This thesis is mainly focused on the equivalence problem for a subclass of Lorentzian manifolds: the degenerate Kundt spacetimes. These spacetimes are not defined uniquely by their scalar curvature invariants. To prove two metrics are diffeomorphic, one must apply Cartan's equivalence algorithm, which is a non-trivial task: in four dimensions Karlhede has adapted the algorithm to the formalism of General Relativity and significant effort has been spent applying this algorithm to particular subcases. No work has been done on the higher dimensional case. First, we study the existence of a non-spacelike symmetry in two well-known subclasses of the N dimensional degenerate Kundt spacetimes: those spacetimes with constant scalar curvature invariants (CSI) and those admitting a covariant constant null vector (CCNV). We classify the CSI and CCNV spacetimes in terms of the form of the Killing vector giving constraints for the metric functions in each case. For the rest of the thesis we fix N=4 and study a subclass of the CSI spacetimes: the CSI-? spacetimes, in which all scalar curvature invariants vanish except those constructed from the cosmological constant. We produce an invariant characterization of all CSI-? spacetimes. The Petrov type N solutions have been classified using two scalar invariants. However, this classification is incomplete: given two plane-fronted gravitational waves in which both pairs of invariants are similar, one cannot prove the two metrics are equivalent. Even in this relatively simple subclass, the Karlhede algorithm is non-trivial to implement. We apply the Karlhede algorithm to the collection of vacuum Type N VSI (CSI-?, ? = 0) spacetimes consisting of the vacuum PP-wave and vacuum Kundt wave spacetimes. We show that the upper-bound needed to classify any Type N vacuum VSI metric is four. In the case of the vacuum PP-waves we have proven that the upper-bound is sharp, while in the case of the Kundt waves we have lowered the upper-bound from five to four. We also produce a suite of invariants that characterize each set of non-equivalent metrics in this collection. As an application we show how these invariants may be related to the physical interpretation of the vacuum plane wave spacetimes.

The existence of Gravitational Waves from binary black holes is one of the most interesting predictions of General Relativity. These ripples in space-time should be visible to ground-based gravitational wave detectors worldwide in the next few years. One such detector, the Laser Interferometer Gravitational-wave Observatory (LIGO) is in the process of being upgraded to its Advanced sensitivity which should make gravitational wave detections routine. Even so, the signals that LIGO will detect will be faint compared to the detector noise, and so accurate waveform templates are crucial. In this thesis, we present a detailed analysis of the accuracy of hybrid gravitational waveforms. Hybrids are created by stitching a long post-Newtonian inspiral to the late inspiral, merger, and ringdown produced by numerical relativity simulations. We begin our investigation with a study of the systematic errors in the numerical waveform, and errors due to hybridization and choice of detector noise. For current NR waveforms, the largest source of error comes from the unknown high-order terms in the post-Newtonian waveform, which we first explore for equal-mass, non-spinning binaries, and also for unequal-mass, non-spinning binaries. We then consider the potential reduction in hybrid errors if these higher-order terms were known. Finally, we investigate the possibility of using hybrid waveforms as a detection template bank and integrating NR+PN hybrids into the LIGO detection pipeline.

This thesis examines the role of shear in inhomogeneous spherically symmetric spacetimes in the field of general relativity. The Einstein field equations are derived for a perfect fluid source in comoving coordinates. By assuming a barotropic equation of state, two classes of nonaccelerating solutions are obtained for the Einstein field equations. The first class has equation of state p = ⅓µ and the second class, with equation of state p = µ, generalises the models of Van den Bergh and Wils (1985). For a particular choice of a metric potential a new class of solutions is found which is expressible in terms of elliptic functions of the first and third kind in general. A class of nonexpanding cosmological models is briefly studied. The method of Lie symmetries of differential equations generates a self-similar variable which reduces the field and conservation equations to a system of ordinary differential equations. The behaviour of the gravitational field in this case is governed by a Riccati equation which is solved in general. Another class of solutions is obtained by making an ad hoc choice for one of the gravitational potentials. It is demonstrated that for a stiff fluid a particular case of the generalised Emden-Fowler equation arises. / Thesis (Ph.D.)-University of Natal, Durban, 1996.

The study of exact solutions to the Einstein and Einstein-Maxwell field equations, by imposing a symmetry requirement on the manifold, has been the subject of much recent research. In this thesis we consider specifically conformal symmetries in static and nonstatic spherically symmetric spacetimes. We find conformally invariant solutions, for spherically symmetric vectors, to the Einstein-Maxwell field equations for static spacetimes. These solutions generalise results found previously and have the advantage of being regular in the interior of the sphere. The general solution to the conformal Killing vector equation for static spherically symmetric spacetimes is found. This solution is subject to integrability conditions that place restrictions on the metric functions. From the general solution we regain the special cases of Killing vectors, homothetic vectors and spherically symmetric vectors with a static conformal factor. Inheriting conformal vectors in static spacetimes are also identified. We find a new class of accelerating, expanding and shearing cosmological solutions in nonstatic spherically symmetric spacetimes. These solutions satisfy an equation of state which is a generalisation of the stiff equation of state. We also show that this solution admits a conformal Killing vector which is explicitly obtained. / Thesis (Ph.D.)-University of Natal, Durban, 1993.

In this thesis we study the physical properties of the manifold in general relativity that admits a conformal motion. The results obtained are general as the metric tensor field is not specified. We obtain the Lie derivative along a conformal Killing vector of the kinematical and dynamical quantities for the general energy-momentum tensor of neutral matter. Equations obtained previously are regained as special cases from our results. We also find the Lie derivative of the energy-momentum tensor for the electromagnetic field. In particular we comprehensively study conformal symmetries in the Bianchi I spacetime. The conformal Killing vector equation is integrated to obtain the general conformal Killing vector and the conformal factor subject to integrability conditions. These conditions place restrictions on the metric functions. A particular solution is exhibited which demonstrates that these conditions have a nonempty solution set. The solution obtained is a generalisation of the results of Moodley (1991) who considered locally rotationally symmetric spacetimes. The Killing vectors are regained as special cases of the conformal solution. There do not exist any proper special conformal Killing vectors in the Bianchi I spacetime. The homothetic vector is found for a nonvanishing constant conformal factor. We establish that the vacuum Kasner solution is the only Bianchi I spacetime that admits a homothetic vector. Furthermore we isolate a class of vectors from the solution which causes the Bianchi I model to degenerate into a spacetime of higher symmetry. / Thesis (M.Sc.)-University of KwaZulu-Natal, 1992.

In this dissertation we study conformal symmetries in the Stephani universe which is a generalisation of the Robertson-Walker models. The kinematics and dynamics of the Stephani universe are discussed. The conformal Killing vector equation for the Stephani metric is integrated to obtain the general solution subject to integrability conditions that restrict the metric functions. Explicit forms are obtained for the conformal Killing vector as well as the conformal factor . There are three categories of solution. The solution may be categorized in terms of the metric functions k and R. As the case kR - kR = 0 is the most complicated, we provide all the details of the integration procedure. We write the solution in compact vector notation. As the case k = 0 is simple, we only state the solution without any details. In this case we exhibit a conformal Killing vector normal to hypersurfaces t = constant which is an analogue of a vector in the k = 0 Robertson-Walker spacetimes. The above two cases contain the conformal Killing vectors of Robertson-Walker spacetimes. For the last case in - kR = 0, k =I 0 we provide an outline of the integration process. This case gives conformal Killing vectors which do not reduce to those of RobertsonWalker spacetimes. A number of the calculations performed in finding the solution of the conformal Killing vector equation are extremely difficult to analyse by hand. We therefore utilise the symbolic manipulation capabilities of Mathematica (Ver 2.0) (Wolfram 1991) to assist with calculations. / Thesis (M.Sc.)-University of Natal, Durban, 1992.

Some branches of quantum gravity demand the existence of higher dimensions and the addition of higher curvature terms to the gravitational Lagrangian in the form of the Lovelock polynomials. In this thesis we investigate some of the classical properties of Lovelock gravity. We first derive the Hamiltonian for Lovelock gravity and find that it takes the same form as in general relativity when written in terms of the Misner-Sharp mass function. We then minimally couple the action to matter fields to find Hamilton’s equations of motion. These are gauge fixed to be in the Painleve-Gullstrand co–ordinates and are well suited to numerical studies of black hole formation. We then use these equations of motion for the massless scalar field to study the formation of general relativistic black holes in four to eight dimensions and Einstein-Gauss-Bonnet black holes in five and six dimensions. We study Choptuik scaling, a phenomenon which relates the initial conditions of a matter distribution to the final observables of small black holes. In both higher dimensional general relativity and Einstein-Gauss-Bonnet gravity we confirm the existence of cusps in the mass scaling relation which had previously only been observed in four dimensional general relativity. In the general relativistic case we then calculate the critical exponents for four to eight dimensions and find agreement with previous calculations by Bland et al but not Sorkin et al who both worked in null co–ordinates. For the Einstein-Gauss-Bonnet case we find that the self-similar behaviour seen in the general relativistic case is destroyed. We find that it is replaced by some other form of scaling structure. In five dimensions we find that the period of the critical solution at the origin is proportional to roughly the cube root of the Gauss-Bonnet parameter and that there is evidence for a minimum black hole radius. In six dimensions we see evidence for a new type of scaling. We also show, from the equations of motion, that there is reason to expect qualitative differences between five and higher dimensions.

In this thesis we consider spherically symmetric cosmological models when the shear is nonzero and also cases when the shear is vanishing. We investigate the role of the Emden-Fowler equation which governs the behaviour of the gravitational field. The Einstein field equations are derived in comoving coordinates for a spherically symmetric line element and a perfect fluid source for charged and uncharged matter. It is possible to reduce the system of field equations under different assumptions to the solution of a particular Emden-Fowler equation. The situations in which the Emden-Fowler equation arises are identified and studied. We analyse the Emden-Fowler equation via the method of Lie point symmetries. The conditions under which this equation is reduced to quadratures are obtained. The Lie analysis is applied to the particular models of Herlt (1996), Govender (1996) and Maharaj et al (1996) and the role of the Emden-Fowler equation is highlighted. We establish the uniqueness of the solutions of Maharaj et al (1996). Some physical features of the Einstein-Maxwell system are noted which distinguishes charged solutions. A charged analogue of the Maharaj et al (1993) spherically symmetric solution is obtained. The Gutman-Bespal'ko (1967) solution is recovered as a special case within this class of solutions by fixing the parameters and setting the charge to zero. It is also demonstrated that, under the assumptions of vanishing acceleration and proper charge density, the Emden-Fowler equation arises as a governing equation in charged spherically symmetric models. / Thesis (M.Sc.)-University of Natal, Durban, 1998.

The construction of the James Webb Space Telescope has brought attention to infrared astronomy and cosmology. The potential information about our universe to be gained by this mission and future infrared telescopes is staggering, but infrared observation faces many obstacles. These telescopes face large amounts of noise by many phenomena, from emission off of the mirrors to the cosmic infrared background. Infrared telescopes need to be designed in such a way that noise is minimized to achieve sufficient signal to noise ratio on high redshift objects. We will investigate current and planned space and ground based telescopes, model the noise they encounter, and discover their limitations. The ultimate goal of our investigation is to compare the sensitivity of these missions in the near and mid IR and to propose new missions. Our investigation is broken down into four major sections: current missions, noise, signal, and proposed missions. In the proposed missions section we investigate historical and current infrared telescopes with attention given to their location and properties. The noise section discusses the noise that an infrared telescope will encounter and set the background limit. The signal section will look at the spectral energy distributions (SED) of a few significant objects in our universe. We will calculate the intensity of the objects at various points on Earth and in orbit. In the final section we use our findings in the signal and noise sections to model integration times (observation time) for a variety of missions to achieve a given signal to noise ratio (SNR).

This thesis is about the structure of the boundary of the universe, i.e., points where the geometric structures of spacetime cannot be continued. In partic­ular, we study the structure of the b-boundary by B. Schmidt. It has been known for some time that the b-boundary construction has several drawbacks, perhaps the most severe being that it is often not Hausdorff separated from interior points in spacetime. In other words, the topology makes it impossible to distinguish which points in spacetime are near the singularity and which points are ‘far’ from it. The non-Hausdorffness of the b-completion is closely related to the concept of fibre degeneracy of the fibre in the frame bundle over a b-boundary point, the fibre being smaller than the whole structure group in a specific sense. Fibre degeneracy is to be expected for many realistic spacetimes, as was proved by C. J. S. Clarke. His proofs contain some errors however, and the purpose of paper I is to reestablish the results of Clarke, under somewhat different conditions. It is found that under some conditions on the Riemann curvature tensor, the boundary fibre must be totally degenerate (i.e., a single point). The conditions are essentially that the components of the Riemann tensor and its first derivative, expressed in a parallel frame along a curve ending at the singularity, diverge sufficiently fast. We also demonstrate the applicability of the conditions by verifying them for a number of well known spacetimes. In paper II we take a different view of the b-boundary and the b-length func­tional, and study the Riemannian geometry of the frame bundle. We calculate the curvature Rof the frame bundle, which allows us to draw two conclusions. Firstly, if some component of the curvature of spacetime diverges along a horizontal curve ending at a singularity, R must tend to — oo. Secondly, if the frame bundle is extendible through a totally degenerate boundary fibre, the spacetime must be a conformally flat Einstein space asymptotically at the corresponding b-boundary point. We also obtain some basic results on the isometries and the geodesics of the frame bundle, in relation to the corresponding structures on spacetime. The first part of paper III is concerned with imprisoned curves. In Lorentzian geometry, the situation is qualitatively different from Riemannian geometry in that there may be incomplete endless curves totally or partially imprisoned in a compact subset of spacetime. It was shown by B. Schmidt that a totally imprisoned curve must have a null geodesic cluster curve. We generalise this result to partially im­prisoned incomplete endless curves. We also show that the conditions for the fibre degeneracy theorem in paper I does not apply to imprisoned curves. The second part of paper III is concerned with the properties of the b-length functional. The b-length concept is important in general relativity because the presence of endless curves with finite b-length is usually taken as the definition of a singular spacetime. It is also closely related to the b-boundary definition. We study the structure of b-neighbourhoods, i.e., the set of points reachable from a fixed point in spacetime on (horizontal) curves with b-length less than some fixed number e > 0. This can then be used to understand how the geometry of spacetime is encoded in the frame bundle geometry, and as a tool when studying the structure of the b-boundary. We also give a result linking the b-length of a general curve in the frame bundle with the b-length of the corresponding horizontal curve. / digitalisering@umu