We present an extension of Geodesics in Lorentzian Manifolds (Semi-Riemannian Manifolds or pseudo-Riemannian Manifolds ). A geodesic on a Riemannian manifold is, locally, a length minimizing curve. On the other hand, geodesics in Lorentzian manifolds can be viewed as a distance between ``events''. They are no longer distance minimizing (instead, some are distance maximizing) and our goal is to illustrate over what time parameter geodesics in Lorentzian manifolds are defined. If all geodesics in timelike or spacelike or lightlike are defined for infinite time, then the manifold is called ``geodesically complete'', or simply, ``complete''. It is easy to show that the magnitude of a geodesic is constant, so one can characterize geodesics in terms of their causal character: if this magnitude is negative, the geodesic is called timelike. If this magnitude is positive, then it is spacelike. If this magnitude is 0, then it is called lightlike or null. Geodesic completeness can be considered by only considering one causal character to produce the notions of spacelike complete, timelike complete, and null or lightlike complete. We illustrate that some of the notions are inequivalent.

In this report, we give a deeper investigation of the loss-compensated recirculating delayed self-heterodyne interferometer (LC-RDSHI) for ultra-narrow linewidth measurement, including the theoretical analysis, experimental implementation, further modification on the system and more applications. Recently, less than 1kHz linewidth fiber lasers have been commercialized. But even the manufacturers face a challenge on accurately measuring the linewidth of such lasers. There is a need to develop more accurate methods to characterize ultra-narrow laser linewidth and frequency noises. Compared with other currently available linewidth measurement techniques, the loss-compensated recirculating delayed-heterodyne interferometer (LC-RDSHI) technique is the most promising one. It overcomes the bottle-neck of the high resolution requirement on the delayed self-heterodyne interferometer (DSHI) by using a short length of fiber delay line. This method does not need another narrower and more stable laser as the reference which is the necessary component in heterodyne detection. The laser spectral lineshape can be observed directly instead of complicated interpretation in frequency discriminator techniques. The theoretical analysis of a LC-RDSHI gives us a guidance on choosing the optimal parameters of the system and assists us to interpret the recorded spectral lineshape. Laser linewidth as narrow as 700Hz has been proved to be measurable by using the LC-RDSHI method. The non-linear curve fitting of Voigt lineshape to separate Lorentzian and Gaussian components was investigated. Voigt curve fitting results give us a clear view on laser frequency noises and laser linewidth nature. It is also shown that for a ultra-narrow linewidth laser, simply taking 20dB down from the maximum value of the beat spectrum and dividing by $2\sqrt{99}$ will over estimate the laser linewidth and coherent length. Besides laser linewidth measurement in the frequency domain, we also implemented time-domain frequency noise measurement by using a LC-RDSHI. The long fiber delay obtained by a fiber recirculating loop provides a higher resolution of frequency noise measurement. However, spectral width broadening due to fiber nonlinearity, environmental perturbations and laser intrinsic 1/f frequency noises are still potential problems in the LC-RDSHI method. A new method by adding a transmitter switch and a loop switch is proposed to minimize the Kerr effect caused by multiple recirculation. / Ph. D.

Un problema clásico en geometría lorentziana es la descripción de las inmersiones isométricas entre los espacios lorentzianos de curvatura constante. En este trabajo nos centramos en la clasificación de las inmersiones isométricas del plano lorentziano en el espacio anti-de Sitter tridimensional. Damos una fórmula de representación de estas inmersiones en términos de pares de curvas (con posibles singularidades) en el plano hiperbólico. Esto nos permite resolver los problemas propuestos por Dajczer y Nomizu en 1981. De entre todas las inmersiones isométricas del plano lorentziano en el espacio anti-de Sitter, algunas de ellas corresponden a toros lorentzianos (los ejemplos más sencillos son los toros de Hopf). Como aplicación de nuestra anterior descripción, probamos que todos estos toros pueden obtenerse a partir de dos curvas cerradas en el espacio hiperbólico. Finalmente, demostramos que los toros de Hopf son los únicos toros llanos lorentzianos inmersos en una amplia familia de sumersiones de Killing lorentzianas tridimensionales. / A classical problem in Lorentzian geometry is the description of the isometric immersions between Lorentzian spaces of constant curvature. We investigate the problem of classifying the isometric immersion from the Lorentz plane into the three-dimensional anti-de Sitter space, providing a representation formula of these isometric immersions in terms of pairs of curves (possibly with singularities) in the hyperbolic plane. We then give an answer to the open problems proposed by Dajczer and Nomizu in 1981. Among all isometric immersions of the Lorentz plane into the anti-de Sitter space, some of them are actually Lorentzian tori (the basic examples are the Hopf tori). As an application of our previous description, we prove that any such torus can be recovered from two closed curves in the hyperbolic plane. Finally, we prove that Lorentzian Hopf tori are the only immersed Lorentzian flat tori in a wide family of Lorentzian three-dimensional Killing submersions.

O Teorema de Representação de Weierstrass clássico, que faz uso da análise complexa para descrever uma superfície mínima imersa no espaço Euclidiano em termos de dados holomorfos, tem sido extremamente útil seja para construir novos exemplos de superfícies mínimas, seja para o estudo das propriedades destas superfícies. Em [24], usando a equação harmônica, os autores determinam uma fórmula de representação para superfícies mínimas, simplesmente conexas, imersas em uma variedade Riemanniana qualquer. Neste caso, a condição de holomorficidade dos dados de Weierstrass consiste em um sistema de equações diferenciais parciais com coeficientes não constantes. Logo, em geral, é complicado determinar soluções explícitas. No entanto, escolhendo adequadamente o espaço ambiente, tais equações se simplificam e a fórmula pode ser usada para produzir novos exemplos de imersões mínimas conformes. No espaço de Lorentz-Minkowski tridimensional uma fórmula de representação tipo-Weierstrass foi provada por Kobayashi, para o caso das imersões mínimas de tipo espaço (ver [18]), e por Konderak no caso das imersões mínimas de tipo tempo (ver [20]). Na demonstração destas fórmulas se utilizam as ferramentas da análise complexa e paracomplexa, respectivamente. Recentemente, em [22] os resultados de Kobayashi e Konderak foram generalizados para o caso de superfícies mínimas (de tipo espaço e de tipo tempo) imersas em 3-variedades Lorentzianas. Nesta dissertação estudaremos as fórmulas de representação de Weierstrass para superfícies mínimas imersas em variedades Riemannianas e Lorentzianas, que foram obtidas nos artigos [18], [20], [22] e [24]. / The classic Weierstrass Representation Theorem, which makes use of complex analysis to describe a minimal surface immersed in the Euclidean space in terms of holomorphic data, has been extremely useful either to construct new examples of minimal surfaces, rather than to study structural properties of these surfaces. In [24], using the standard harmonic equation, the authors determine a representation formula for simply connected immersed minimal surfaces in a Riemannian manifold. In this case, the holomorphicity condition of the Weierstrass data is a system of partial differential equations with nonconstant coefficients. Therefore, in geral, it is very difficult to determine explicit solutions. However, for particular ambient spaces, these equations become simpler and the formula can be used to produce new examples of conformal minimal immersions. In the three-dimensional Lorentz-Minkowski space a Weierstrass-type representation formula was proved by Kobayashi for spacelike minimal immersions (see [18]), and by Konderak for the case of timelike minimal immersions (see [20]). In the demonstration of these formulas are used the tools of complex and paracomplex analysis, respectively. Recently, in [22] the results of Kobayashi and Konderak were generalized to the case of (spacelike and timelike) minimal surfaces immersed in 3-Lorentzian manifolds. In this dissertation, we will study the Weierstrass representation formula for immersed minimal surfaces in Riemannian and Lorentzian manifolds, that was obtained in the articles [18], [20], [22] and [24].

O Teorema de Representação de Weierstrass clássico, que faz uso da análise complexa para descrever uma superfície mínima imersa no espaço Euclidiano em termos de dados holomorfos, tem sido extremamente útil seja para construir novos exemplos de superfícies mínimas, seja para o estudo das propriedades destas superfícies. Em [24], usando a equação harmônica, os autores determinam uma fórmula de representação para superfícies mínimas, simplesmente conexas, imersas em uma variedade Riemanniana qualquer. Neste caso, a condição de holomorficidade dos dados de Weierstrass consiste em um sistema de equações diferenciais parciais com coeficientes não constantes. Logo, em geral, é complicado determinar soluções explícitas. No entanto, escolhendo adequadamente o espaço ambiente, tais equações se simplificam e a fórmula pode ser usada para produzir novos exemplos de imersões mínimas conformes. No espaço de Lorentz-Minkowski tridimensional uma fórmula de representação tipo-Weierstrass foi provada por Kobayashi, para o caso das imersões mínimas de tipo espaço (ver [18]), e por Konderak no caso das imersões mínimas de tipo tempo (ver [20]). Na demonstração destas fórmulas se utilizam as ferramentas da análise complexa e paracomplexa, respectivamente. Recentemente, em [22] os resultados de Kobayashi e Konderak foram generalizados para o caso de superfícies mínimas (de tipo espaço e de tipo tempo) imersas em 3-variedades Lorentzianas. Nesta dissertação estudaremos as fórmulas de representação de Weierstrass para superfícies mínimas imersas em variedades Riemannianas e Lorentzianas, que foram obtidas nos artigos [18], [20], [22] e [24]. / The classic Weierstrass Representation Theorem, which makes use of complex analysis to describe a minimal surface immersed in the Euclidean space in terms of holomorphic data, has been extremely useful either to construct new examples of minimal surfaces, rather than to study structural properties of these surfaces. In [24], using the standard harmonic equation, the authors determine a representation formula for simply connected immersed minimal surfaces in a Riemannian manifold. In this case, the holomorphicity condition of the Weierstrass data is a system of partial differential equations with nonconstant coefficients. Therefore, in geral, it is very difficult to determine explicit solutions. However, for particular ambient spaces, these equations become simpler and the formula can be used to produce new examples of conformal minimal immersions. In the three-dimensional Lorentz-Minkowski space a Weierstrass-type representation formula was proved by Kobayashi for spacelike minimal immersions (see [18]), and by Konderak for the case of timelike minimal immersions (see [20]). In the demonstration of these formulas are used the tools of complex and paracomplex analysis, respectively. Recently, in [22] the results of Kobayashi and Konderak were generalized to the case of (spacelike and timelike) minimal surfaces immersed in 3-Lorentzian manifolds. In this dissertation, we will study the Weierstrass representation formula for immersed minimal surfaces in Riemannian and Lorentzian manifolds, that was obtained in the articles [18], [20], [22] and [24].

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, we address several problems related to the bosonic classical and algebraic quantum field theories in curved spacetime. In particular, the main question is: how do the theories change under finite global variations of the spacetime metric tensor? To answer this question a new deformation tool, the paracausal deformation, is developed and studied on its own as a new approach to investigate the structure of the space of globally hyperbolic metric tensors associated with a smooth manifold $\M$. Then the classical M{\o}ller maps are constructed to compare solutions of the hyperbolic PDEs defining the classical field theories and the quantum M{\o}ller $*$-isomorphisms follow to compare the CCR quantum algebras associated to the propagation of the quantum fields on the different background geometries. These maps possess the important property of preserving Hadamard states, providing a new way to implement the deformation argument used to prove the existence of such states in general globally hyperbolic spacetime. Moreover, the algebraic quantization of the Proca field, i.e the massive spin 1 field, on a general globally hyperbolic spacetime is for the first time studied in detail: by employing techniques coming from microlocal analysis and spectral theory a Hadamard state is constructed on ultrastatic spacetimes and then the M{\o}ller operator is used to prove the existence of such states in general globally hyperbolic spacetimes. A discussion about the definition of Hadamard states for the massive vector fields closes the work. The thesis is based on two works on algebraic quantization of bosonic field theories and Hadamard states: \cite{Norm}, \cite{Proca}. The papers are co-authored by my supervisor Prof. Valter Moretti (UniTN) and cosupervisor Simone Murro (UniGe). The first \cite{DefArg1} has not been included since, at the time it was written, the paracausal deformation, the construction of M{\o}ller operators, the right approach to intertwine the causal propagators and all the other tools developed in the subsequent works were still at a rough stage.

In 2020, Huh, Matherne, Mészáros, and St. Dizier established the Lorentzian property of normalized Schur polynomials and conjectured the Lorentzian nature of other Schur-type symmetric polynomials. More recently in 2022, Matherne, Morales, and Selover proved that chromatic symmetric functions of indifference graphs of abelian Dyck paths are Lorentzian. In this thesis, we study the more general class of Lorentzian polynomials that is also invariant under the standard permutation action on variables. Throughout this work, we give exposition to the classical theory of symmetric polynomials and Lorentzian polynomials. Then we present several fundamental results on symmetric Lorentzian polynomials and highlight potential avenues for future research. / År 2020 bevisade Huh-Matherne-Mészáros-St.Dizier att normaliserade schur polynom är lorentziska och antog att andra symmetriska polynom av Schur-typ också är det. År 2022 bevisade Matherne-Morales-Selover att kromatiska symmetriska funktioner för indifferensgrafer av abeliska Dyck-paths är lorentziska. Motiverade av dessa resultat studerar vi den mer allmänna klassen av lorentziska polynom som också är invarianta under standardpermutationsverkan på variabler. I avhandlingen ger vi några grundläggande resultat om symmetriska lorentziska polynom och pekar på möjliga framtida riktningar.

Dans cette thèse des actions conformes de cohomogénéité un sur l'univers d'Einstein tridimensionel sont classifiées. Notre stratégie est d'établir dans un premier temps quel peut être le groupe de transformations conformes impliqué, à conjugaison près. Nous décrivons aussi la topologie et la nature causale des orbites d'une telle action. / In this thesis, the conformal actions of cohomogeneity one on the three-dimensional Einstein universe are classified. Our strategy in this study is to determine the representation of the acting group in the group of conformal transformations of Einstein universe up to conjugacy. Also, we describe the topology and the causal character of the orbits induced by cohomogeneity one actions in Einstein universe.

Surface plasmon waves are transverse magnetic electromagnetic waves propagating along a dielectric-metal interface. These waves can be excited by resonant absorption of electromagnetic radiation leading to surface plasmon resonance (SPR) at the interface. The resonance is characterised by a reduction in the intensity of the reflected light at the interface due to strong coupling of incident optical radiation to surface plasmons. This gives rise to a minimum at a sharply defined angle of incidence, referred to as SPR angle or plasmon angle. The phenomenon of SPR has been extensively used in the past to develop reflective type optical devices for sensing applications on account of the high dielectric function dependent sensitivity of the SPR angle. Basically, devices which exhibit this phenomenon have a structure consisting of a metal film sandwiched between two dielectrics. The reflectivity of such a device is theoretically modelled based on either theory of thin films (Fresnel's model) or theory of resonance (Lorentzian model). These models have very effectively predicted the behaviour of such devices based on the shift in SPR angle due to the dielectric function variations. We have been investigating the SPR device for intensity based metrological applications utilising its high angular sensitive reflectivity, with fixed SPR angle. In these intensity based applications or measurements, direct and simple expressions connecting intensity variation to angular change are unavailable in the literature and quantitative estimation or data inversion is based on either curve fitting or iterative methods. Fresnel and Lorentzian models have commonly been used in the experiments but data inversion through the Fresnel model is computationally complex and the Lorentzian model, although less complicated, gives erroneous results due to its approximate nature. In order to obtain a simple expression between intensity variation and the angular change, we have re-looked at the two existing models in order to derive an expression which has the simplicity of the Lorentzian model and the accuracy of the Fresnel model in the experiments with fixed plasmon angles. These efforts have been particularly directed to understand the relationship between intensity variation and meteorologically important properties of such devices. This thesis is an attempt to summarize the computational results which have led us to some novel experimental methodologies which have been used to exploit these devices for inverse type, illumination specific, SPR based applications. The work presented in this thesis is organised in six chapters. Chapter 1, gives an overview of optical sensing, theory of surface plasmons, excitation schemes for surface plasmons, development of the SPR device and its characterisation. It also includes a brief literature review in the area of surface plasmon resonance, covering both the theoretical and experimental aspects. The objectives of the work and the scope of the thesis are also presented. Chapter 2 presents the existing models of SPR device, based on Fresnel's and the Lorentzian models. These models allow reflectance calculations from knowledge of either the optical parameters that describe the layers or the parameters of the waves that propagate through them. Using these models, the inverse problem of estimating either the angle of incidence or the optical constants of the layers of the sensors utilizing the intensity based measurements is generally difficult. In order to solve this problem where the plasmon angles are fixed, a modified formalism for the angle scanned SPR spectrum of a three-layered SPR sensor is presented in this chapter. The new formalism regroups the wave vector parameters of Lorentzian resonance theory into a set of non-dimensional parameters 1, 4K and R. The new reflectivity index (1), which is the ratio of reflectance to the absorptance, has been introduced to help explain the physical processes underlying the device operation in the high sensitivity region of the characteristics. The parameter 4Kis a constant of the device and it depends on the dielectric constants of the device. This is a new SPR index and is identified at a point where reflectance and absorptance match. Parameter R is related to the loss mechanisms in the device and will be explained in detail in Chapter 3. This simple model links the new reflectivity index (1) to the angular detune from SPR angle (ΔƟ) and it brings out a parabolic variation of ΔƟ with 1. In this chapter the mathematical derivation of the proposed model is presented and the significance of the new parameters 1, 4Kand Rare discussed. Chapter 3 evaluates the characteristic nature of errors associated with the predictions from the proposed model and presents methods for neutralizing them. It is demonstrated with the help of the function K which is linearly dependant on 1, that the proposed model predicts the reflectance from the wave vector parameters as accurately as the Fresnel's model. This R parameter explains the slowly varying nature of the radiative loss with the angle of incidence and this variation contributes significantly to the SPR characteristics. As a consequence, it is found that the SPR characteristics can be represented as a sum of two primary functions which are parabolic and linear, respectively, and this leads to the easy explanation of the SPR characteristics. The present chapter also discusses a new observation that the angle-scanned SPR spectrum can be accurately described using a straight line in intercept form. The intercept value depends on 4Kand the slope depends on K. In addition to this, this chapter discusses practical methods for estimation of the intercept and the slope of such a straight line which are functions of the key wave vector parameters. A detailed discussion on the proposed model highlighting its advantages for inverse type, illumination specific, SPR-based applications with fixed SPR angle is also presented. Chapter 4 describes the applications of the proposed model for optical constant measurements. The first part highlights a new approach for the determination of the dielectric constants of the metal film used for the optimised- or nearly-optimised SPR sensors using the proposed model. In the complex dielectric constant, the real part is calculated from the SPR angle and the imaginary part from 4K. A discussion on the dielectric constant study of silver and gold metal film is presented. The advantages of the proposed approach such as its simplicity and direct methodology are then discussed. The second part of the chapter also proposes a new approach to carry out measurements on the absorbance of the medium with enhanced sensitivity utilising the parameter 4K It describes a computational study on the variation of 4K values with the dielectric function and highlights the relationship of 4K variation due to the imaginary part of the dielectric function (absorption) of the samples. The physical processes causing a change in the value of 4Kdue to absorption is also discussed along with some computational results. Chapter 5 reports the study carried out to bring out the importance of the new index,4K in metrological applications. Based on the new model, the effect of the laser beam divergence on SPR curve is studied. This chapter first of all discusses the design of the SPR device and the new methods for the development and characterisation of such a device. Details of the experimental procedure for laser divergence evaluation are proposed along with some of the significant computational results. Furthermore, a few applications such as focal length measurement of optical lenses, micro-displacement measurement based on the divergence of the laser beam are also reported. Since the SPR characteristics can be represented easily using the new model, the angular dependent intensity variation can be utilised for some metrological applications with simple data processing. In this context, the high angular sensitivity of the SPR device is studied and some applications such as micro-displacement measurement, pressure measurement and optical wedge angle measurement are included to highlight the above advantages. The last chapter, Chapter 6, gives a summary and conclusions of the work presented in the thesis. The scope for future investigations is also included in this chapter.