of the corresponding operators.