Hawking radiation is a phenomenon where the combination of geometry of spacetime around a black hole and quantum effects near its event horizon causes particle emission. Stephen Hawking was one of the first to make computations and conclude that this is valid for every black hole in general. Therefore, the goal of the project was to understand how the presence of a black hole changes geometry of spacetime, explore some of its peculiar properties and, finally, connect it to Hawking radiation. It turns out that one way to describe geometry around a black hole is to use the Schwarzchild metric which fully describes surroundings of a non-rotating and uncharged black hole. Using the so called Klein-Gordon equation and some additional computations one then sees that there’s indeed a particle emission. However, the radiation appears to be observer dependent which is due to curvature near event horizon. Hawking radiation has temperature which happens to be extremely small to detect, but this result reveals the fact that black holes radiate faster as they shrink. However, the time it takes for an arbitrary black hole to evaporate is much longer than the age of the Universe. Encountering those and some other challenges Hawking radiation remains hypothetical. / Hawkingstrålning är ett fenomen där kombinationen av geometri av rumtid runt ett svart hål och kvantmekaniska effekter nära dess händelsehorisont leder till partikel emission. Stephen Hawking var bland de första att göra beräkningar och dra slutsatsen att detta är giltigt för alla svarta hål. Syftet med projektet var därför att förstå hur närvaron av ett svart hål ändrar geometri av rumtid, undersöka dess vissa speciella egenskaper samt anknyta det till Hawkingstrålning. Det visar sig att ett sätt att beskriva geometri kring ett svart hål är att använda Schwarzchild metriken som helt beskriver omgivningen av ett icke roterande och oladdat svart hål .Använder man sig av så kallade Klein-Gordon ekvationen och några ytterligare beräkningar så kommer man till slutsaten att det verkligen finns enemission av partiklar. Emissionen verkar dock vara observatörsberoende på grund av krökning nära händelsehorisont. Hawkingstrålning har temperatur som visar sig vara extremt liten för att upptäcka, men resultaten avslöjar faktumet att svarta hål strålar ut snabbare då de krymper. Tiden det tar för ett godtyckligt svart hål att koka bort är dock mycket längre än åldern of Universum. På grund av dessa och några andra utmanningar återstår Hawkingstrålning hypotetiskt.

Zusammenfassung In den letzten Jahren haben Schwarze Löcher viel Aufmerksamkeit auf sich gezogen, insbesondere wegen ihrer ungewöhlichen quantentheoretischen Eigenschaften. Ein in diesem Zusammenhang interessantes Modell ist das geeichte SL(2,R)/U(1)-Wess-Zumino-Novikov-Witten-Modell, das im Rahmen der Stringtheorie als Euklidisches zweidimensionales Schwarzes Loch interpretiert werden kann. Die vorliegende Arbeit analysiert die klassischen Eigenschaften dieses Modells, um so die Grundlage für quantentheoretische Untersuchungen zu schaffen. Ausgangspunkt ist eine allgemeine Betrachtung über geeichte Wess-Zumino-Novikov-Witten-Modelle (WZNW-Modelle). Herkömmlicherweise werden sie mit Hilfe von Eichfeldern formuliert, deren Bewegungsgleichungen rein algebraisch sind. In der vorliegenden Arbeit werden die Eichfelder aus den Modellen eliminiert. Dabei entsteht eine Klasse von nichtlinearen integrablen konformen Feldtheorien, für deren Bewegungsgleichung eine explizite Lax-Paar-Darstellung abgeleitet wird. Diese Ergebnisse werden auf das geeichte SL(2,R)/U(1)-WZNW-Modell spezialisiert. Zum Vergleich wird auch die Eliminierung des Eichfeldes durch explizite Pfadintegration untersucht, die jedoch aufgrund mathematischer Ambiguitäten nicht zu einem abschließenden Ergebnis geführt wird. Das klassische geeichte SL(2,R)/U(1)-WZNW-Modell wird sowohl in einem unendlich ausgedehnten Minkowski-Raum als auch mit räumlich periodischen Randbedingungen untersucht. Letzteres ist für die stringtheoretische Interpretation des Modells wichtig. Es werden die nichtlinearen Bewegungsgleichungen und ihre allgemeine Lösung angegeben. Diese enthält Parameterfunktionen. Es wird ein Verfahren abgeleitet, um die Parameterfunktionen aus vorgegebenen Anfangsbedingungen zu bestimmen. Mit Hilfe dieses Verfahrens werden die Poissonklammern der Parameterfunktionen aus den kanonischen Poissonklammern der physikalischen Felder berechnet. Es wird gezeigt, daß es eine nichtlokale kanonische Transformation der nichtlinearen physikalischen Felder auf freie Felder gibt. Die entsprechende Bäcklund-Transformation wird angegeben. / Abstract In recent years, Black Holes have attracted much attention, in particular, because of their unusual quantum-theoretical properties. An interesting model, in this context, is the SL(2,R)/U(1) gauged Wess-Zumino-Novikov-Witten model, which can be interpreted stringtheoretically as Euclidean two-dimensional Black Hole. The present dissertation analyzes the classical properties of this model, in order to prepare the basis for quantum-theoretical investigations. First, gauged Wess-Zumino-Novikov-Witten (WZNW) models are intoduced in general. Usually, they are formulated including gauge fields, whose equations of motion are purely algebraic. In the present dissertation, the gauge fields are eliminated from the models. A class of non-linear integrable field theories arises, whose equations of motion can be represented by Lax pairs explicitly. These results are specialized to the SL(2,R)/U(1) gauged WZNW model. For comparison, the elimination of the gauge field by explicit path integration is also investigated. But due to mathematical ambiguities, this investigation does not lead to a final result. The classical SL(2,R)/U(1) gauged WZNW model is investigated in an infinitely extended Minkowski space-time as well as with spatially periodic boundary conditions. The latter is important for the stringtheoretical interpretation of the model. The non-linear equations of motion and their general solution are given. A procedure is derived to determine the parameter functions of the general solution from given initial conditions of the equations of motion. By means of this procedure the Poisson brackets of the parameter functions are calculated from the canonical Poisson brackets of the physical fields. It is shown that there is a non-local canonical transformation of the non-linear physical fields onto free fields. The corresponding Backlund transformation is presented.

This thesis considers the problem of calculating and observing the mixing of modes of positive and negative frequency in inhomogeneous, dispersive media. Scattering of vacuum modes of the electromagnetic field at a moving interface in the refractive index of a dielectric medium is discussed. Kinematics arguments are used to demonstrate that this interface may, in a regime of linear dispersion, act as the analogue of the event horizon of a black hole to modes of the field. Furthermore, a study of the dispersion of the dielectric shows that five distinct configurations of modes of the inhomogeneous medium at the interface exist as a function of frequency. Thus it is shown that the interface is simultaneously a black- and white-hole horizon-like and horizonless emitter. The role, and importance, of negative-frequency modes of the field in mode conversion at the horizon is established and yields a calculation of the spontaneous photonic flux at the interface. An algorithm to calculate the scattering of vacuum modes at the interface is introduced. Spectra of the photonic flux in the moving and laboratory frame, for all modes and all realisable increase in the refractive index at the interface are computed. As a result of the various mode configurations, the spectra are highly structured in intervals with black-hole, white-hole and no horizon. The spectra are dominated by a negative-frequency mode, which is the partner in any Hawking-type emission. An experiment in which an incoming positive-frequency wave is populated with photons is assembled to observe the transfer of energy to outgoing waves of positive and negative frequency at the horizon. The effect of mode conversion at the interface is clearly shown to be a feature of horizon physics. This is a classical version of the quantum experiment that aims at validating the mechanism of Hawking radiation.

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.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-14T21:49:47Z No. of bitstreams: 1 PATRÍCIO JOSÉ FÉLIX DA SILVA - DISSERTAÇÃO PPGF 2009..pdf: 1514686 bytes, checksum: b72b139b4e01b55657953090b7322867 (MD5) / Made available in DSpace on 2018-08-14T21:49:47Z (GMT). No. of bitstreams: 1 PATRÍCIO JOSÉ FÉLIX DA SILVA - DISSERTAÇÃO PPGF 2009..pdf: 1514686 bytes, checksum: b72b139b4e01b55657953090b7322867 (MD5) Previous issue date: 2009-03-09 / CNPq / Capes / Um sistema de coordenadas tem a função de localizar os eventos do espaço-tempo com respeito a um sistema de referência. A construção do sistema de coordenadas depende crucialmente da noção de simultaneidade associada ao referencial. No entanto, não existe uma maneira natural, ou privilegiada, de definir simultaneidade para referenciais não inerciais, mesmo no espaço-tempo de Minkowski. Cada procedimento conduz a diferentes sistemas de coordenadas. Neste trabalho, discutimos alguns métodos bem conhecidos da literatura especializada. Estudamos as coordenadas de Rindler, de Fermi-Walker, as coordenadas de Radar e as coordenadas de Emissão (ou GPS). O sistema de coordenadas de Rindler é um dos sistemas de grande destaque porque permite simular algumas propriedades da geometria do Buraco Negro num espaço-tempo plano. As coordenadas de Rindler estão associadas a uma família de observadores uniformemente acelerados que obedecem à relação a=1/ρ, onde a é a aceleração própria do observador e ρ a sua posição inicial com respeito a algum sistema de referência inercial. Neste trabalho, propomos um método para construção de sistemas de coordenadas adaptados a observadores cuja a celeração depende da posição inicial segundo a regra a=a0/ρn, onde n ∈ N e a0 é uma constante, usando o princípio da localidade. O caso n = 1 recupera as coordenadas de Rindler. Os outros casos nos permitem discutir a relação entre a geometria não-Euclidiana das secções espaciais e referenciais acelerados,como originariamente proposto por Einstein. Além disso, com a generalização podemos simular o comportamento de observadores estáticos tanto nas proximidades do horizonte de um Buraco Negro (n=1) quanto em regiões afastadas (n=2). / The main role of a coordinate systein is to localize the event-s of spacetime with respect to a frame of reference. The construetion of a coordinate systein depeuds crucially on the notíon of simultaneity associated to the frame of reference. However, there is no natural manner of defining simultaneity adapted to non-inertial frames of reference, even in the case of Minkowski spacetime. Each procedure leads to different coordinate systems. In thls work. we discuss some well-known methods found in the Literatura. We study the Rindler coordinates. Fermi-Walker coordinates. Radar coodinadates and Emission (or GPS) coordinates. The system of Rindler coordinates has great interest because it simulates in a flat spacetime some aspects of a Black Hole's geometry. We can say that Rindler coordinates are adapted to a family of uniformly accelerated observeis which obey the relatiou a = i, where a is the proper acceieration and p is the initial position with respect to some inertial system. In this work, we also propose a method in order to construct coordinate systems adapted to observers whose accelerations depend on the initial position according to the formula a = where n e N and a» is a constant, by using the locality principie. The case TI = 1 reproduces the Rindler coordinates. The other cases allow us to verify a connection between non-Euciideaii geometry of the spatial sections and non-inertial frames of reference, as it was originally suggested by Einstein. With this generalization we can also simulate the behavior of static observers in the vicinity of a Black Hole"s Horizon (TI = 1) and also in distant regions (n - 2)

The aim of this thesis is to create collections of examples for the subject Astronomy taught for students of pedagogical faculties, studying this discipline as a part of physics courses. Due to very different mathematical knowledge of students I have chosen typical and not much difficult examples oriented to several branches of astronomy. Each part of examples begins with a self-contained theoretical introduction. The difficulty rises gradually from trivial to more complicated examples. The examples are mainly focused on motions in radial gravitational fields.

The mildly active nuclear region in the galaxy M81 (henceforth, M81‹) is one of the nearest low-luminosity active galactic nuclei (LLAGN) whose structure is marginally resolved when probed with Very Long Baseline Interferometry (VLBI). Motivated by the way resolved radio sources usually appear on the smallest scales, i.e., a core with a one- sided jet structure, we developed a strictly one-sided, asymmetric triangular model, which we call ASYM, with brightness distribution along a line segment on the sky, with maximum brightness at one end of the segment fading linearly to zero at the other end. The ASYM model is compared and contrasted with an elliptical Gaussian model (hereafter, GAUS), by fitting existing VLBI data of M81‹ at 39 epochs between 1993 and 2003 at 8.4 and 5.0 GHz with the two models. Contrary to what we envisioned, we find that for 77% of our epochs, a simple GAUS model fits the visibility data of M81‹ at 8.4 GHz better (i.e., has a lower reduced 2) than the ASYM model. We conclude that M81‹ is not strictly a one-sided, asymmetric jetted source; as is thought to be the case for the majority of AGN observed at VLBI scales. Our results imply that M81‹ is mostly symmetrical with a significant jet counterpart which cannot be overlooked. / School of Interdisciplinary Research and Graduate Studies (SIRGS) / M. Sc. (Astronomy)

A white dwarf is most likely the end stage of a low mass star like our Sun, which results when the parent star consumes all the hydrogen in its core, thus bringing fusion to a halt. It is a dense and compact object, where the inward gravitational pull is balanced by the outward pressure arising due to the motion of its constituent degenerate electrons. The theory of non-magnetized and non-rotating white dwarfs was formulated extensively by S. Chandrasekhar in the 1930s, who also proposed a maximum possible mass for this objects, known as the Chandrasekhar limit (Chandrasekhar 1935)1. White dwarfs are believed to be the progenitors of extremely bright explosions called type Ia supernovae (SNeIa). SNeIa are extremely important and popular astronomical events, which are hypothesized to be triggered in white dwarfs having mass close to the famous Chandrasekhar limit ∼ 1.44M⊙. The characteristic nature of the variation of luminosity with time of SNeIa is believed to be powered by the decay of 56Ni to 56Co and, finally, to 56Fe. This feature, along with the consistent mass of the exploding white dwarf, is deeply linked with their utilization as “standard candles” for cosmic distance measurement. In fact, SNeIa measurements were instrumental in establishing the accelerated nature of the current expansion of the universe (Perlmutter et al. 1999). However, several recently observed peculiar SNeIa do not conform to this traditional explanation. Some of these SNeIa are highly over-luminous, e.g. SN 2003fg, SN 2006gz, SN 2007if, SN 2009dc (Howell et al. 2006; Scalzo et al. 2010), and some others are highly under-luminous, e.g. SN 1991bg, SN 1997cn, SN 1998de, SN 1999by, SN 2005bl (Filippenko et al. 1992; Taubenberger et al. 2008). The luminosity of the former group of SNeIa implies a huge Ni-mass (often itself super-Chandrasekhar), invoking highly super-Chandrasekhar white dwarfs, having mass 2.1 − 2.8M⊙, as their most plausible progenitors (Howell et al. 2006; Scalzo et al. 2010). On the other hand, the latter group produces as low as ∼ 0.1M⊙ of Ni (Stritzinger et al. 2006), which rather seem to favor sub-Chandrasekhar explosion scenarios. In this thesis, as the title suggests, we have endeavored to establish the existence of exotic, super- and sub-Chandrasekhar limiting mass white dwarfs, in order to explain the aforementioned peculiar SNeIa. This is an extremely important puzzle to solve in order to comprehensively understand the phenomena of SNeIa, which in turn is essential for the correct interpretation of the evolutionary history of the universe. Effects of magnetic field: White dwarfs have been observed to be magnetized, having surface fields as high as 105 − 109 G (Vanlandingham et al. 2005). The interior field of a white dwarf cannot be probed directly but it is quite likely that it is several orders of magnitude higher than the surface field. The theory of weakly magnetized white dwarfs has been investigated by a few authors, however, their properties do not starkly contrast with that of the non-magnetized cases (Ostriker & Hartwick 1968). In our venture to find a fundamental basis behind the formation of super-Chandrasekhar white dwarfs, we have explored in this thesis the impact of stronger magnetic fields on the properties of white dwarfs, which has so far been overlooked. We have progressed from a simplistic to a more rigorous, self-consistent model, by adding complexities step by step, as follows: • spherically symmetric Newtonian model with constant (central) magnetic field • spherically symmetric general relativistic model with varying magnetic field • model with self-consistent departure from spherical symmetry by general relativis-tic magnetohydrodynamic (GRMHD) numerical modeling. We have started by exploiting the quantum mechanical effect of Landau quanti-zation due to a maximum allowed equipartition central field greater than a critical value Bc = 4.414 × 1013 G. To begin with, we have carried out the calculations in a Newtonian framework assuming spherically symmetric white dwarfs. The primary ef-fect of Landau quantization is to stiffen the equation of state (EoS) of the underlying electron degenerate matter in the high density regime, and, hence, yield significantly super-Chandrasekhar white dwarfs having mass much & 2M⊙ (Das & Mukhopadhyay 2012a,b). Consequently, we have proposed a new mass limit for magnetized white dwarfs which may establish the aforementioned peculiar, over-luminous SNeIa as new standard candles (Das & Mukhopadhyay 2013a,b). We have furthermore predicted possible evo-lutionary scenarios by which super-Chandrasekhar white dwarfs could form by accretion on to a commonly observed magnetized white dwarf, by invoking the phenomenon of flux freezing, subsequently ending in over-luminous, super-Chandrasekhar SNeIa (Das et al. 2013). Before moving on to a more complex model, we have justified the assumptions in our simplistic model, in the light of various related physics issues (Das & Mukhopad-hyay 2014b), and have also clarified, and, hence, removed some serious misconceptions regarding our work (Das & Mukhopadhyay 2015c). Next, we have considered a more self-consistent general relativistic framework. We have obtained stable solutions of magnetostatic equilibrium models for white dwarfs pertaining to various magnetic field profiles, however, still in spherical symmetry. We have showed that in this framework, a maximum stable mass as high as ∼ 3.3M⊙ can be realized (Das & Mukhopadhyay 2014a). However, it is likely that the anisotropic effect due to a strong magnetic field may cause a deformation in the spherical structure of the white dwarfs. Hence, in order to most self-consistently take into account this departure from spherical symmetry, we have constructed equilibrium models of strongly magnetized, static, white dwarfs in a general relativistic framework, first time in the literature to the best of our knowledge. In order to achieve this, we have modified the GRMHD code XNS (Pili et al. 2014), to apply it in the context of white dwarfs. Interestingly, we have found that signifi-cantly super-Chandrasekhar white dwarfs, in the range ∼ 1.7 − 3.4M⊙, are obtained for many possible field configurations, namely, poloidal, toroidal and mixed (Das & Mukhopadhyay 2015a). Furthermore, due to the inclusion of deformation caused by a strong magnetic field, super-Chandrasekhar white dwarfs are obtained for relatively lower central magnetic field strengths (∼ 1014 G) compared to that in the simplistic model — as correctly speculated in our first work of this series (Das & Mukhopadhyay 2012a). We have also found that although the characteristic deformation induced by a purely toroidal field is prolate, the overall shape remains quasi-spherical — justifying our earlier spherically symmetric assumption while constructing at least some models of strongly magnetized white dwarfs (Das & Mukhopadhyay 2014a). Indeed more accurate and extensive numerical analysis seems to have validated our analytical findings. Thus, very interestingly, our investigation has established that magnetized white dwarfs can indeed have mass that significantly exceeds the Chandrasekhar limit, irre-spective of the origin of the underlying magnetic effect — a discovery which is not only of theoretical importance, but also has a direct astrophysical implication in explaining the progenitors of the peculiar, over-luminous, super-Chandrasekhar SNeIa. Effects of modified Einstein’s gravity: A large array of models has been required to explain the peculiar, over- and under- luminous SNeIa. However, it is unlikely that nature would seek mutually antagonistic scenarios to exhibit sub-classes of apparently the same phenomena, i.e., triggering of thermonuclear explosions in white dwarfs. Hence, driven by the aim to establish a unification theory of SNeIa, we have invoked in the last part of this thesis a modification to Einstein’s theory of general relativity in white dwarfs. The validity of general relativity has been tested mainly in the weak field regime, for example, through laboratory experiments and solar system tests. However, the question remains, whether general relativity requires modification in the strong gravity regime, such as, the expanding universe, the region close to a black hole and neutron star. For instance, there is evidence from observational cosmology that the universe has undergone two epochs of cosmic acceleration, the theory behind which is not yet well understood. The period of acceleration in the early universe is known as inflation, while the current accelerated expansion is often explained by invoking a mysterious dark energy. An alternative approach to explain the mysteries of inflation and dark energy is to modify the underlying gravitational theory itself, as it conveniently avoids involving any exotic form of matter. Several modified gravity theories have been proposed which are extensions of Einstein’s theory of general relativity. A popular class of such theories is known as f (R) gravity (e.g. see de Felice & Tsujikawa 2010), where the Lagrangian density f of the gravitational field is an arbitrary function of the Ricci scalar R. In the context of astrophysical compact objects, so far, modified gravity theories have been applied only to neutron stars, which are much more compact than white dwarfs, in order to test the validity of such theories in the strong field regime (e.g. Cooney et al. 2010; Arapoˇglu et al. 2011). Moreover, a general relativistic correction itself does not seem to modify the properties of a white dwarf appreciably when compared to Newtonian calculations. Our venture of exploring modified gravity in white dwarfs in this thesis, is a first in the literature to the best of our knowledge. We have exploited the advantage that white dwarfs have over neutron stars, i.e., their EoS is well established. Hence, any change in the properties of white dwarfs can be solely attributed to the modification of the underlying gravity, unlike in neutron stars, where similar effects could be produced by invoking a different EoS. We have explored a popular, yet simple, model of f (R) gravity, known as the Starobinsky model (Starobinsky 1980) or R−squared model, which was originally pro-posed to explain inflation. Based on this model, we have first shown that modified gravity reproduces those results which are already explained in the paradigm of general relativity (and Newtonian framework), namely, low density white dwarfs in this context. This is a very important test of the modified gravity model and is furthermore necessary to constrain the underlying model parameter. Next, depending on the magnitude and sign of a single model parameter, we have not only obtained both highly super-Chandrasekhar and highly sub-Chandrasekhar limiting mass white dwarfs, but we have also established them as progenitors of the peculiar, over- and under-luminous SNeIa, respectively (Das & Mukhopadhyay 2015b). Thus, an effectively single underlying the-ory unifies the two apparently disjoint sub-classes of SNeIa, which have so far hugely puzzled astronomers. To summarize, in the first part of the thesis, we have established the enormous significance of magnetic fields in white dwarfs in revealing the existence of significantly super-Chandrasekhar white dwarfs. These super-Chandrasekhar white dwarfs could be ideal progenitors of the peculiar, over-luminous SNeIa, which can, hence, be used as new standard candles of cosmic distance measurements. In the latter part of the thesis, we have established the importance of a modified theory of Einstein’s gravity in revealing both highly super- and highly sub-Chandrasekhar limiting mass white dwarfs. We have furthermore demonstrated how such a theory can serve as a missing link between the peculiar, super- and sub-Chandrasekhar SNeIa. Thus, the significance of the current thesis lies in the fact that it not only questions the uniqueness of the Chandrasekhar mass-limit for white dwarfs, but it also argues for the need of a modified theory of Einstein’s gravity to explain astrophysical observations.

Les comportements critiques des systèmes de mécanique statistique en 2 dimensions ou de mécanique quantique en 1+1 dimensions, ainsi que certains aspects des systèmes sans interactions en 2+1 dimensions, sont efficacement décrits par les méthodes de la théorie des champs conforme et de l'intégrabilité, dont le développement a été spectaculaire au cours des 40 dernières années. Plusieurs problèmes résistent cependant toujours à une compréhension exacte, parmi lesquels celui de la transition entre plateaux dans l'Effet Hall Quantique Entier. La raison principale en est que de tels problèmes sont généralement associés à des théories non unitaires, ou théories conformes logarithmiques, dont la classification se révèle être d'une grande difficulté mathématique. Se tournant vers la recherche de modèles discrets (chaînes de spins, modèles sur réseau), dans l'espoir en particulier d'en trouver des représentations en termes de modèles exactement solubles (intégrables), on se heurte à la deuxième difficulté représentée par le fait que les théories associées sont la plupart du temps non compactes, ou en d'autres termes qu'elles donnent lieu à un continuum d'exposants critiques. En effet, le lien entre modèles discrets et théories des champs non compactes est à ce jour loin d'être compris, en particulier il a longtemps été cru que de telles théories ne pouvaient pas émerger comme limites continues de modèles discrets construits à partir d'un ensemble compact de degrés de libertés, par ailleurs les seuls qui donnent a accès à une construction systématique de solutions exactes.Dans cette thèse, on montre que le monde des modèles discrets compacts ayant une limite continue non compacte est en fait beaucoup plus grand que ce que les quelques exemples connus jusqu'ici auraient pu laisser suspecter. Plus précisément, on y présente une solution exacte par ansatz de Bethe d'une famille infinie de modèles(les modèles $a_n^{(2)}$, ainsi que quelques résultats sur les modèles $b_n^{(1)}$, où il est observé que tous ces modèles sont décrits dans un certain régime par des théories conformes non compactes. Parmi ces modèles, certains jouent un rôle important dans la description de phénomènes physiques, parmi lesquels la description de polymères en deux dimensions avec des interactions attractives et des modèles de boucles impliqués dans l'étude de modèles de Potts couplés ou dans une tentative de description de la transition entre plateaux dans l'Effet Hall par un modèle géométrique compact.On montre que l'existence insoupçonnéede limite continues non compacts pour de tels modèles peut avoir d'importantes conséquences pratiques, par exemple dans l'estimation numérique d'exposants critiques ou dans le résultats de simulations de Monte Carlo. Nos résultats sont appliqués à une meilleure compréhension de la transition theta décrivant l'effondrement des polymères en deux dimensions, et des perspectives pour une potentielle compréhension de la transition entre plateaux en termes de modèles sur réseaux sont présentées. / The critical points of statistical mechanical systems in 2 dimensions or quantum mechanical systems in 1+1 dimensions (this also includes non interacting systems in 2+1 dimensions) are effciently tackled by the exact methods of conformal fieldtheory (CFT) and integrability, which have witnessed a spectacular progress during the past 40 years. Several problems have however escaped an exact understanding so far, among which the plateau transition in the Integer Quantum Hall Effect,the main reason for this being that such problems are usually associated with non unitary, logarithmic conformal field theories, the tentative classification of which leading to formidable mathematical dificulties. Turning to a lattice approach, andin particular to the quest for integrable, exactly sovable representatives of these problems, one hits the second dificulty that the associated CFTs are usually of the non compact type, or in other terms that they involve a continuum of criticalexponents. The connection between non compact field theories and lattice models or spin chains is indeed not very clear, and in particular it has long been believed that the former could not arise as the continuum limit of discrete models built out of acompact set of degrees of freedom, which are the only ones allowing for a systematic construction of exact solutions.In this thesis, we show that the world of compact lattice models/spin chains with a non compact continuum limit is much bigger than what could be expected from the few particular examples known up to this date. More precisely we propose an exact Bethe ansatz solution of an infinite family of models (the so-called $a_n^{(2)}$ models, as well as some results on the $b_n^{(1)}$ models), and show that all of these models allow for a regime described by a non compact CFT. Such models include cases ofgreat physical relevance, among which a model for two-dimensional polymers with attractive interactions and loop models involved in the description of coupled Potts models or in a tentative description of the quantum Hall plateau transition by somecompact geometrical truncation. We show that the existence of an unsuspected non compact continuum limit for such models can have dramatic practical effects, for instance on the output of numerical determination of the critical exponents or ofMonte-Carlo simulations. We put our results to use for a better understanding of the controversial theta transition describing the collapse of polymers in two dimensions, and draw perspectives on a possible understanding of the quantum Hall plateautransition by the lattice approach.

The idea of holography [1, 2] finds a concrete realization in form of the AdS/CFT correspondence [3, 4]. This duality relates a field theory with conformal symmetries to quantum gravity living in one higher dimension. In this thesis we study aspects of black hole quasinormal modes, higher spin theories and entanglement entropy in the context of this duality. In almost all cases we have been able to subject the duality to some precision tests. Quasinormal modes encode the spectrum of black holes and the time-scale of pertur- bations therein [5]. From the dual CFT viewpoint they are the poles of retarded Green's function (or peaks in the spectral function) [6]. Quasinormal modes were previously studied for scalar, gauge field and fermion fluctuations [7]. We solve for these quasinormal modes of higher spin (s _ 2) fields in the background of the BTZ black hole [8, 9]. We obtain an exact solution for a field of arbitrary spin s (integer or half-integer) in the BTZ background. This implies that the BTZ is perhaps the only known black hole background where such an analysis can be done analytically for all bosonic and fermionic fields. The quasinormal modes are shown to match precisely with the poles of the corresponding Green's function in the CFT living on the boundary. Furthermore, we show that one-loop determinants of higher spin fields can also be written as a product form [10] in terms of these quasinormal modes and this agrees with the same obtained by integrating the heat-kernel [11]. We then turn our attention to dualities relating higher-spin gravity to CFTs with W algebra symmetries. Since higher spin gravity does go beyond diffeomorphism invariance, one needs re_ned notions of the usual concepts in differential geometry. For example, in general relativity black holes are defined by the presence of the horizon. However, higher spin gravity has an enlarged group of symmetries of which the diffeomorphisms form a subgroup. The appropriate way of thinking of solutions in higher spin gravity is via characterizations which are gauge invariant [12, 13]. We study classical solutions embedded in N = 2 higher spin supergravity. We obtain a general gauge-invariant condition { in terms of the odd roots of the superalgebra and the eigenvalues of the holonomy matrix of the background { for the existence of a Killing spinor such that these solutions are supersymmetric [14]. We also study black holes in higher spin supergravity and show that the partition function of these black holes match exactly with that obtained from a CFT with the same asymptotic symmetry algebra [15]. This involved studying the asymptotic symmetries of the black hole and thereby developing the holographic dictionary for the bulk charges and chemical potentials with the corresponding quantities of the CFT. We finally investigate entanglement entropy in the AdS3/CFT2 context. Entanglement entropy is an useful non-local probe in QFT and many-body physics [16]. We analytically evaluate the entanglement entropy of the free boson CFT on a circle at finite temperature (i.e. on a torus) [17]. This is one of the simplest and well-studied CFTs. The entanglement entropy is calculated via the replica trick using correlation functions of bosonic twist operators on the torus [18]. We have then set up a systematic high temperature expansion of the Renyi entropies and determined their finite size corrections. These _nite size corrections both for the free boson CFT and the free fermion CFT were then compared with the one-loop corrections obtained from bulk three dimensional handlebody spacetimes which have higher genus Riemann surfaces (replica geometry) as its boundary [19]. One-loop corrections in these geometries are entirely determined by the spectrum of the excitations present in the bulk. It is shown that the leading _nite size corrections obtained by evaluating the one-loop determinants on these handlebody geometries exactly match with those from the free fermion/boson CFTs. This provides a test for holographic methods to calculate one-loop corrections to entanglement entropy. We also study conformal field theories in 1+1 dimensions with W-algebra symmetries at _nite temperature and deformed by a chemical potential (_) for a higher spin current. Using OPEs and uniformization techniques, we show that the order _2 correction to the Renyi and entanglement entropies (EE) of a single interval in the deformed theory is universal [20]. This universal feature is also supported by explicit computations for the free fermion and free boson CFTs { for which the EE was calculated by using the replica trick in conformal perturbation theory by evaluating correlators of twist fields with higher spin operators [21]. Furthermore, this serves as a verification of the holographic EE proposal constructed from Wilson lines in higher spin gravity [22, 23]. We also examine relative entropy [24] in the context of higher-spin holography [25]. Relative entropy is a measure of distinguishability between two quantum states. We confirm the expected short-distance behaviour of relative entropy from holography. This is done by showing that the difference in the modular Hamiltonian between a high-temperature state and the vacuum matches with the difference in the entanglement entropy in the short-subsystem regime.