In recent years, there has been an upsurge in interest in three-dimensional theories of gravity. In particular, two theories of massive gravity in three dimensions hold strong promise in the search for fully consistent theories of quantum gravity, an understanding of which will shed light on the problems of quantum gravity in four dimensions. One of these theories is the “old” third-order theory of topologically massive gravity (TMG) and the other one is a “new” fourth-order theory of massive gravity (NMG). Despite this increase in research activity, the problem of finding and classifying solutions of TMG and NMG remains a wide open area of research. In this thesis, we provide explicit new solutions of massive gravity in three dimensions and suggest future directions of research. These solutions belong to the Kundt class of spacetimes. A systematic analysis of the Kundt solutions with constant scalar polynomial curvature invariants provides a glimpse of the structure of the spaces of solutions of the two theories of massive gravity. We also find explicit solutions of topologically massive gravity whose scalar polynomial curvature invariants are not all constant, and these are the first such solutions. A number of properties of Kundt solutions of TMG and NMG, such as an identification of solutions which lie at the intersection of the full nonlinear and linearized theories, are also derived. / text

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.

In this thesis, Newman-Penrose techniques are used to obtain some new exact solutions to Einstein's field equations of general relativity and to assist in the physical interpretation of some exact radiative space-times. Attention is restricted to algebraically special space-times with a twist-free, repeated principal null congruence. In particular, the Robinson-Trautman type N solutions, which describe expanding gravitational waves, are investigated for all possible values of the cosmological constant A and the Gaussian curvature parameter E. The wave surfaces are always (hemi-)spherical, with successive surfaces displaced along time-like, space-like or null lines, depending on E. Explicit sandwich waves of this class are studied in Minkowski, de Sitter or anti-de Sitter backgrounds and a particular family of such solutions, which can be used to represent snapping or decaying cosmic strings, is considered in detail. The singularity and global structure of the solutions is also presented. In the remaining part of the thesis, the complete family of space-times with a non-expanding, shear-free, twist-free, geodesic principal null congruence (Kundt waves), that are of algebraic type III and for which the cosmological constant (Ac) is non-zero, is presented. The possible presence of an aligned pure radiation field is also assumed. These space-times generalise the known vacuum solutions of type N with arbitrary Ac and type III with Ac = O. It is shown that there are two, one and three distinct classes of solutions when Ac is respectively zero, positive and negative and, in these cases, the wave surfaces are plane, spherical or hyperboloidal in Minkowski, de Sitter or anti-de Sitter backgrounds respectively. The singularities which occur in these space-times are interpreted in terms of envelopes of these wave surfaces. Again, by considering functions of the retarded time which "cross-over" between canonical types, sandwich waves are also studied. The limiting cases of these, giving rise to shock or impulsive waves, are also considered.

In this work we study various aspects of the behaviour of free test particles in Einstein's general relativity and analyze specific physical properties of the background spacetimes. In the first part we investigate geodesic motions in the four-dimensional constant curvature spacetimes, i.e., Minkowski and (anti-)de Sitter universe, with an expanding impulsive gravitational wave. We derive the simple refraction formulae for particles crossing the impulse and describe the effect of nonvanishig cosmological constant. In the second part of this work we present a general method useful for geometrical and physical interpretation of arbitrary spacetimes in any dimension. It is based on the systematic analysis of the relative motion of free test particles. The equation of geodesic deviation is rewritten with respect to the natural orthonormal frame. We discuss the contributions given by a specific algebraic structure of the curvature tensor and the matter content of the universe. This formalism is subsequently used for investigation of the large class of nontwisting spacetimes. In particular, we analyse the motions in the nonexpanding Kundt and expanding Robinson--Trautman family of solutions.

El objetivo global del estudio que esta memoria de tesis presenta consiste en evaluar la viabilidad de aplicar nuevos materiales absorbentes del sonido que intentan, por un lado, ser una alternativa a los clásicamente utilizados (como las lanas minerales) y, por otro, ofrecer una solución a una de las problemáticas actuales más importantes, como es el dar salida a los productos de desecho de las industrias textiles mediante el reciclado. Para ello, se han estudiado algunos de los parámetros que permiten caracterizar el comportamiento acústico de los materiales absorbentes del sonido (tales como la impedancia acústica, el coeficiente de absorción acústica y la resistencia específica al flujo, entre otros). También se han descrito diversos métodos de medida experimentales utilizados para obtener estos parámetros acústicos. De los métodos presentados en este trabajo, se ha centrado la atención en los basados en el tubo de impedancia acústica (o tubo de Kundt). Esta técnica presenta ventajas, como la de requerir sólo un pequeño espacio en laboratorio así como probetas de los materiales a estudiar no demasiado grandes. El estudio se ha asentado sobre tres pilares fundamentales: modelización matemática, caracterización acústica de materiales y simulación numérica. En primer lugar, tras repasar los principales modelos y teorías utilizados en la evaluación acústica de los materiales absorbentes sonoros de tipo poroso y fibroso, se propone un nuevo modelo matemático y se demuestra su validez para el tipo de materiales estudiados. Con respecto a la caracterización acústica, se han realizado diversas campañas de mediciones con el fin de obtener el coeficiente de absorción acústica y la resistencia específica al flujo de diversos materiales. Por último se aplica, mediante un programa informático basado en el método de los elementos finitos, la técnica de la simulación numérica con el fin de contrastar los resultados obtenidos experimentalmente, así como para la evaluación de un / Juliá Sanchis, E. (2008). Modelización, simulación y caracterización acústica de materiales para su uso en acústica arquitectónica [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/2932 / Palancia

Within this work we have been interested in the frame approach to analysis of the field equations in the context of theories of gravity, in particular, the Einstein General Relativ- ity and Quadratic theory of gravity. As the starting point we have summarised the least action principle formulation of the General Relativity and introduced the Quadratic grav- ity extending the classic Einstein-Hilbert action by adding quadratic curvature terms. The Quadratic gravity field equation have been rewritten into the form separating the Ricci tensor contribution. As a next step we have reviewed the Newman-Penrose formal- ism on a purely geometrical level and discussed employing the field equations constraints. While in the case of General Relativity it is quite trivial, in the Quadratic gravity it be- comes much more involved, however, the General Relativity procedure can be followed even here. As an illustration, we have formulated the constraints on the gravitational field in the cases of the spherically symmetric spacetimes and so-called pp-waves both in the GR as well as Quadratic gravity. 1

In the review part of the thesis we summarize various modified theories of gravity, especially those that are characterized by additional curvature invariants in the Lagrangian density. Further, we review non-twisting geometries, especially their Kundt subclass. Finally, from the principle of least action we derive field equations for the case with the Lagrangian density corresponding to an arbitrary function of the curvature invariants. In the original part of the thesis we explicitly express particular components of the field equations for non-gyratonic Kundt geometry in generic quadratic gravity in arbitrary dimension. Then we discuss how this, in general fourth order, field equations restrict the Kundt metric in selected geome- trically privileged situations. We also analyse the special case of Gauss-Bonnet theory. 1