Search

In this thesis we look at the problem of nding and classifying stationary and biaxisymmetric solutions in ve-dimensional theories of gravity, using particular hidden symmetries. We consider three theories: the electrostatic sector of Einstein-Maxwell, vacuum gravity and minimal supergravity (Einstein-Maxwell gravity with a Chern-Simons term). For electrostatic solutions to Einstein-Maxwell theory, the equations on the metric and Maxwell eld possess a SL(2;R) symmetry. This allows one to derive transformations which either charge a solution or immerse it in an electric Melvin background. By considering a neutral static black lens seed and performing these two transformations with appropriately tuned transformation parameters, we construct the rst example of a regular black lens in Einstein Maxwell theory with topologically trivial asymptotics. For vacuum gravity we consider asymptotically at solutions. The vacuum Einstein equations are integrable in the sense that they can be reformulated as the integrability condition for an auxiliary linear system of PDEs. Taking these PDEs, one can integrate them over the event horizons, the axes of symmetry and in nity. By carefully considering continuity conditions between these solutions, one may actually solve for metric data on the horizons and the axes in terms of some geometrically de ned moduli, subject to a set of polynomial constraints. This represents a very useful tool for answering the existence problem, reducing it to the much more tractable question of whether a particular system of polynomials (subject to some inequalities) has any solutions. Using this polynomial system we provide a constructive uniqueness proof for the Kerr (using analogous four-dimensional results), Myers-Perry and black ring solutions. We also prove, through a combination of analytic and numerical methods, that the \simplest" L(n; 1) black lens cannot exist by showing that it must possess a conical singularity on one of the axes. Finally we consider the case of asymptotically at solutions in minimal supergravity. As with the vacuum, this is an integrable theory and so a similar analysis can be performed with exactly analogous results, although with rather more complicated polynomial systems determining the existence of solutions. A notable feature of minimal supergravity, not present in the vacuum theory, is the existence of regular solitons - in this context these are non-trivial solutions without black hole regions. We begin the exploration of the moduli space of these solitons by rst studying the case of at space.

Gravitational wave (GW) observations made with the LIGO and Virgo interferometers are transforming our understanding of the astrophysics of stellar origin compact objects (CO). The planned Laser Interferometer Space Antennae (LISA) will extend these observations to lower frequencies, thus higher mass systems and will have similar transformative potential. However, this potential can only be realised through the development of new waveform models and data analysis tools. This is the focus of this thesis. The primary GW source discussed in this work are Extreme mass-ratio inspirals (EMRIs), the inspiral of a small CO of mass $1M_{\odot} - 100M_{\odot}$ into a super-massive black hole (SMBH) with mass $10^{(4-7)}M_{\odot}$. Through analytical and numerical means, we develop tools that model circular and equatorial adiabatic inspiral-transition-plunge trajectories and resultant EMRI waveforms, regardless of the central SMBH spin. From a Fisher matrix (FM) approach we show analytically and verify numerically that constraints on the spin parameter from LISA observations of near-extremal EMRIs surpass those of moderately spinning EMRIs by $\sim$ 3 orders of magnitude. Using the Cutler-Vallisneri (CV) formalism that is based off the FM, we investigate accuracy requirements of low eccentricity Schwarzschild based EMRI waveforms. We determine what level of numerical precision the components of the gravitational-self-force must be computed in order to not significantly bias parameter estimates. The CV formalism is then extended to account for an arbitrary number of waveforms with modelling errors and an astrophysical foreground and detector noise. The formalism is illustrated with simplified, yet realistic, scenarios appropriate to third-generation ground-based and the space-based detectors. In the final piece of this work, we present the impact that data gaps have on parameter estimation in the context of LISA. Data gaps could potentially introduce correlations amongst noise components in the frequency domain. The likelihood, signal-to-noise ratio and FM are generalised to take these correlations into account. We conclude by presenting a case study, exemplifying the detrimental effect of mis-modelling the noise as a stationary process rather than the correct non-stationary process due to the presence of data gaps.

This thesis presents a framework in which to explore kinematical symmetries beyond the standard Lorentzian case. This framework consists of an algebraic classification, a geometric classification, and a derivation of the geometric properties required to define physical theories on the classified spacetime geometries. The work completed in substantiating this framework for kinematical, super-kinematical, and super-Bargmann symmetries constitutes the body of this thesis. To this end, the classification of kinematical Lie algebras in spatial dimension D = 3, as presented in [3,4], is reviewed; as is the classification of spatially-isotropic homogeneous spacetimes of [5]. The derivation of geometric properties such as the non-compactness of boosts, soldering forms and vielbeins, and the space of invariant affine connections is then presented. We move on to classify the N = 1 kinematical Lie superalgebras in three spatial dimensions, finding 43 isomorphism classes of Lie superalgebras. Once these algebras are determined, we classify the corresponding simply-connected homogeneous (4|4)-dimensional superspaces and show how the resulting 27 homogeneous superspaces may be related to one another via geometric limits. Finally, we turn our attention to generalised Bargmann superalgebras. In the present work, these will be the N = 1 and N = 2 super-extensions of the Bargmann and Newton-Hooke algebras, as well as the centrally-extended static kinematical Lie algebra, of which the former three all arise as deformations. Focussing solely on three spatial dimensions, we find 9 isomorphism classes in the N = 1 case, and we identify 22 branches of superalgebras in the N = 2 case.

Modern theoretical physics has benefited from a rapid growth in mathematical technology. In particular, technology developed in one field can be quickly adapted for use in another. Two key techniques developed for simplifying calculations of Feynman diagrams are spinor-helicity and the double copy. This thesis will discuss how they can be applied to general relativity. Spinor-helicity is used in particle physics to simplify expressions. In D > 4 this is done by observing that the residual symmetry of the little group is non-trivial. We adapt this technology to classify higher-dimensional spacetimes in the style of the D = 4 Petrov classification. Focusing on D = 5, our scheme naturally reproduces the full structure previously seen in both the CMPP and de Smet classifications, and resolves long-standing questions concerning their relationship. We review the exact classical double copy introduced for stationary Kerr-Schild spacetimes. We consider a time-dependent generalisation: the accelerating, radiating point particle. This Kerr-Schild solution has a non-trivial stress-energy tensor which we interpret as the radiative part of the field and find the corresponding single copy. Using Bremsstrahlung as an example, we determine a scattering amplitude describing the radiation which is consistent with the quantum double copy. This indicates a profound connection between exact classical solutions and the double copy. The double copy relates YM and gravity amplitudes through the observation that numerators of Feynman diagrams can be made to obey a Jacobi relation mirroring the colour charges. This additional structure can be adapted for use in classical perturbative calculations. The double copy maps to N = 0 supergravity requiring careful treatment of the dilaton. Using the Janis-Newman-Winicour family of naked singularities as an example we demonstrate how to construct spacetime metrics through a systematic perturbative expansion.

This thesis presents a classification of all asymptotically at, supersymmetric and biaxisymmetric (i.e. possessing a U(1)2-symmetry) soliton and black hole solutions to five-dimensional minimal supergravity. In particular, by combining local constraints from supersymmetry of the solutions with global constraints for stationary and bi-axisymmetric spacetimes, we show that any solution must be multi-centred with a Gibbons{Hawking base. We also find a refinement of the allowed horizon topologies of this class of solutions, to one of S3, S1 X S2 or a lens space L(p; 1). We construct the general, smooth solution associated with each possible rod structure, thereby finding a large moduli space of black hole spacetimes with noncontractible 2-cycles in the domain of outer communication. This includes examples for each of the allowed horizon topologies. In the absence of a black hole we obtain a classification of the known "bubbling" soliton spacetimes. We then move on to a systematic analysis of the subclass of three-centred solutions contained in the constructed moduli space, focusing on the special case of single black hole solutions. This class is composed of seven regular black hole solutions. We find that four of these can have the same conserved charges as the original spherical, supersymmetric black hole, the BMPV solution. These consist of a black lens with L(3; 1) horizon topology and three distinct families of spherical black holes with nontrivial topology outside the horizon. The former provides the first example of a nonspherical black hole with the same conserved charges as the BMPV black hole. Moreover, of these four solutions, three can have a greater entropy than the BMPV black hole near the BMPV upper spin bound. One of these is a previously known spherical black hole with nontrivial topology; the other two are new examples of a spherical black hole with nontrivial topology and an L(3; 1) black lens.

In this thesis we study the time evolution of Rényi and entanglement entropies of thermal states in Conformal Field Theory (CFT). These quantities are usually hard to compute but Ryu-Takayanagi (RT) and Hubeny-Rangamani-Takayanagi (HRT) proposals allow us to find the same quantities using calculations in general relativity. We will introduce main concepts of holography, quantum information and conformal field theory that will be used to derive the results of this thesis. In the first part of the thesis, we explicitly compute entanglement entropy of the rotating BTZ black hole by directly applying HRT proposal and finding lengths of spacelike geodesics. Rényi entropy of thermal state perturbed by a local quantum quench is computed by mapping correlators on two glued cylinders to the plane for field theory containing a single free boson and for 2d CFTs in the large c limit. We consider Thermofield Double State (TFD) which is an entangled state in direct product of two 2D CFTs. It is conjectured to be holographically equivalent to the eternal BTZ black hole. TFD state is perturbed by a local quench in one CFT and mutual information between two intervals in two CFTs is computed. We find when mutual information vanishes and interpret this as scrambling time, i.e. time scale required for the system to thermalize. This field theory result is modelled with a massive free falling particle in the BTZ black hole. We have computed the back-reaction of the particle on the metric of BTZ and used RT proposal to find holographic entanglement entropy. Finally, we generalize this calculation to the case of rotating BTZ with inner and outer horizons. It is dual to the CFT with different temperatures for left and right moving modes. We calculate mutual information and scrambling time and find exact agreement between results in the gravity and those in the CFT.

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.