Introduction to General Relativity

An Elementary Guide to the State of the Art in the Quantum Information Field Introduction to Quantum Physics and Information Processing guides beginners in understanding the current state of research in the novel, interdisciplinary area of quantum information. Suitable for undergraduate and beginning graduate students in physics, mathematics, or engineering, the book goes deep into issues of quantum theory without raising the technical level too much. The text begins with the basics of quantum mechanics required to understand how two-level systems are used as qubits. It goes on to show how quantum properties are exploited in devising algorithms for problems that are more efficient than the classical counterpart. It then explores more sophisticated notions that form the backbone of quantum information theory. Requiring no background in quantum physics, this text prepares readers to follow more advanced books and research material in this rapidly growing field. Examples, detailed discussions, exercises, and problems facilitate a thorough, real-world understanding of quantum information.

As gravitational wave astronomy prepares for the first detections of gravitational waves from compact-object binary inspirals, theoretical work is required on the study of (i) gravitational-wave sources, (ii) the signals emitted by those sources, and (iii) the searches for those signals in detector data. This thesis describes work on all three fronts. (i) We discuss intermediate-mass-ratio inspirals (IMRIs) of black holes or neutron stars into intermediate-mass black holes (IMBHs) that could be detected with Advanced LIGO. We analyze different mechanisms of IMRI formation and compute IMRI event rates of up to tens of events per year for Advanced LIGO. We study the spin evolution of IMBHs that grow through a series of minor mergers. We explore how a deviation of an IMRI's central body from a Kerr black hole influences geodesics, including the possibility of chaotic orbital dynamics. We also address the scientific consequences of extreme-mass-ratio inspiral (EMRI) detections by LISA for astrophysics and general relativity, and the difficulties associated with detecting and analyzing EMRI signals. (ii) We study the periodic standing-wave approximation (PSWA), which can potentially provide accurate waveforms in the last inspiral cycles of a comparable-mass black-hole binary. Using a simple model, we find that the solution to Einstein's equations for inspiraling black holes can be recovered to a high accuracy by the addition a perturbative radiation-reaction field to the standing-wave, noninspiraling solution. (iii) We demonstrate the utility of searching for and analyzing tracks in time-frequency spectrograms of a gravitational-wave signal as a means of estimating the parameters of a massive black-hole binary inspiral, as observed by LISA.

Starting from Newton's gravitational theory, we give a general introduction into the spherically symmetric solution of Einstein's vacuum field equation, the Schwarzschild(–Droste) solution, and into one specific stationary axially symmetric solution, the Kerr solution. The Schwarzschild solution is unique and its metric can be interpreted as the exterior gravitational field of a spherically symmetric mass. The Kerr solution is only unique if the multipole moments of its mass and its angular momentum take on prescribed values. Its metric can be interpreted as the exterior gravitational field of a suitably rotating mass distribution. Both solutions describe objects exhibiting an event horizon, a frontier of no return. The corresponding notion of a black hole is explained to some extent. Eventually, we present some generalizations of the Kerr solution.

The notion of black holes has changed mightily over the past couple of centuries, especially in conjunction with Einstein's theory of general relativity, offering a scaffold for understanding its extreme gravitational properties. This paper will provide an overview of the theoretical principles underlying black holes, with particular emphasis on two important answers to Einsteins field equations: the Schwarzschild solution, which outlines uncharged and non-rotating black holes, and the Kerr solution, which characterizes rotating black holes. We are focusing our conversation on Sagittarius A* (Sgr A*), the enormous black hole situated at the heart of the Milky Way galaxy. This work will demonstrate that, based on observational dataespecially gravitational waves and dual collimated outflowsthere is substantial justification for considering Sgr A* modelled as a Kerr black hole, which indicates significant angular momentum. This carries more general implications for understanding black hole dynamics and what they mean for galaxy formation. More detailed measurements, such as those taken via radio velocity observations and future gravitational wave detectors.

Plane walls (including plane domain walls) without reflection symmetry are studied in the framework of Einstein's general relativity. Using the distribution theory, all the Einstein field equations and Bianchi identities are split into two groups: one holding in the regions outside of the wall and the other holding at the wall. The Einstein field equations at the wall are found to take a very simple form, and given explicitly in terms of the discontinuities of the metric coefficients and their derivatives. The Bianchi identities at the wall are also given explicitly. Using the latter, the interaction of a plane wall with gravitational waves and some specific matter fields is studied. In particular, it is found that, when a gravitational plane wave passes through a wall, if the wall has no reflection symmetry, the phenomena, such as reflection, stimulation, or absorption, in general, occur. It is also found that, unlike for gravitational waves, a massless scalar wave or an electromagnetic wave continuously passes through a wall without any reflection. The repulsion and attraction of a plane wall are also studied. It is found that the acceleration of an observer who is at rest relative to the wall usually consists of threemore » parts: one is due to the force produced by the wall, the second is due to the force produced by the space-time curvature, which is zero if the wall has reflection symmetry, and the last is due to the accelerated motion of the wall. As a result, a repulsive (attractive) plane wall may not be repulsive (attractive) at all. Finally, the collision and interaction among the walls are studied.« less

Chapter 12 - Curvature, Strong Gravity, and Gravitational Waves

This thesis presents recent research into analytic topics in the classical theory of General Relativity. It is a thesis in two parts. The first part features investigations into the spectrum of perturbed, rotating black holes. These include the study of near horizon perturbations, leading to a new generic frequency mode for black hole ringdown; an treatment of high frequency waves using WKB methods for Kerr black holes; and the discovery of a bifurcation of the quasinormal mode spectrum of rapidly rotating black holes. These results represent new discoveries in the field of black hole perturbation theory, and rely on additional approximations to the linearized field equations around the background black hole. The second part of this thesis presents a recently developed method for the visualization of curved spacetimes, using field lines called the tendex and vortex lines of the spacetime. The works presented here both introduce these visualization techniques, and explore them in simple situations. These include the visualization of asymptotic gravitational radiation; weak gravity situations with and without radiation; stationary black hole spacetimes; and some preliminary study into numerically simulated black hole mergers. The second part of thesis culminates in the investigation of perturbed black holes using these field line methods, which have uncovered new insights into the dynamics of curved spacetime around black holes.

Most of us sometimes have to face a student asking: 'What do I need to get started on this'. (In my case 'this' would typically be a topic in general relativity.) After thinking about it for quite a while, and consulting candidate texts again and again, a few days later I usually end up saying: read this chapter in book I (but without going too much detail), then that chapter in book II (but ignore all those comments), then the first few sections of this review paper (but do not try to work out equations NN to NNN), and then come back to see me. In the unlikely event that the student comes back without changing the topic, there follows quite a bit of explaining on a blackboard over the following weeks.

This text on general relativity and its modern applications is suitable for an intensive one-semester course on general relativity, at the level of a Ph.D. student in physics. Assuming knowledge of classical mechanics and electromagnetism at an advanced undergraduate level, basic concepts are introduced quickly, with greater emphasis on their applications. Standard topics are covered, such as the Schwarzschild solution, classical tests of general relativity, gravitational waves, ADM parametrization, relativistic stars and cosmology, as well as more advanced standard topics like vielbein-spin connection formulation, trapped surfaces, the Raychaudhuri equation, energy conditions, the Petrov and Bianchi classifications and gravitational instantons. More modern topics, including black hole thermodynamics, gravitational entropy, effective field theory for gravity, the PPN expansion, the double copy and fluid-gravity correspondence, are also introduced using the language understood by physicists, without too abstract mathematics, proven theorems, or the language of pure mathematics.

Originally published in 1970, this book gives a comprehensive account of the properties of ice, the connections between them and the way in which they derive from the structure of the water molecule and the small mass of the proton. The properties are discussed in terms of quantum mechanics and solid state theory with emphasis on physical principles rather than on theoretical models. The book is intended as an exemplification of the principles of chemical physics for beginning graduate students in physics of physical chemistry and as a text and reference book on the properties of ice for research workers in glaciology, cloud physics, meteorology and associated fields. Although the author assumes a familiarity with fundamental physics, he has taken some trouble to make his account self-contained by reference to the underlying principles in every case or by more detailed discussion where the application is not a standard one.

The conference brought together people working in mathematical general relativity, a field which lies at the interface of analysis, differential geometry and physics. There was a mix of established workers in the subject with participants at the start of their careers and researchers from neighbouring fields. The Einstein equations are at the heart of general relativity theory. They form a system of evolution equations and their solutions are conveniently parametrized by initial data. These initial data are required to satisfy constraint equations and these were the topic of several lectures at the conference. The talks of Corvino and Pollack were concerned with recent progress in methods for constructing solutions of the constraints. An important concept for solutions of the constraints is that of mass, including quasilocal mass. This was a central theme of the talks of Degeratu, Huisken and Shi. Huisken introduced a striking new approach to the definition of mass based on isoperimetric inequalities. Dain presented a variational characterization of the extreme Reissner–Nordström solution, thus relating the study of the constraint equations to the theory of black holes. The talk of Wohlfarth was rather outside the main area of the conference. He described a new concept of geometry motivated by string theory which may come to enrich the circle of ideas within mathematical relativity. Many of the talks at the conference were on evolution equations related to general relativity. Dafermos and Finster presented results related to the dynamical stability of black holes. Dafermos described new results on the rate of decay of solutions of the wave equation on the Schwarzschild spacetime while Finster explained applications of methods of functional analysis to the wave equation on the Kerr spacetime. Struwe's talk concerned uniqueness for supercritical nonlinear wave equations which from the point of view of the Einstein equations are an important example to compare with. The Maxwell equations are another important comparison system and Bauer showed how they can be used to understand more about radiation formulae. Velázquez gave an introduction to singularity formation in the Keller–Segel model, a parabolic system coming from mathematical biology. This is a possible source of insight for obtaining a rigorous understanding of critical collapse in general relativity. The talks of Andréasson, Choptuik and Lindblom dealt with various aspects of the application of numerical techniques to the study of the Einstein evolution equations. Choptuik showed impressive new simulations of coalescing black holes due to Pretorius which could hardly have been imagined just a year ago. Cosmology is at present a very active area of research in general relativity. This is in part due to the challenge of understanding the observed accelerated expansion of our universe. Mathematics is beginning to make its mark in this subject and this was reflected by talks of Heinzle, Rendall and Tod. Mathematical relativity is a meeting point for many ideas and the abstracts which follow give some idea of the variety of the subject. In fact the spectrum of topics discussed by the participants at the conference was much wider than those for which talks could be scheduled. By limiting the number of presentations it was possible to leave plenty of time for people to exchange insights. The lively interactions observed make us hopeful that this conference has given a boost to the development of the subject in the next few years.

This chapter presents some of the basic concepts of general relativity. Topics discussed include notation and conventions, special relativity, manifolds and tensors, the metric tensor, Lie derivatives and Killing fields, coordinate transformationsm, covariant derivatives and geodesics, curvature, Bianchi identities and the Einstein tensor, general relativity, matter and the stress-energy tensor, the Einstein field equations, weak fields and gravitational waves, the Schwarzschild solution and black holes, black holes with charge and angular momentum, and causal structure, singularities and black holes.

According to the preface, this is intended as a textbook for a variety of readers, including resident physicians and beginning graduate students in medical physics. The number of chapters devoted to the different topics covered will give some indication of the relative importance the author assigns to each subject. The first three chapters are concerned with the structure of matter, radioactive decay, and interactions of particulate radiations, while the next four deal with the production of x rays, x-ray circuits and generators, and the fundamental interactions of x and gamma rays with matter. Dosimetry, radiation quality, interaction of radiations with body tissues, radiation fields, and other aspects of the physics of radiotherapy are discussed in Chapters 8-16. The following four chapters cover the physics of nuclear medicine, including radiation detectors, data analysis and handling, gamma-ray spectrometry, measurement of internal radioactivity, and clinical use of radioactive nuclides. Chapters 21 through 24 deal with the physics of diagnostic radiology, including intensifying screens, conventional and television fluoroscopy, cinefluorography, stereoscopic and subtraction radiography, microradiography, computers in diagnostic radiol-

This book describes the theory of electroweak interactions, starting from a level understandable to students with only a first degree in physics. This theory, the Glashow–Salam–Weinberg (GSW) model, unifies the weak and electromagnetic forces of nature and gives a detailed description of the interactions between quarks and leptons, the basic building blocks of matter. The various experimental tests of the model that have been made and that are planned are described in detail, with reference to the fact that all results obtained so far are in agreement with the model. The interactions of quarks by the strong force, the theory of quantum chromodynamics, are also discussed. The GSW model, together with quantum chromodynamics, constitute the so-called 'standard model'. Theories proposing further unification of the forces of nature are outlined. This is a rapidly moving subject, and this up-to-date book will be of great value to researchers and beginning graduate students in high energy physics.

The Laser Interferometric Gravitational-Wave Observatory (LIGO) is designed to detect the Gravitational Waves (GW) predicted by Albert Einstein’s general theory of relativity. The advanced LIGO project is ongoing an upgrade to increase the detection sensitivity by more than a factor of 10, which will make the events detection a routine occurrence. In addition to using higher power lasers, heavier test mass, and better isolation systems, several new designs and techniques are proposed in the long-term upgrade, such as modifying the optics configuration to reduce the quantum noise, active noise cancellation of the Newtonian noise, optimizing the coating structure, and employing non-Guassian laser beams etc. In the first part of my thesis (Chapters 2 and 3), I apply statistical mechanics and elastostatics to the LIGO coated mirrors, and study the thermal fluctuations that dominate advanced LIGO’s most sensitive frequency band from 40 Hz to 200 Hz. In particular, in Chapter 2, I study the so-called coating Brownian noise, fluctuations of mirrors coated with multiple layers of dielectrics due to internal friction. Assuming coating materials to be isotropic and homogeneous, I calculate the cross spectra of Brownian fluctuations in the bulk and shear strains of the coating layers, as well as fluctuations in the height of the coating-substrate interface. The additional phase shifting and back-scattering caused by photo elastic effects are also considered for the first time. In Chapter 3, I study whether it is realistic to adopt higher-order Laguerre-Gauss modes in LIGO, in order to mitigate the effect of mirror thermal noise. We investigate the effect on the detector’s contrast defect caused by the mode degeneracy. With both analytical calculation and numerical simulation, we show that with this approach, the detector’s susceptibility to mirror figure errors is reduced greatly compared to using the nondegenerate modes, therefore making it unacceptable for LIGO requirements. For the future GW detectors, with much lower noises and higher sensitivity, this might be used to investigate the quantum behaviors of macroscopic mechanical objects. In recent years the linear optomechanical systems with cavity modes coupling to a mechanical oscillator have been studied extensively. In the second part of my thesis (Chapter 4), I study the interaction between a single photon and a high-finesse cavity with a movable mirror, in the so-called strong coupling regime, where the recoil of the photon can cause significant change in the momentum of the mirror. The results are applied to analyze the case with a Fabry-Perot cavity. We also present that with engineering the photon wave function, it is possible to prepare the oscillator into an arbitrary quantum state.