Thèses sur le sujet « Post Newtoniano »

La stima dei parametri fisici degli eventi di onde gravitazionali rilevati da LIGO e Virgo si basa su modelli di forme d'onda analitiche, eventualmente calibrati da simulazioni di Relatività Numerica. Il modello EOB (Effective-One-Body) è uno dei principali modelli analitici per l’analisi dei segnali gravitazionali emessi da buchi neri e stelle di neutroni. La qui presente tesi, a grandi linee, è un lavoro di “potenziamento” di questo modello, e in particolare I) lo miglioriamo nei suoi vari settori, con l'obiettivo finale di costruire un modello che includa tutte le informazioni fisiche disponibili: in particolare, quelle riguardanti i multipoli subdominanti, utili per ottimizzare la risoluzione angolare dell'analisi dati degli esperimenti di onde gravitazionali [ 1, 4, 5, 7]; II) usarlo per sviluppare approssimanti Post-Newtoniani veloci, di ordine elevato, molto utili per le analisi bayesiane inerenti le pipeline LIGO e Virgo [3]; III) utilizzarlo per studiare, nel caso delle stelle binarie di neutroni, l’influenza degli effetti di self-spin sulla loro forma d'onda [2]. Uno degli elementi costitutivi centrali del modello EOB è la forma d'onda multipolare Post-Newtoniana (PN) fattorizzata e circolarizzata introdotta in [8] per binarie senza spin. In[4], estendiamo fino a l= 6 (cioè a multipoli alti) l'approccio di Nagar e Shah [9], poiché ha un accordo con la relatività numerica (utilizzata come punto di riferimento per la calibrazione) migliore del suo precursore [8]. In [4], questo approccio è stato aggiornato al caso di una particella rotante intorno ad un buco nero di Schwartzschild. Lo stesso approccio è stato altresì adattato in [5] al fine di aggiornare il modello EOB quadrupolare TEOBResumS, un codice C ++ [1] disponibile nella LIGO Advanced Library (LAL) e citato nel catalogo [10], a una sua naturale versione multipolare [7]. Fatto salvo ciò, mediante la tecnica di espansione EOB-PN definita in [11], il lavoro svolto in [3] conduce a un approssimante fenomenologico di ordine 5.5PN veloce e preciso che, includendo più informazioni sulle correzioni test-particle rispetto a quello standard a 3.5PN, ottimizza la stima dei parametri di marea dell'analisi dei dati BNS. Nel lavoro [2], incorporiamo in TEOBResumS i termini di self-spin dipendenti dall’equazione di stato (EOS) all'ordine next-to-next-to-leading (NNLO), insieme ad altri effetti (bilineari, cubici e quartici) al leading-order (LO). Qui, con la stessa cassetta degli attrezzi usata in[3], studiamo la dipendenza degli effetti di self spin legati all’EOS, e dimostriamo che le correzioni di ordine NLO e NNLO, a livello di interazione monopolo-quadrupolo, producono effetti di accelerazione di fase più marcati rispetto al corrispondente contributo LO; oltre a questo, si osserva altresì che una volta inclusi gli effetti di auto spin al NLO (3PN) nel Taylor F2, essi sono più attrattivi rispetto a quanto previsto dalla corrispondente descrizione EOB. Infine, abbiamo ottenuto un approssimante TaylorF2 che fornisce una rappresentazione analiticamente semplificata, ma fedele all'EOB, degli effetti di self-spin (effetti di coda inclusi) che può essere utile per migliorare gli attuali modelli di forma d'onda PN (o fenomenologici) per l’inspiralling di stelle di neutroni. Referenze: References: [1] Phys. Rev., D98(10):104052, 2018. [2] Phys. Rev., D99:044007, 2019. [3] Phys. Rev., D99:124051, 2019. [4] Phys. Rev., D97(8): 084016, 2018. [5] Phys. Rev., D100(10):104056, 2019. [6] Phys. Rev., D99(4):044051, 2019. [7] Phys. Rev. D 102, 024077 (2020) [8]Phys. Rev., D79:064004, 2009. [9] Phys. Rev., D94(10):104017, 2016. [10] Phys. Rev. X 9, 031040 [11] Phys. Rev., D95(12):124001, 2017.<br>The parameter estimation of gravitational wave events detected by LIGO and Virgo relies on analytical waveforms models, possibly calibrated (or informed) by Numerical Relativity simulations. The effective-one-body (EOB) model is one of the main analytical models available that can be efficiently used for analyzing both black hole and neutron star binaries. In this script we I) improve it in its various sectors, with the final aim to build a model that includes all the physical information available: in particular, the higher subdominant multipoles information, that is useful to optimize GW data analysis’ angular resolution [1, 4, 5, 7]; II) use it to develop high-order fast PN approximants for Bayesian analysis in LIGO and Virgo pipelines [3]; III) use it to study the self-spin effects of binary Neutron Stars on their own waveform [2]. One of the central building blocks of the EOB model is the factorized and resummed (circularized) multipolar post-Newtonian (PN) waveform introduced in Ref. [8] for nonspinning binaries. In Ref. [4], we extend up to ` = 6 (i.e. to high multipoles) the resummation approach of Nagar and Shah [9], since it has a better analytical/numerical relativity agreement than its precursor [8]. Ref. [4], updated to the case of a spinning particle of Schwartzschild problem in Ref. [5], has been used in order to update the spin-aligned, quadrupolar EOB model TEOBResumS, a C++ code [1] available in the LIGO Advanced Library (LAL) and cited in the GW catalogue [10], to a multipolar version [7]. Therefore, following the EOB-PN expansion technique defined in [11], Ref. [3] leads to a fast and accurate 5.5PN phenomenological approximant that, by including more point-mass information than the standard 3.5PN one, optimizes the tidal-parameter estimation of BNS data analysis. In Ref [2], we incorporate the EOS-dependent selfspin terms in TEOBResumS at next-to-next-to-leading (NNLO) order, together with other (bilinear, cubic and quartic) nonlinear-in-spin effects (at leading order, LO). Here, with the same toolbox used in Ref. [3], we study the EOS dependence of the self-spin effects and show that the next-to-leading order (NLO) and NNLO monopole-quadrupole corrections yield increasingly phase-accelerating effects compared to the corresponding LO contribution; that the standard TaylorF2 post-Newtonian (PN) treatment of NLO (3PN) EOS-dependent self-spin effects makes their action stronger than the corresponding EOB description; and, finally, we obtain a tail-augmented TaylorF2 approximant that yields an analytically simplified, EOB-faithful, representation of the EOS-dependent self-spin phasing that can be useful to improve current PN-based (or phenomenological) waveform models for inspiralling neutron star binaries. References: [1] Phys. Rev., D98(10):104052, 2018. [2] Phys. Rev., D99:044007, 2019. [3] Phys. Rev., D99:124051, 2019. [4] Phys. Rev., D97(8): 084016, 2018. [5] Phys. Rev., D100(10):104056, 2019. [6] Phys. Rev., D99(4):044051, 2019. [7] Phys. Rev. D 102, 024077 (2020) [8]Phys. Rev., D79:064004, 2009. [9] Phys. Rev., D94(10):104017, 2016. [10] Phys. Rev. X 9, 031040 [11] Phys. Rev., D95(12):124001, 2017.

The post-Newtonian (PN) perturbative framework has been successful in understanding the slow-motion, weak fi eld limit of Einstein's theory of gravity on solar system scales, and for isolated astrophysical systems. The parameterized post-Newtonian (PPN) formalism extended the PN framework to put very tight constraints on deviations from Einstein's theory on the aforementioned scales and systems. In this work, we extended and applied the post-Newtonian formalism to cosmological scales. We fi rst used it to construct a cosmological model to understand the effect of regularly arranged point sources on the background expansion. Here we found that at higher orders we obtained a small radiation-like correction to the standard Friedmann-Lemaitre-Robertson-Walker (FLRW) equations, for a matter-dominated universe. This radiation-like correction was purely due to the inhomogeneity of our model, and the non-linearity of Einstein's eld equations. We also extended the post-Newtonian formalism to include other forms of matter that are cosmologically relevant, such as radiation and a cosmological constant, and studied the non-linear effects they might have on the background expansion. Then we constructed an extension of the parameterized post-Newtonian formalism (PPN) to cosmological scales. We used it to parameterize the background expansion of the universe as well as rst-order perturbations in cosmology, using four functions of time. In the future, this could allow us to put constraints on deviations from Einstein's theory of gravity on cosmological scales. We gave examples of how our parameterization would work for dark energy models and scalar-tensor and vector-tensor theories of gravity. In the nal part of this work, we studied how light propagation behaves in an inhomogeneous post-Newtonian cosmology with matter and a cosmological constant. We used it to understand the effect that inhomogeneities would have on observables such as angular diameter distances as compared to those that are expected from a homogeneous and isotropic FLRW universe.

A method for obtaining a smaller bound for the post-Newtonian parameter alpha 2 is outlined. alpha2 is a parameter in the post-Newtonian formalism associated with violation of Lorentz invariance. The current bound found by Nordvedt, alpha2 < 1.2 x 10 -7, was obtained using solar system data. We show that, by analysing millisecond pulsar profiles and limiting the amount of secular evolution, a smaller bound can be obtained.

This thesis is devoted to the post-Newtonian (PN) approximation of General Relativity in Cosmology. We analysed the procedure for passing from GR to Newtonian theory in both the Eulerian and the Lagrangian approaches to gravitational dynamics and also the connection between the two. In the GR framework we provided the transformation equations from the Poisson gauge to the synchronous and comoving gauge for the metric in the Newtonian approximation, being completely non-linear in the standard perturbation theory. We fully recovered equations and solution for the metric both in the relativistic perturbation theory up to second order and in the Newtonian limit at any order. We then extend our transformation to the PN approximation and our results are consistent with GR up to second order in perturbation theory.  We specialise the PN Lagrangian dynamics to globally plane-parallel configurations. We obtained the analytical solution of the Einstein field equations. The solution is non-perturbative in the standard sense, exact up to PN order and extends the Zel’dovich approximation which, in turn, is exact for non-linear plane-parallel dynamics in Newtonian gravity. An application of our solution in the context of the back-reaction proposal is eventually given, providing a PN estimation of kinematical back-reaction, mean spatial curvature, average scale-factor and expansion rate.

The Post-Newtonian limit of a fourth-order metric theory of gravity is discussed, based on the most general quadratic Lagrangian. This approach involves the use of the perturbation expansion of the metric tensor, extended up to the fourth-order and expressed in terms of three gravitational potentials. The analysis concerns the search and the resolution, where possible, of systems of coupled fourth-order differential equations, obtained by varying the conditions placed on the coupling constants appearing in the Lagrangian. The solutions are computed using the Green’s function method. Finally, the gravitational potentials thus obtained are compared with the Newtonian one derived in General Relativity in the weak-field and low-veocity limit.

La détection ainsi que l’analyse des ondes gravitationnelles émises par les systèmes binaires d’objets compacts reposent sur notre capacité à faire des prédictions précises au sein de la théorie de la relativité générale. Dans cette thèse, nous utilisons la théorie post-newtonienne (PN), et en particulier le formalisme connu sous le nom de Blanchet-Damour-Iyer, afin d’étudier de tels systèmes. La finalité des différents calculs réalisés au sein de cette thèse est d’obtenir la phase du signal gravitationnel à l’ordre 4,5PN, et les résultats que nous présentons nous rapprochent fortement de cet objectif. Tout d’abord, nous calculons les sillages d’ondes à l’ordre 3 dans le champ radiatif, ce qui nous permet d’obtenir le coefficient 4,5PN du flux d’énergie émis par des systèmes binaires compacts sans spin dans le cas d’orbites circulaires. Puis, nous calculons la dernière ambiguïté apparaissant dans les équations du mouvement de deux corps compacts sans spin à l’ordre 4PN, ce qui nous permet d’obtenir la première dérivation à partir de principes fondamentaux de ce résultat. Nous étudions alors en détail les différentes quantités conservées générées par cette dynamique. Enfin,nous présentons un premier résultat préliminaire du quadrupôle de masse source à l’ordre 4PN, ce qui constitue l’une des étapes cruciales dans l’obtention de la phase à l’ordre 4.5PN<br>The detection and the analysis of gravitational waves emitted by compact binary systems rely on our ability to make accurate predictions within general relativity. In this thesis, we use the post-Newtonian (PN) formalism, and in particular the Blanchet-Damour-Iyer framework, to study the dynamics and the emission of gravitational waves of such systems. The different computations that we performed are motivated by our aim to obtain the phase of the gravitational wave signal at the 4.5PN order. In that regard, crucial steps have been achieved within this thesis. First of all, we compute the third-order tail effects in the radiation field, yielding the 4.5PN coefficient of the energy flux for binaries of non-spinning objects in circular orbits. Besides, we determine the remaining ambiguity of the 4PN Lagrangian of two spinless compact bodies. This result completes the first derivation from first principles of the 4PN equations of motion. Then we comprehensively study the conserved quantities of the 4PN dynamics. Finally, we provide a preliminary result of the 4PN source mass quadrupole, which constitutes one of the crucial steps towards the computation of the 4.5PN phase

One of the most exciting predictions of Einstein's theory of gravitation that have not yet been proven experimentally by a direct detection are gravitational waves. These are tiny distortions of the spacetime itself, and a world-wide effort to directly measure them for the first time with a network of large-scale laser interferometers is currently ongoing and expected to provide positive results within this decade. One potential source of measurable gravitational waves is the inspiral and merger of two compact objects, such as binary black holes. Successfully finding their signature in the noise-dominated data of the detectors crucially relies on accurate predictions of what we are looking for. In this thesis, we present a detailed study of how the most complete waveform templates can be constructed by combining the results from (A) analytical expansions within the post-Newtonian framework and (B) numerical simulations of the full relativistic dynamics. We analyze various strategies to construct complete hybrid waveforms that consist of a post-Newtonian inspiral part matched to numerical-relativity data. We elaborate on exsisting approaches for nonspinning systems by extending the accessible parameter space and introducing an alternative scheme based in the Fourier domain. Our methods can now be readily applied to multiple spherical-harmonic modes and precessing systems. In addition to that, we analyze in detail the accuracy of hybrid waveforms with the goal to quantify how numerous sources of error in the approximation techniques affect the application of such templates in real gravitational-wave searches. This is of major importance for the future construction of improved models, but also for the correct interpretation of gravitational-wave observations that are made utilizing any complete waveform family. In particular, we comprehensively discuss how long the numerical-relativity contribution to the signal has to be in order to make the resulting hybrids accurate enough, and for currently feasible simulation lengths we assess the physics one can potentially do with template-based searches.<br>Eine der aufregendsten Vorhersagen aus Einsteins Gravitationstheorie, die bisher noch nicht direkt durch ein Experiment nachgewiesen werden konnten, sind Gravitationswellen. Dies sind winzige Verzerrungen der Raumzeit selbst, und es wird erwartet, dass das aktuelle Netzwerk von groß angelegten Laserinterferometern im kommenden Jahrzehnt die erste direkte Gravitationswellenmessung realisieren kann. Eine potentielle Quelle von messbaren Gravitationswellen ist das Einspiralen und Verschmelzen zweier kompakter Objekte, wie z.B. ein Binärsystem von Schwarzen Löchern. Die erfolgreiche Identifizierung ihrer charakteristischen Signatur im Rausch-dominierten Datenstrom der Detektoren hängt allerdings entscheidend von genauen Vorhersagen ab, was wir eigentlich suchen. In dieser Arbeit wird detailliert untersucht, wie die komplettesten Wellenformenmodelle konstruiert werden können, indem die Ergebnisse von (A) analytischen Entwicklungen im post-Newtonschen Verfahren und (B) numerische Simulationen der voll-relativistischen Bewegungen verknüpft werden. Es werden verschiedene Verfahren zur Erstellung solcher "hybriden Wellenformen", bei denen der post-Newtonsche Teil mit numerischen Daten vervollständigt wird, analysiert. Existierende Strategien für nicht-rotierende Systeme werden vertieft und der beschriebene Parameterraum erweitert. Des Weiteren wird eine Alternative im Fourierraum eingeführt. Die entwickelten Methoden können nun auf multiple sphärisch-harmonische Moden und präzedierende Systeme angewandt werden. Zusätzlich wird die Genauigkeit der hybriden Wellenformen mit dem Ziel analysiert, den Einfluss verschiedener Fehlerquellen in den Näherungstechniken zu quantifizieren und die resultierenden Einschränkungen bei realen Anwendungen abzuschätzen. Dies ist von größter Bedeutung für die zukünftige Entwicklung von verbesserten Modellen, aber auch für die korrekte Interpretation von Gravitationswellenbeobachtungen, die auf Grundlage solcher Familien von Wellenformen gemacht worden sind. Insbesondere wird diskutiert, wie lang der numerische Anteil des Signals sein muss, um die Hybride genau genug konstruieren zu können. Für die aktuell umsetzbaren Simulationslängen wird die Physik eingeschätzt, die mit Hilfe von Modell-basierten Suchen potentiell untersucht werden kann.

2013 - 2014<br>In this Thesis we report a general review of Extended Theories of Gravity and the fundamental aspects of General Relativity. We show the technicality of development of field equation with respect to Newtonian, Post-Newtonian approach and the post-Minkowskian limit. We analyse also the problem of how conformally transformed models behave in the weak field limit approximation. This issue could be extremely relevant in order to select conformally invariant physical quantities. The photon deflection is considered in the framework of the Newtonian Limit of a general class of f (R, Rαβ Rαβ, RαβγδRαβγδ) - Gravity where f is an unspecific function of the Ricci scalar R, Ricci tensor squareRαβ Rαβ and Riemann tensor square RαβγδRαβγδ. Studying in the weak-field approximation - Newtonian and Post-Newtonian limit - the geodesic and Lense-Thirring processions by using the recent experimental results of the Gravity Probe B and LARES satellite and using the damping of the orbital period of coalescing stellar binary systems, we impose constraints on the free parameters of such models of Extended Theories of Gravity. [edited by author]<br>XIII n.s.

Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Physics, 2008.<br>Includes bibliographical references (p. 51).<br>The purpose of this thesis is to investigate both the qualitative and quantitative effects of including higher order amplitude terms in the post-Newtonian expansion for gravitational waves on parameter estimation of inspiraling binary systems. First, we review the mechanism behind gravitational wave production and the formalism behind our estimation of parameters given a specific waveform. Then, we use a Monte Carlo simulation of 1000 separate binary systems with random position, and orientation parameters and fixed mass ratios between binary objects to generate gravitational waveforms measurements from a detector model which mimics data received from the proposed LISA mission. After that we numerically estimate how well those parameters are determined. The data presented compares median values of accuracy defined as Aý/ý for parameters ( of luminosity distance, chirp mass, and reduced mass, as well as the major and minor axes of a localization ellipse between waveform models which include only the leading quadrupole harmonic amplitude contribution, and the .5PN amplitude harmonic correction to the quadrupole. Our results show that, for all the Monte Carlo simulations run, there is a substantial global improvement in accuracy of the estimated parameters when higher order .5PN amplitude terms are included in the waveform model. The largest improvement shown comes from the range of masses between 105 and 106 solar masses, which is the ideal reception band for the LISA detector array. This improvement can eventually be applicable to aid in the location of binary sources for confirmation of direct gravitational wave observation. We conclude from these results that it is indeed advantageous to include higher order terms in the post-Newtonian expansion for gravitational wave models in order to obtain more accurate parameter estimates.<br>by Terrence J. Torres.<br>S.B.

Cette thèse porte sur la compréhension de systèmes dynamiques liés aux films liquides minces entraînés par la gravité; principalement des modèles d’eau peu profonde et des équations de lubrification. D’un point de vue modélisation, nous avons obtenu d’une part des modèles à 3 équations des eaux peu profondes pour les fluides newtoniens à viscosité constante décrivant l’évolution de la hauteur du fluide, de sa vitesse et d’une nouvelle variable définie équivalent eau taux de cisaillement. De tels modèles parviennent à capturer des régimes instables et à donner des résultats numériques satisfaisants pour l’étude de seuil d’instabilité et de la phase de l’onde à une distance modérée du seuil. D’autre part, une équation de lubrification et un modèle d’eau peu profonde ont été obtenu à partir d’un modèle de fluide non-Newtonien connu sous le nom de fluide bi-visqueux- dont la rhéologie se rapproche de celle des fluides pseudo plastiques et de Bingham. Concernant la partie analyse, l’existence globale de solutions faibles non négatives pour les équations de lubrification, telles que l’équation de Derrida-Lebowitz-Speer-Spohn, partant des solutions faibles non négatives pour les équations appropriées d’eau peu profonde visqueuse (en manipulant les termes de traînée et la formule de capillarité), est montrée. La nouveauté dans ce travail est de montrer que l’entropie BF qui a été introduite pour les équations de lubrification dans le contexte des films minces est encodée dans l’entropie BD introduite pour Navier-Stokes compressible avec viscosités dépendant de la densité. On s’intéresse également aux solutions dissipatives pour des systèmes incompressible et compressible de type Oldroyd ainsi qu’aux solutions faibles pour un système dégénéré de type Bingham<br>The work in this thesis falls in the category of comprehending dynamical systems relatedto thin liquid films driven by gravity; mainly shallow water models and lubrication equations.From a modeling point of view, we have derived from one hand 3-equation shallow water modelsfor Newtonian fluids with constant viscosity that describe the evolution of the fluid’s height, velocityand a new defined variable equivalent to the shear rate. Such models succeed in capturingunstable regimes and in giving satisfactory numerical results for the instability threshold andthe wave speed at moderate distance from threshold. On the other hand, a lubrication equationand a shallow water model were derived for a non Newtonian fluid- known as a bi-viscousfluid- whose rheology approximates that of pseudo plastic and Bingham fluids. Concerning theanalysis part, the global existence of nonnegative weak solutions for lubrication equations, suchas the Derrida-Lebowitz-Speer-Spohn equation, is proved starting from nonnegative weak solutionsfor appropriate viscous shallow-water equations (playing with drag terms and capillarityformula). The novelty in this work is to show that the BF entropy which is introduced forlubrication equations in the context of thin films is encoded inside the BD entropy introducedfor compressible Navier Stokes equations with density dependent viscosity. We also investigatethe dissipative solution of Navier Stokes system of Oldroyd-B rheology, as well as the globalweak solution for degenerate lake system of Bingham rheology

Orientador: Patricio A. Letelier Sotomayor<br>Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica<br>Made available in DSpace on 2018-08-13T05:26:41Z (GMT). No. of bitstreams: 1 Steklain_AndreFabiano_D.pdf: 11096709 bytes, checksum: 482e5ffb56f964f7786da54ec1791864 (MD5) Previous issue date: 2009<br>Resumo: Neste trabalho a dinâmica do problema de Hill é analisada utilizando-se duas metodologias diferentes. Na primeira metodologia, ainda no contexto da mecânica newtoniana, utilizamos potenciais que reproduzem efeitos da relatividade geral. Foram utilizados os potenciais de Paczynski-Wiita e um dos potenciais de Artemova, Bjornsson e Novikov (ABN). Estes potenciais reproduzem os efeitos que surgem no contexto da métrica de Schwarzschild (horizonte de eventos) e da métrica de Kerr (efeito Lense-Thirring), respectivamente. Na segunda metodologia as equações de movimento são obtidas a partir da relatividade geral, utilizando a métrica aproximada de um sistema binário obtida a partir de uma expansão pós-newtoniana de primeira ordem (1PN). A análise da dinâmica envolveu o estudo da estabilidade das órbitas fechadas, utilizando ferramentas clássicas como seções de Poincaré e expoentes de Lyapunov. Foram estudadas também trajetórias não limitadas utilizando escape fractal. Dentre os resultados obtidos destacam-se dois fatos. No caso do potencial ABN, existe uma influência da rotação na estabilidade das órbitas. No caso relativístico existe um limite para o qual o sistema, em geral caótico, se torna estável, diferentemente do que se poderia esperar de acordo com os potenciais pseudo-Newtonianos, em particular considerando o potencial de Paczynski-Wiita.<br>Abstract: In this work the Hill problem dynamics is analyzed using two different approaches. In the first approach, still in the realm of Newtonian mechanics, we use potentials that reproduce General Relativity effects. We use the Paczynski-Wiita and one of the Artemova, Bj¨ornsson e Novikov (ABN) potentials. These potentials reproduce effects that arise in the context of the Schwarzschild metric (event horizon) and of the Kerr metric (Lense-Thirring effect), respectively. On the second approach the equations of motion are obtained using general relativity, from the approximate metric of a binary system obtained from post-Newtonian expansions up to first order (1PN). In the analysis of the dynamics we study the stability of bounded orbits using classical tools, like Poincare sections and Lyapunov exponents. We also study open trajectories using Fractal Escape analysis. From our results we remark that two features. For the ABN potential there is an influence of the rotations on the stability of the orbits. In general relativity there is a limit where the system, in general chaotic, become stable, in disagreement with the pseudo-Newtonian potentials, in particular the Paczy'nski-Wiita potential.<br>Doutorado<br>Doutor em Matemática Aplicada

本文データは平成22年度国立国会図書館の学位論文(博士)のデジタル化実施により作成された画像ファイルを基にpdf変換したものである<br>Kyoto University (京都大学)<br>0048<br>新制・課程博士<br>博士(理学)<br>甲第5895号<br>理博第1602号<br>新制||理||893(附属図書館)<br>UT51-95-D214<br>京都大学大学院理学研究科物理学第二専攻<br>(主査)教授 佐藤 文隆, 教授 益川 敏英, 教授 九後 太一<br>学位規則第4条第1項該当

In this thesis, we review some basic properties of the Einstein-aether gravity. We derive the field equations from an action and study a subclass of this theory corresponding to the Einstein-Maxwell like theory. We also show that the G&ouml<br>del type metrics are also exact solutions of this theory. Furthermore, we determine the observational constraints on the dimensionless preferred parameters of this theory using the parametrized post- Newtonian formalism. We stress that none of calculations and discussions are original in this thesis.

Pour extraire l'empreinte laissee par le passage d'une onde gravitationnelle dans le signal bruite d'un des futurs detecteurs (VIRGO/LIGO), il est necessaire de connaitre avec une grande precision la forme de cette onde. Or, les equations de la Relativite Generale (RG), qui regissent cette forme, sont d'une grande complexite et ne sont pas solubles analytiquement pour des systemes dynamiques tels que les binaires d'etoiles a neutrons. Celles ci representent pourtant les sources les plus prometteuses de rayonnement gravitationnel. On a donc recours a des methodes d'approximation. Dans cette these nous avons etudie l'approximation post-Newtonienne (developpement perturbatif en puissance de 1/c) dans le cadre de la RG. Dans un premier temps nous avons etudie cette approximation d'un point de vue general afin de demontrer un theoreme qui permet d'affirmer que, sous certaines hypotheses, le developpement post-Newtonien du champ de gravitation d'une source isolee auto-gravitante peut etre itere a tout les ordres en 1/c, ce qui n'avait jamais ete fait jusqu'a present. Nous avons montre que ce developpement se raccorde a tous les ordres a une solution exterieur, valable notamment a l'infini, et satisfaisant aux conditions d'absence de radiation rentrante. Dans un deuxieme temps, sous l'impulsion de Luc Blanchet et en collaboration avec Gilles Esposito-Farese, je me suis interesse au probleme de l'energie post-Newtonienne a l'ordre 1/c^6 d'un systeme binaire de corps compacts etendus. Nous nous interessons notamment au probleme du calcul d'un coefficient indetermine a l'ordre 3PN du a une incompletude des methodes de regularisation. Pour lever cette indetermination, nous nous sommes interesses au calcul de cette meme energie pour des corps etendus lorsque l'on tient compte de la structure interne. Nous presentons dans cette these les raisons possibles de l'indetermination, la methode et les problemes poses par l'extension finie des corps.

Avec la naissance de l’"astronomie gravitationnelle", vient l’opportunité inédite de tester la relativité générale et ses alternatives dans un régime de champ fort jamais observé jusqu’alors : celui de la coalescence d’un système binaire d’objets compacts. Cette thèse propose d’étudier le problème du mouvement ainsi que du rayonnement gravitationnel d’un tel système en gravités modifiées, en y adaptant et en généralisant certains développements analytiques clés de la relativité générale. On montre d’abord comment étendre le formalisme "effective-one-body" (EOB) à une large classe de gravités modifiées, parmi lesquelles les théories scalaire-tenseur. Dans ces dernières, l’interaction gravitationnelle est modifiée par l’ajout d’un degré de liberté scalaire (sans masse) à la relativité générale. Le lagrangien à deux corps correspondant étant connu à l’ordre post-post-keplerien, nous construisons un hamiltonien EOB associé, décrivant le mouvement d’une particule test dans des champs effectifs. Ceci permet de simplifier la dynamique à deux corps et d’en définir une resommation ; et ainsi, d’en explorer le régime de champ fort, près de la coalescence du système. On "s’attaque" ensuite, et pour la première fois, à la description analytique d’un système binaire de trous noirs "chevelus", afin d’obtenir les formes d’ondes gravitationnelles (EOB) associées ; et ce, sur l’exemple simple des théories Einstein-Maxwell-dilaton, qui généralisent les théories scalaire-tenseur par l’ajout d’un champ vectoriel (sans masse). Pour ce faire, on calcule le lagrangien à deux corps à l’ordre post-keplerien ainsi que le flux d’énergie rayonnée à l’infini à l’ordre quadrupolaire. Tout comme en relativité générale, ces développements reposent sur la description de la trajectoire des trous noirs par les lignes d’univers de particules ponctuelles, décrites par une action "skeleton" généralisant celle, géodésique, de la relativité générale. Enfin, à l’aide des "superpotentiels" de Katz, que l’on généralise pour définir la masse (nœtherienne) d’un trou noir à "cheveux" vectoriel et scalaire, on montre que la première loi de la thermodynamique qui en découle est particulièrement adaptée, lorsqu’un trou noir est membre d’un système binaire, pour en décrire les réajustements éventuels sous l’influence d’un compagnon lointain. La thermodynamique des trous noirs est alors utilisée pour interpréter et discuter du domaine de validité de leur "skeletonisation"<br>With the birth of "gravitational wave astronomy" comes the opportunity to test general relativity and its alternatives in a strong field regime that had never been observed so far: that of the coalescence of a compact binary sytem. This thesis studies the problem of motion and gravitational radiation from such systems in modified gravities, by adapting some of the key analytical tools that were first developed in the context of general relativity. First, we show how to widen the "effective-one-body" (EOB) formalism to a large class of modified gravities, including, e.g., scalar-tensor theories. In the latter, the gravitational interaction is described by supplementing general relativity with a (massless) scalar degree of freedom. The corresponding two-body lagrangian being known at post-post-keplerian order, we build an associated EOB hamiltonian, which describes the motion of a test particle orbiting in effective external fields. This enables to simplify and resum the two-body dynamics; and hence, to explore the strong-field regime near merger. We then "tackle", for the first time, the analytical description of "hairy" binary black hole systems, and obtain their (EOB) gravitational waveform counterparts in Einstein-Maxwell-dilaton theories, which generalize scalar-tensor theories by means of a (massless) vector field. To that end, we derive the two-body lagrangian at post-keplerian order as well as the energy flux radiated at infinity at quadrupolar order. As in general relativity, our developments rely on the phenomenological description of the black hole’s trajectories as worldlines of point particles that are, in turn, described by a "skeleton" action generalizing that of general relativity. Finally, we develop a formalism based on Katz’ "superpotentials" to define the mass (as a nœther charge) of a black hole that is endowed with vector and scalar "hair". We then deduce the first law of thermodynamics, which is particularly suitable to describe its readjustments when interacting with a faraway companion. Black hole thermodynamics is lastly shown to be a powerful tool to interpret and discuss the scope of their "skeletonization"

The motivation from this work stems from the idea of Dvali and Gomez that the end-state of the gravitational collapse is a Bose-Einstein condensate at the critical point, constituted by a large number of soft, off-shell and maximally packed gravitons. This approach goes beyond the semiclassical picture and considers black holes as purely quantum objects. The result is a model without a central singularity that expresses crucial quantities, such as the self-coupling constant of gravitons and the mass of the black hole in terms of only one parameter, the number of gravitons N, and is able to account for Bekenstein entropy and Hawking radiation. Recently, an effective quantum description of the static gravitational potential for a spherically symmetric system up to post-Newtonian order has been constructed, relying on a toy model of scalar gravitons. This model allows to reproduce the classical Newtonian potential by employing a coherent state, establishing a connection between the corpuscular model and post-Newtonian corrections. These works constitute the starting point of this thesis. After recovering physical units in the Lagrangian for the field up to post-Newtonian order and finding its equations of motion, we move on to its linearisation, that models the field as a Newtonian background plus a small perturbation. The perturbation is parametrized by a spherical wave ansatz in the WKB approximation, where the wavelength of the wave is thought to be much smaller than the scale on which the background varies. After writing the equations of motion for the perturbation, we then find the correspondent dispersion relation, up to the first order in the graviton self-coupling, and make considerations about the behaviour of the perturbations. The end result is that the perturbations decay in time and end up reabsorbed in the background. If on the other hand the opposite, long-wavelength limit is considered, the perturbations get amplified and might signal an instability.

La tesi riguarda la formazione di strutture a larga scala nell'universo, cioè l'origine degli addensamenti di materia che hanno portato alla formazione dei cluster di galassie. La maggior parte del lavoro ha riguardato aspetti non lineari della Teoria delle Perturbazioni Cosmologiche, trattando in particolare il periodo di transizione tra epoca della radiazione e epoca della materia. In questo contesto si è considerato un modello non-standard di materia, analizzando il ruolo dell'indice barotropico nell'evoluzione del contrasto di densità . La nota approssimazione Meszaros è stata generalizzata ad una analisi non lineare che ha permesso di trovare la skewness della distribzione di materia, un importante indice di non-Gaussianità rilevabile dai dati osservativi. Nel contesto delle perturbazioni cosmologiche è stata formulata la teoria Post-Newtoniana (1PN) con lo scopo di ottenere un set di equazioni valido per ogni range di distanze, in particolare per le scale intermedie. I risultati finali coincidono sia con la teoria lineare relativistica per grandi scale sia con la teoria non lineare Newtoniana per piccole scale; quest'ultima connessione fornisce una chiara visione della relazione fra Relatività Generale e teoria Newtoniana.<br>The subject matter of this thesis is the formation of large-scale structure in the universe, describing the clustering of matter in galaxies and clusters of galax- ies. Most of the study has dealt with the non-linear evolution of cosmological fluctuations, focusing on the scalar sector of perturbation theory. The period of transition between the radiation era and the matter era has been largely examined, extending the already known linear results to a non-standard matter model and to a non-linear analysis. The obtained second order solutions for the matter fluctuations variables have been used to find the skewness of the density and velocity distributions, an important statistic estimator measuring the level of non-Gaussianity of a statistic ensamble. In the contest of cosmological perturbations a complete Post-Newtonian (1PN) treatment is presented with the aim of obtain a set of equations suitable in particular for the intermediate scales. The final result agrees with both the non linear Newtonian theory of small scales and the linear general relativistic theory of large scales. Analyzing the limit cases of our approach to 1PN cosmology, we have clarified the link between the Newtonian theory of gravity and General Relativity.

La première partie de cette thèse traite des théories de gravité massive. L'étude de ces théories a connu un regain d'intérêt depuis la découverte de l'accélération de l'expansion de l'univers, car elles pourraient expliquer cette dernière sans avoir à recourir à une constante cosmologique. La découverte, en 2010 d'une théorie cohérente de gravité massive, dite dRGT, a ouvert un vaste et prometteur champ d'investigation. Dans cette thèse nous déterminons, dans une formulation métrique et covariante, la linéarisation autour d'espace-temps arbitraires de ces théories, et de leur extension bimétrique. Ce travail nous permet ensuite de compter par une méthode lagrangienne le nombre de degrés de liberté qui se propagent. La seconde partie de cette thèse s'inscrit dans le cadre des ondes gravitationnelles en relativité générale et porte plus précisément sur la dynamique de systèmes binaires d'objets compacts. Ce travail est important dans la perspective de leur détection par les détecteurs interférométriques d'ondes gravitationnelles terrestres et spatial. Nous étudions le problème de la dynamique de systèmes binaires d¿objets compacts en relativité générale, à l¿aide de la méthode d'approximation dites des développements post-newtoniens (PN). Nous dérivons les équations du mouvement à l'ordre $4$PN en coordonnées harmoniques. Nous utilisons une méthode basée sur une action de Fokker adaptée au formalisme post-newtonien, en dérivant notamment les effets de sillage d'onde qui apparaissent à $4$PN<br>The first part of this thesis deals with massive gravity theories. There has been a renewal of interest in these theories since the discovery of the acceleration of the expansion of the universe, because they could explain it without having to resort to a cosmological constant. The discovery in 2010 of a coherent theory of massive gravity, named dRGT, has opened a vast and promising field of investigation. In this thesis we determine, in a metric and covariant formulation, the linearization around arbitrary backgrounds of these theories and their bimetric extension. This result then allows us to count with a Lagrangian method the number of degrees of freedom that are propagating. The second part of this thesis concerns gravitational waves in general relativity and especially the dynamics of coalescing compact binary systems. This work is important in view of their detection by interferometric detectors, both terrestrial and spacial. We study the dynamics of compact binary systems in general relativity, using the approximation method based on post-Newtonian developments (PN). We derive the equations of motion to $4$ PN order in harmonic coordinates. We use a method based on a Fokker action adapted to the post-Newtonian formalism, in particular deriving the tail effects appearing at $4$PN

La première partie de cette thèse s'inscrit dans le cadre de la modélisation des ondes gravitationnelles en provenance des systèmes binaires coalescents de trous noirs, dans la perspective de leur détection par les antennes gravitationnelles terrestres LIGO/VIRGO et spatiale LISA. Nous étudions la dynamique relativiste de tels systèmes binaires d'objets compacts à l'aide de deux méthodes d'approximation en relativité générale : les développements post-newtoniens, et le formalisme de la force propre, une extension naturelle de la théorie des perturbations d'un trou noir ; nous démontrons la cohérence des résultats ainsi obtenus dans leur domaine de validité commun. Dans un second temps, nous combinons ces deux méthodes perturbatives afin d'estimer l'effet de recul gravitationnel lors de la coalescence de deux trous noirs de Schwarzschild ; nos résultats sont en très bon accord avec ceux obtenus par des simulations en relativité numérique. La seconde partie de cette thèse traite du problème de la matière noire en astrophysique. L'hypothèse de la matière noire rend compte de nombreuses observations indépendantes de l'échelle des amas de galaxies jusqu'aux échelles cosmologiques. Les observations à l'échelle galactique sont toutefois en bien meilleur accord avec la phénoménologie de la dynamique newtonienne modifiée (MOND), qui postule une modification des lois de la gravité en l'absence de matière noire. Nous proposons une troisième alternative : conserver la théorie de la gravitation standard, mais doter la matière noire d'une propriété de polarisabilité dans un champ gravitationnel, de façon à rendre compte de la phénoménologie de MOND à l'échelle des galaxies.

Includes bibliographical references (leaves 168-179).<br>xiii, 179 leaves ; 30 cm.<br>Title page, contents and abstract only. The complete thesis in print form is available from the University Library.<br>Finds that the post-Newtonian approximation seems to be a better approximation of the general relativistic theory than the standard Newtonian theory.<br>Thesis (Ph.D.)--Adelaide University, Dept. of Physics and Mathematical Physics, 2001

Includes bibliographical references (leaves 168-179).<br>xiii, 179 leaves ; 30 cm.<br>Finds that the post-Newtonian approximation seems to be a better approximation of the general relativistic theory than the standard Newtonian theory.<br>Thesis (Ph.D.)--Adelaide University, Dept. of Physics and Mathematical Physics, 2001

碩士<br>淡江大學<br>物理學系<br>86<br>First, we introduce the energy-momentum tensor of the spinning particles and use it to derive the Papapetrou's spin equation. In the post-Newtonian approximation method, we expand this tensor to the post-Newtonian order and derive corresponding equations of motion. In the chapter 4,the N spinning particle results are reduced to the case of a binary system. To see how the spin effects come in, the equations of motion, the central position,the total energy and the total angular momentum are evaluated including spin effects. Finally, consider a general elliptical orbit, and obtain the velocities, spin vectors,and position in all three directions One find that, in particular, the velocity and the displacement in the z-direction are non- zero. Only in the post-Newtonian or higher orders, the spin effects cause the binary system to leave the plane of the Newtonian orbit. Most stars possess spin. If we want an accurate result, we can not but consider the spin effects.

碩士<br>國立清華大學<br>物理系<br>101<br>We calculate the post-Newtonian limit in the teleparallel equivalent of General Relativity (TEGR) with a scalar eld which non-minimally couples to gravity. The background is set up at Minkowski spacetime, and both the tetrad eld and energy momentum tensor are perturbed in the parametrized post-Newtonian formalism. The comparisons of the post- Newtonian result with general relativity and solar system experiments are discussed.

The Numerical INJection Analysis (NINJA) project is a collaborative effort between members of the numerical relativity and gravitational wave data analysis communities. The purpose of NINJA is to study the sensitivity of existing gravitational-wave search and parameter-estimation algorithms using numerically generated waveforms, and to foster closer collaboration between the numerical relativity and data analysis communities. The first NINJA project used only a small number of injections of short numerical-relativity waveforms, which limited its ability to draw quantitative conclusions. The goal of the NINJA-2 project is to overcome these limitations with long post-Newtonian - numerical relativity hybrid waveforms, large numbers of injections, and the use of real detector data. We report on the submission requirements for the NINJA-2 project and the construction of the waveform catalog. Eight numerical relativity groups have contributed 63 hybrid waveforms consisting of a numerical portion modelling the late inspiral, merger, and ringdown stitched to a post-Newtonian portion modelling the early inspiral. We summarize the techniques used by each group in constructing their submissions. We also report on the procedures used to validate these submissions, including examination in the time and frequency domains and comparisons of waveforms from different groups against each other. These procedures have so far considered only the $(ell,m)=(2,2)$ mode. Based on these studies we judge that the hybrid waveforms are suitable for NINJA-2 studies. We note some of the plans for these investigations.

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.

Το αντικείμενο της εργασίας αυτής είναι η μελέτη των μετανευτώνειων προσεγγίσεων στο πλαίσιο της Γενικής Σχετικότητας, με έμφαση στους αστέρες νετρονίων. Λόγω του οτι, η μελέτη των αστέρων νετρονίων βασίζεται στην υδροδυναμική περιγραφή της ύλης, το κύριο ενδιαφέρον μας αφορά στον τρόπο που εισάγονται οι μετανευτώνειες προσεγγίσεις στις υδροδυναμικές εξισώσεις της Γενικής Σχετικότητας. Στο πρώτο κεφάλαιο παρουσιάζονται ορισμένα θεωρητικά στοιχεία γύρω απο φυσικά χαρακτηριστικά και τις ιδιότητες των συμπαγών αστέρων. Στο δεύτερο κεφάλαιο παρουσιάζονται τα γενικά χαρακτηριστικά των καταστατικών εξισώσεων των συμπαγών αστέρων, με έμφαση στην πολυτροπική καταστατική εξίσωση. Στο τρίτο κεφάλαιο παρουσιάζεται η θεωρία των μετανευτώνειων προσεγγίσεων, καθώς και οι μετανευτώνειες εξισώσεις πρώτης τάξης της υδροδυναμικής, στη Γενική Σχετικότητα όπως εισήχθησαν απο τον Chandrasekhar. Στο τέταρτο κεφάλαιο παρουσιάζεται η διαταρακτική μέθοδος που χρησιμοποιείται απο τους Fahlman και Anand, για την μελέτη των περιστρεφόμενων πολυτρόπων στο πλαίσιο της πρώτης μετανευτώνειας προσέγγισης στη Γενική Σχετικότητα. Στο πέμπτο κεφάλαιο παρουσιάζεται η μετανευτώνεια προσέγγιση δεύτερης τάξης όπως διατυπώθηκε απο τους Chandrasekhar και Nutku. Στο έκτο, και τελευταίο, κεφάλαιο παρουσιάζεται ένα υπολογιστικό αλγεβρικό πακέτο για μετανευτώνειους υπολογισμούς στη Γενική Σχετικότητα, το PROCRUSTES. Με την βοήθεια του πακέτου αυτού υπολογίσαμε διάφορες ποσότητες στη δεύτερη μετανευτώνεια προσέγγιση, όπως τον τανυστή ενέργειας - ορμής, 𝑇𝑖𝑗, τον τανυστή Ricci, 𝑅𝑖𝑗, τις εξισώσεις κίνησης, Τ𝑖𝑗;𝑗 = 0, και άλλες. Το PROCRUSTES είναι ένα πολύ χρήσιμο εργαλείο στη μετανευτώνεια μελέτη καθώς μπορεί κανείς να παράξει τις περίπλοκες εκφράσεις διαφόρων ποσοτήτων σε ελάχιστο χρόνο και χωρίς την πιθανότητα λάθους. Επίσης, με την βοήθεια του πακέτου αυτού, υπολογίσαμε τις αναλυτικές εκφράσεις των εξισώσεως κίνησης, Τ𝑖𝑗;𝑗 = 0, στην δεύτερη μετανευτώνεια προσέγγιση. Με κατάλληλη μετατροπή των εκφράσεων αυτών, μπορούμε να εφαρμόσουμε την μέθοδο των Fahlman και Anand, με σκοπό την μελέτη των περιστερόμενων πολυτρόπων στη μετανευτώνεια προσέγγιση δεύτερης τάξης. Στο τέλος της εργασίας παρατίθεται ένα συμπλήρωμα με τη δομή του προγράμματος και ορισμένες απο τις ποσότητες που υπολογίστηκαν στο πλαίσιο της εργασίας.<br>The main subject of my master thesis is the study of,the post-Newtonian approximations (PNA) in General Relativity (G-R), mainly those that concern the neutron stars. Owing to the study of neutron stars is on the hydrodynamic description of matter, our main interest lies uponthe way the PNA affects the hydrodynamic equations of G-R. In the first chapter there is presented the main theory around the physical attributes of compact stars. The second chapter reffers to the general features of the equations of state (EoS) of compact stars, giving emphasis to the polytropic EoS. In the yhird chapter, there is presented the theory of the PNA. There are also presented the the hydrodynamic equations in the PNA as the were introduced by Chandrasekhar. The fourth chapter is dedicated to the presentation of the method that was introduced by Fahlman and Anand, on the study of rotating polytropes in the PNA to G-R. The fifth chapter focuses on the second PNA as it formulated by Chandrasekhar and Nutku. During the sixth and final chapter there is presented a computer algebra package for post-Newtonian calculations in G-R, the PROCRUSTES. Whith the aid of thiw package, we calculated several quantities in the second PNA, as for example the E-M tensor 𝑇𝑖𝑗„ the Ricci tensor 𝑅𝑖𝑗 and the equations of motion (EoM) Τ𝑖𝑗 ;𝑗 = 0. PROCRUSTES is a very useful tool for the post-Newtonian study, as someone can produce the compicated equations of several quantities in no time and eliminating the possibility of making some mistake during the calculation. Moreover, using PROCRUSTES we calculated the expressions of the EoM’s in the second PNA. Under appropriate transformation of these expressions, we are able to apply the method of Fahlman and Anand onto these equations with a view to the study of rotating polytropes in the second PNA. At the end of this work there is quoted a supplement with the structure of the programm we used, along with some of the quantities that were calculated during this work.

Κύριο αντικείμενο μελέτης της παρούσας μεταπτυχιακής διπλωματικής εργασίας είναι οι περιστρεφόμενοι αστέρες νετρονίων. Λόγω του ότι οι κλασικές διαταρακτικές μέθοδοι που εφαρμόζονται για την εύρεση της ακτίνας ενός περιστρεφόμενου πολυτροπικού μοντέλου περιορίζονται, από την επιφάνεια του αστέρα, αναπτύξαμε μία μέθοδο για τον υπολογισμό ποσοτήτων πέραν αυτού του ορίου. Αυτή η γενικευμένη μέθοδος χρησιμοποιεί τις μετανευτώνειες παραμέτρους ως όρους διαταραχής. Υλοποιώντας έναν κώδικα σε γλώσσα προγραμματισμού Fortran, υπολογίσαμε εκτεταμένους πίνακες ποσοτήτων και σταθερών. Μέσω της γενικευμένης αυτής μεθόδου επιτυγχάνεται η εύρεση της ακριβούς τιμής της ακτίνας ενός τέτοιου μοντέλου καθώς και ο καθορισμός της κρίσιμης παραμέτρου περιστροφής, η οποία αποτελεί μία μετανευτώνεια παράμετρο. Ο υπολογισμός της κρίσιμης παραμέτρου διαταραχής επιτυγχάνεται με ευκολία, κυρίως εκ του λόγου ότι η μέθοδος έχει υπολογίσει εκτεταμένους πίνακες συναρτησιακών τιμών. Οι υπολογιζόμενες κρίσιμες παράμετροι διαταραχής είναι μεγαλύτερες των αντιστοίχων τιμών της βιβλιογραφίας (κυρίως σε σύγκριση με τους Fahlman-Anand [55]), και φαίνεται να συμφωνούν καλύτερα με τις τιμές που υπολογίζονται από τις λεγόμενες επαναληπτικές μεθόδους. Τα αποτελέσματα επαληθεύουν με μεγάλη ακρίβεια τιμές συναρτήσεων και παραμέτρων σε σύγκριση με την κλασική βιβλιογραφία. Η παρούσα εργασία χωρίζεται σε πέντε μέρη, τα οποία αναπτύσσονται στα κεφάλαια 1, 2, 3, 4 και 5. Στο πρώτο κεφάλαιο, περιγράφεται ο αστέρας νετρονίων ως αστροφυσικό αντικείμενο. Δίνεται βάρος τόσο στη δομή του όσο και στα φυσικά χαρακτηριστικά του. Η ύπαρξη των αστέρων νετρονίων είναι απόλυτα συνδεδεμένη με τους πάλσαρς. Αυτοί αποτελούν ένα «ζωντανό» παράδειγμα περιστρεφομένων αστέρων νετρονίων; έτσι, γίνεται αναφορά στις φυσικές ιδιότητες και στις διεργασίες που πραγματοποιούνται σε αυτούς. Στο δεύτερο κεφάλαιο, αναφερόμαστε στις καταστατικές εξισώσεις που διέπουν το εσωτερικό των αστέρων νετρονίων, και στην έννοια του πολυτρόπου. Αφενός μεν, διότι δεν μπορεί να πραγματοποιηθεί μία μελέτη για αυτούς τους αστέρες χωρίς να υιοθετηθεί κάποια καταστατική εξίσωση, αφετέρου δε διότι μία από τις πλέον ενδεικτικές για την περιγραφή τους (και την οποία εμείς υιοθετούμε) είναι αυτή του πολυτρόπου. Επιπλέον, αναλύουμε τις εξισώσεις που διέπουν το αδιατάρακτο πολυτροπικό μοντέλο, όπως και αυτές που περιγράφουν το αντίστοιχο διαταραγμένο, σύμφωνα με τη θεωρία που ανέπτυξε ο Chandrasekhar. Στο τρίτο κεφάλαιο, χρησιμοποιούμε τη Γενική Θεωρία της Σχετικότητας στη μελέτη του πολυτροπικού μοντέλου, εστιάζοντας κυρίως στον τρόπο με τον οποίο τροποποιείται η κλασική θεώρηση, στο πώς μετασχηματίζονται η βασικές ποσότητες του μοντέλου, και στο πώς προκύπτουν οι σχέσεις της μετανευτώνειας προσέγγισης. Εξάγουμε τις μετανευτώνειες εξισώσεις της υδροδυναμικής και αναπτύσσουμε το διαταρακτικό μοντέλο επίλυσης, από το οποίο προκύπτουν οι εξισώσεις που επιλύουμε αριθμητικά. Στο τέταρτο κεφάλαιο, κάνουμε αναφορά στις διάφορες αριθμητικές μεθόδους που έχουν αναπτυχθεί για την μελέτη του σχετικιστικά περιστρεφόμενου πολυτροπικού μοντέλου. Στο πέμπτο κεφάλαιο, παρουσιάζουμε πίνακες αποτελεσμάτων και ενδιαφέρουσες γραφικές παραστάσεις. Δίνουμε επίσης ορισμένες αλγοριθμικές λεπτομέρειες για το πρόγραμμά μας. Συγκεκριμένα, γενικεύουμε τη μέθοδο των μετανευτωνείων προσεγγίσεων και αναλύουμε τα πλεονεκτήματα της. Ακολούθως, παραθέτουμε μία περιγραφή της αριθμητικής διαπραγμάτευσης της μεθόδου και την πορεία υλοποίησής της. Τέλος, παρατίθενται οι πίνακες των αποτελεσμάτων και τα τελικά συμπεράσματα.<br>In the present Thesis, we study rotating neutron stars. Due to the fact that the classical perturbation methods implemented to compute the radius of a polytropic rotating model are restricted by the star's surface, we develop a method for continuing integrations beyond this limit. This general approach utilises the postnewtonian parameters in terms of disturbance. By the application of a code written in Fortran, we calculate extensive tables of quantities and constants. Furthermore, we compute the radius as well as the critical rotation parameter, which plays the role of a postnewtonian term. This Thesis is organized in five chapters. In the first chapter, the neutron star is presented as an astrophysical object. Its structure and physical characteristics are of a great importance. Moreover, the existence of neutron stars is linked to pulsars, which are "living" examples of rotating neutron stars. Therefore, the physical characteristics of these objects are discussed in this chapter. The second chapter refers to the equations that describe the structure of the neutron stars and to the concept of polytropes. First, due to the difficulty in implementing a study for these stars without the adoption of any equation of state as well as due to the most indicative one for their description which is that of the polytrope. Second, the equations that refer to the undistorted and those that describe the corresponding distorted configurations are analysed in this chapter, in accordance with Chandrasekhar's perturbation theory. In the third chapter, the General Theory of Relativity is used to the study of the polytropic model, focusing on how the classical theory is corrected, on how the basic model's quantities are transformed and on how the equations of the postnewtonian approach are derived. The equations to be solved result from the latter ones. Furthermore, a a discussion on the various numerical methods that have been developed for studying the relativistic rotating polytopric model is given in the fourth chapter. In the fifth chapter of this Thesis, a number of tables illustrating results as well as some interesting diagrams are included. Certain algorithmic details for our program are given. We also discuss the generalisation of the postnewtonian approach and its advantages.

Title: Matter Models in General Relativity with a Decreasing Number of Sym- metries Author: Norman Gürlebeck Institute: Institute of theoretical physics Supervisor: Prof. RNDr. Jiří Bičák, DrSc., dr.h.c. Abstract: We investigate matter models with different symmetries in general relativity. Among these are thin (massive and massless) shells endowed with charge or dipole densities, dust distributions and rotating perfect fluid solutions. The electromagnetic sources we study are gravitating spherical symmetric condensers (including the implications of the energy conditions) and arbitrary gravitating shells endowed with a general test dipole distribution. For the latter the Israel formalism is extended to cover also general discontinuous tangential components of the electromagnetic test field, i.e., surface dipole densities. The formalism is applied to two examples and used to prove some general properties of dipole distributions. This is followed by a discussion of axially symmetric, stationary rigidly rotating dust with non-vanishing proper volume. The metric in the interior of such a configuration can be determined completely in terms of the mass density along the axis of rotation. The last matter models we consider are non-axially symmetric, stationary and rotating perfect fluid solutions. This is done with a...

Tesis (Doctor en Física)--Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física, 2011.<br>Los datos iniciales para agujeros negros extremos surgen como un límite singular en ciertas familias de datos para agujeros negros no extremos. En esta tesis se identifica dicho límite y se estudian la existencia, unicidad y propiedades básicas de estos datos. Se encuentra que uno de los finales asintóticamente planos de los datos en la familia se transforma en asiontóticamente cilíndrico al tomar el límite extremo. Por otro lado, se encuentra que existe una familia de datos iniciales que poseen el mismo momento angular y la misma área del final cilíndrico que Kerr extremo. Esta familia está arbitrariamente próxima a Kerr, preserva la simetría axial y la maximalidad. Finalmente se prueba una desigualdad entre el área del horizonte y el momento angular de un agujero negro axialmente simétrico de vacío con constante cosmológica.<br>María Eugenia Gabach Clément.<br>Ecuaciones de vínculo de Einstein -- Método conforme --Agujeros negros estacionarios -- Existencia de datos iniciales extremos de Bowen-York --Desigualdad entre momento angular y área de horizonte -- Espacios funcionales -- Definiciones y teoremas.

Tesis (Lic. en Física )--Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física, 2011.<br>Se estudian las perturbaciones lineales de las ecuaciones de Einstein alrededor de soluciones esféricamente simétricas. Se explica por qué no hay pérdida de generalidad al restringirse a perturbaciones axialmente simétricas. Esto motiva la aplicación del formalismo de reducción dimensional (por el vector de Killing axial) a la métrica perturbada. Aplicando a vacío (Schwarzschild), emerge naturalmente una descomposición de las ecuaciones en las que las cantidades desacoplan en dos modos. Esta propiedad encuentra su correspondencia con el formalismo tradicional donde la división viene dada por perturbaciones que tienen paridad definida bajo la aplicación del mapa antipodal. Las ecuaciones de Zerilli y Regge-Wheeler se recuperan de un modo covariante cuadridimensional, y asociadas a escalares del espacio-tiempo con una interpretación geométrica definida.<br>Ezequiel Boero.

Binary systems comprising of compact objects like neutron stars (NS) and/or black holes (BH) lose their energy and angular momentum via gravitational waves (GW). Radiation reaction due to the emission of GW results in a gradual shrinking of the binary orbit and an accompanying gradual increase in the orbital frequency. The preliminary phase of the binary evolution when the radiation-reaction time-scale is much larger than the orbital time-scale is called the inspiral phase. GW emitted during the final stages of the inspiral phase constitute one of the most important sources for the ground-based laser interferometric GW detectors like LIGO, VIRGO and the proposed space-based detector LISA. For the ground-based detectors, NS and/or stellar mass BH binaries are primary sources, while for LISA super-massive BH (SMBH) binaries are potential targets. Inspiralling compact binaries (ICB) are among the prime targets for interferometric detectors because using approximation schemes in general relativity (GR) like the post-Minkowskian (PM) and the post-Newtonian (PN) approximations one can compute the GW emitted by them with sufficient accuracy both for their detection and parameter estimation leading to GW astronomy. The extreme weakness of gravitational interactions implies that if a GW signal from an ICB is incident on a detector, it will be buried in the noisy detector output. Therefore, sophisticated data analysis techniques are required for detecting the signal in presence of the dominant noise and also estimating the parameters of the signal. From the pre-calculated theoretical waveforms called templates, one already knows the structure of the waveform from an ICB. The technique for detecting signals which are of known form in a noisy detector is matched filtering. This technique consists of cross-correlating the output of a noisy detector assumed to contain the signal of known form with a set of templates. It then finds an ‘optimal’ template that would produce, on average, the highest signal-to-noise ratio (SNR). The efficient performance of matched filtering as a data-analysis strategy for GW signals from ICB presupposes very accurate theoretical templates. Slight mismatches between the signal and the template will result in a loss of signal to noise ratio. Computing very accurate theoretical templates and including effects such as eccentricity are challenging tasks for the theoreticians. This thesis addresses some of the issues related to the waveform modelling of the ICB and their implications for GW data analysis. It is known theoretically that compact binaries reduce their eccentricity through the emission of GW. When GW signals from prototype ICB reach the GW detector bandwidth, their orbits are almost circular. Hence one usually models the binary orbit to be circular for computation of the search templates. The waveform from an ICB in a circular orbit is, at any given PN order of approximation, a linear combination of a finite number of harmonics of the orbital frequency. At the lowest order of approximation, called the Newtonian order, the waveform comprises a single harmonic at twice the orbital frequency. Inclusion of higher order PN corrections lead to the appearance of higher harmonics of the orbital frequency. Since the amplitudes of the higher harmonics contain higher powers of the PN expansion parameter, relative to the Newtonian order, they are referred to as amplitude corrections. The phase of each harmonic, determined by the orbital phase, is known upto 3.5PN order (nPN is the order of approximation equivalent to terms ~(v/c)2n beyond the Newtonian order, where v denotes the binary’s orbital velocity and c is the speed of light). Matched filtering is more sensitive to the phase of the signal rather than its amplitude, since the correlation builds up as long as the signal and the template remain in phase. Motivated by this fact, search templates so far have been a waveform model involving only the dominant harmonic (at twice the orbital frequency), although the phase evolution itself is included upto the maximum available PN order. Such waveforms, in which all amplitude corrections are neglected, but the phase is treated to the maximum available order, are called restricted waveforms (RWF) and these are generally used in the data-analysis of ground-based detectors and also simulated searches for the planned LISA. However, recent studies, in the case of ground-based interferometers, showed that going beyond the RWF approximation could improve the efficiency of detection as well as parameter estimation of the inspiral signal. After a brief overview of the properties of GW and their detection strategies in chapter 1, in chapters 2 and 3, we investigate the implications of going beyond the RWF, in the context of the planned space-based Laser Interferometric Space Antenna (LISA). The sensitivity of ground-based detectors is limited by seismic noise below 20Hz. On the other hand, the space-based LISA will be designed to be sensitive to GWs of frequency (10−4 _1)Hz. The most important source in this frequency band are supermassive BH (SMBH) binaries. There is strong observational evidence for the existence of SMBH with masses in the range of in most galactic nuclei. Mergers of such galaxies result in SMBH binaries whose evolution is governed by the emission of GW. Observation of the GW from SMBH binaries at high redshifts is one of the major science goals of LISA. These observations will allow us to probe the evolution of SMBHs and structure formation and provide an unique opportunity to test General Relativity (and its alternatives) in the strong field regime of the theory. Observing SMBH coalescences with high (100-1000) SNR is crucial for performing all the aforementioned tests. The LISA bandwidth (10−4_ 1)Hz determines the range of masses accessible to LISA because the inspiral signal would end when the system’s orbital frequency reaches the mass-dependent last stable orbit (LSO). In the test-mass approximation, the angular velocity ι at LSO is given by where M is the total mass of the binary. Search templates using the RWF, which contains only the dominant harmonic at twice the orbital frequency, cannot extract power in the signal beyond This further implies that the frequency range [0.1, 100] mHz corresponds to the range for the total mass of BH binaries that would be accessible to LISA. In chapter 2, we show that inclusion of higher harmonics will enhance the mass-range of LISA (for the same frequency range) and allow for the detection of SMBH binaries with total masses higher than The template employed in chapter 2 includes amplitude corrections upto 2.5PN order, while keeping the phase upto 3.5PN order. We call this template the full waveform (FWF). The FWF defined above contains higher harmonics of the orbital frequency, the highest of them being 7 times the orbital frequency. For a SMBH binary with total mass the dominant harmonic at LSO is less than the lower cut-off of the LISA bandwidth. Therefore, if one uses the RWF as a search template, this system is ‘invisible’ to LISA. However, the seventh harmonic can still enter the LISA bandwidth and produce a significant SNR and thus allow its detection. With the FWF, LISA can observe sources which are favoured by astronomical observations, but not observable with the RWF. More specifically, with the inclusion of all known harmonics LISA will be able to observe SMBH coalescences with total mass (and mass-ratio 0.1) for a low frequency cut-off of 10−4Hz (10−5Hz) with an SNR up to ~ 60 (~30) at a distance of 3 Gpc. The orbital motion of LISA around the Sun induces frequency, phase and amplitude modulations in the observed GW signal. These modulations carry information about both the source’s location and orientation. Determination of the angular coordinates of the source also allows determination of the luminosity distance of SMBH binaries. Therefore, SMBH binaries are often referred to as GW “standard sirens” (analogous to the electromagnetic “standard candles”). LISA would also be able to measure the “redshifted” masses of the component black holes with good accuracy for sources up to redshifts of a few. However, GW observations alone cannot provide any information about the redshift of the source. If the host galaxy or galaxy cluster is known one can disentangle the redshift from the masses by optical measurement of the redshift. This would not only allow one to extract the “physical” masses, but also provide an exciting possibility to study the luminosity distance-redshift relation providing a totally independent confirmation of the cosmological parameters. Further, this combined observation can be used to map the distribution of black hole masses as a function of redshift. Another outstanding issue in present day cosmology in which LISA can play a role is the dark energy and its physical origin. Probing the equation-of-state-ratio (w(z)) provides an important clue to the question of whether dark energy is truly a cosmological constant (i.e., w = -1). Assuming the Universe to be spatially flat, a combination of WMAP and Supernova Legacy Survey (SNLS) data yields significant constraints on Without including the spatial flatness as a prior, WMAP, large-scale structure and supernova data place a stringent constraint on the dark energy equation of state, For this to be possible, LISA should (a) measure the luminosity distance to the source with a good accuracy and (b) localize the coalescence event on the sky with good angular resolution so that the host galaxy/galaxy cluster can be uniquely identified. Based on analysis with the RWF, it is found that LISA’s angular resolution is not good enough to identify the source galaxy or galaxy cluster, and that other forms of identification would be needed. Secondly, weak lensing effects would corrupt the distance estimation to the same level as LISA’s systematic error. In chapter 3, we study the problem of parameter estimation in the context of LISA, but using the FWF. We investigate systematically the variation in parameter estimation with PN orders by critically examining the role of higher harmonics in the fast GW phasing and their interplay with the slow modulations induced due to LISA’s motion. More importantly, we explore the improvement in the estimation of the luminosity distance and the angular parameters due to the inclusion of higher harmonics in the waveform. We translate the error in the angular resolution to obtain the number of galaxies (or galaxy clusters) within the error box on the sky. We find that independent of the angular position of the source on the sky, higher harmonics improve LISA’s performance on both counts raised in earlier works based on the RWF. We show that the angular resolution enhances typically by a factor of ~2-500 (greater at higher masses) and the error on the estimation of the luminosity distance goes down by a factor of ~ 2-100 (again, larger at higher masses). For many possible sky positions and orientations of the source, the inaccuracy in our measurement of the dark energy would be at the level of a few percent, so that it would only be limited by weak lensing. We conclude that LISA could provide interesting constraints on cosmological parameters, especially the dark energy equation-of-state, and yet circumvent all the lower rungs of the cosmic distance ladder. Having emphasized the need to consider the FWF as a more powerful template, in chapter 4 we calculate a higher order term in the amplitude corrections of the waveform. In chapters 2 & 3, the FWF incorporated amplitude corrections upto 2.5PN order. In chapter 4 the waveform is calculated upto 3PN order. Recent progress in Numerical Relativity (NR) has resulted in computation of the late inspiral and subsequent merger and ringdown phases of the binary evolution (where PN theory does not hold good) by a full-fledged numerical integration of the Einstein field equations. A new field has emerged recently consisting of high-accuracy comparisons between the PN predictions and the numerically-generated waveforms. Such comparisons and matching to the PN results have proved currently to be very successful. They clearly show the need to include high PN corrections not only for the evolution of the binary’s orbital phase but also for the modulation of the gravitational amplitude. This leads to one more motivation for the work in this chapter: providing the associated spin-weighted spherical harmonic decomposition to facilitate comparison and match of the high PN prediction for the inspiral waveform to the numerically-generated waveforms for the merger and ringdown. For the computation of waveforms from the inspiralling compact binaries one needs to solve the two-body problem in general relativity. The nonlinear structure of general relativity prevents one from obtaining a general solution to this problem. The two-body problem is tackled using the multipolar post-Minkowskian (MPM) wave generation formalism. The MPM formalism describes the radiation field of any isolated post-Newtonian source. The radiation field is first of all parametrized by means of two sets of radiative multipole moments. These moments are then related (by means of an algorithm for solving the non-linearities of the field equations) to the so-called canonical moments which constitute some useful intermediaries for describing the external field of the source. The canonical moments are then expressed in terms of the operational source moments obtained by matching to a PN source and are given by explicit integrals extending over the matter source and gravitational field. The extension of the waveform by half a PN order requires as inputs the relations between the radiative, canonical and source multipole moments for general sources at 3PN order. We also require the 3PN extension of the source multipole moments in the case of compact binaries. The waveform in the far-zone consists of two types of terms, instantaneous and hereditary. The instantaneous terms are determined by the dynamical state of the binary at the retarded time. The hereditary terms, on the other hand, depend on the entire past history of the source. These terms originate from the nonlinear interactions between the various multipole moments and also from backscattering off the curved spacetime generated by the waves themselves. In this chapter, we compute the contributions of all the instantaneous and hereditary terms (which include tails, tails-of-tails and memory integrals) up to 3PN order. The end results of this chapter are given in terms of both the 3PN plus and cross polarizations and the separate spin-weighted spherical harmonic modes. Though most of the sources will be in circular orbits by the time the GWs emitted by the system enter the sensitivity band of the laser interferometers, astrophysical scenarios such as Kozai mechanism could produce binaries which have nonzero eccentricity. Studies have shown that filtering the signal from an eccentric binary with circular orbit templates could significantly degrade the SNR. For constructing a phasing formula for eccentric binaries one has to compute the energy and angular momentum fluxes carried away by the GWs and then compute how the orbital elements evolve with time under gravitational radiation reaction. The far-zone energy and angular momentum fluxes, like the waveform, contain both instantaneous and hereditary contributions. The complete 3PN energy flux and instantaneous terms in the 3PN angular momentum flux are already known. In chapter 5, the hereditary terms in the 3PN angular momentum flux from an ICB moving in quasi-elliptical orbits are computed. A semi-analytic method in the frequency domain is used to compute the hereditary contributions. At 3PN order, the quasi-Keplerian representation of elliptical orbits at 1PN order is required. To calculate the tail contributions we exploit the doubly periodic nature of the motion to average the 3PN fluxes over the binary’s orbit. The hereditary part of the angular momentum flux provided here has to be supplemented with the instantaneous part to obtain the final input needed for the construction of templates for binaries moving in elliptical orbits, a class of sources for both the space based detectors and the ground based ones. Using the hereditary contributions in the 3PN energy flux, we also compute the 3PN accurate hereditary contributions to the secular evolution of the orbital elements of the quasi-Keplerian orbit description.