This book presents a detailed study of the Lanczos potential in general relativity by using tetrad formalisms. It demonstrates that these formalisms offer some simplifications over the tensorial methods, and investigates a general approach to finding the Lanczos potential for algebraic space-time by translating all the tensorial relations concerning the Lanczos potential into the language of tetrad formalisms and using the Newman-Penrose and Geroch-Held-Penrose formalisms. In addition, the book obtains the Lanczos potential for perfect fluid space-time, and applies the results to cosmological models of the universe. In closing, it highlights other methods, apart from tetrad formalisms, for finding the Lanczos potential, as well as further applications of the Newman-Penrose formalism. Given its scope, the book will be of interest to pure mathematicians, theoretical physicists and cosmologists, and will provide common ground for communication among these scientific communities.

The ultimate proofs that black holes exist have been obtained very recently thanks to the detection of gravitational waves from their coalescence and due to material orbiting at a distance of some gravitational radii imaged by optical interferometry or X-ray reverberation mapping. This book provides three comprehensive and up-to-date reviews covering the gravitational wave breakthrough, our understanding of accretion and feedback in supermassive black holes and the relevance of black holes for the Universe since the Big Bang. Neil J. Cornish presents gravitational wave emission from black hole mergers and the physics of detection. Andrew King reviews the physics of accretion on to supermassive black holes and their feedback on host galaxies. Tiziana Di Matteo addresses our understanding of black hole formation at cosmic dawn, the emergence of the first quasars, black hole merging and structure formation. The topics covered by the 48th Saas-Fee Course provide a broad overview of the importance of black holes in modern astrophysics.

Der Wunsch nach einem Verstandnis von Einsteins Theorien ist unter naturwissenschaftlich Interessierten weit verbreitet - und bleibt doch meist unerfullt. Dieses Buch bietet nun eine einzigartige neue Chance: Mit anschaulichen Gedankenexperimenten, exakten Abbildungen, treffenden Analogien und mit strikt auf Mittelschulmathematik beschrankten Rechenschritten werden Sie behutsam in die immer wieder faszinierende Welt der Relativitatstheorien gefuhrt. Sachlich, grundlich und dennoch faszinierend werden die Zeitdehnung, das Zwillingsparadoxon, Schwarze Locher oder die Rotverschiebung des Lichts dargestellt, daneben viele weitere relativistische Effekte, die Ihnen hier erstmals mit ganz einfachen mathematischen Werkzeugen zuganglich gemacht werden.

..". das Buch ist sehr empfehlenswert: Der gesamte Text ist klar, ausfuhrlich und verstandlich geschrieben."

"Ein ausserst gelungenes Buch also, das halt, was es im Untertitel verspricht ..."

Stephan Edinger, Sterne und Weltraum, Mai 2008"

These three lectures cover a certain aspect of complexity and black holes, namely the relation to the second law of thermodynamics. The first lecture describes the meaning of quantum complexity, the analogy between entropy and complexity, and the second law of complexity. Lecture two reviews the connection between the second law of complexity and the interior of black holes. Prof. L. Susskind discusses how firewalls are related to periods of non-increasing complexity which typically only occur after an exponentially long time. The final lecture is about the thermodynamics of complexity, and "uncomplexity" as a resource for doing computational work. The author explains the remarkable power of "one clean qubit," in both computational terms and in space-time terms. This book is intended for graduate students and researchers who want to take the first steps towards the mysteries of black holes and their complexity.

This exploration of the global structure of spacetime within the context of general relativity examines the causal and singular structures of spacetime, revealing some of the curious possibilities that are compatible with the theory, such as 'time travel' and 'holes' of various types. Investigations into the epistemic and modal structures of spacetime highlight the difficulties in ruling out such possibilities, unlikely as they may seem at first. The upshot seems to be that what counts as a 'physically reasonable' spacetime structure in modern physics is far from clear.

A quirky, funny, and accessible blend of science and art that delves into the heart of Einstein’s theory of relativity

It was a link to Albert Einstein’s 1905 paper―an early attempt at explaining his revolutionary ideas on space, time, and matter―that drew Tanya Bub into his imaginative vision of the world. What particularly struck her was how Einstein interwove words and math to create clear visuals illustrating his theories. As an artist, she naturally started doodling as she worked her way through his concepts, creating drawings that intuitively demonstrated Einstein’s core principles.

In Reimagining Time, Tanya Bub teams up with her father, the distinguished physicist Jeffrey Bub, to create a quirky and accessible take on one of science’s most revolutionary discoveries. Blending original art and text, they guide readers―even nonmathematicians―through Einstein’s theory of special relativity to reveal truths about our universe: time is relative, lengths get shorter with motion, energy and mass are interchangeable, and the universe has a speed limit.

This volume presents a selection of articles based on inspiring lectures held at the "Capri" Advanced Summer School, an original event conceived and promoted by Leonida Antonio Gizzi and Ralph Assmann that focuses on novel schemes for plasma-based particle acceleration and radiation sources, and which brings together researchers from the conventional accelerator community and from the high-intensity laser-matter interaction research fields. Training in these fields is highly relevant for ultra-intense lasers and applications, which have enjoyed dramatic growth following the development of major European infrastructures like the Extreme Light Infrastructure (ELI) and the EuPRAXIA project. The articles preserve the tutorial character of the lectures and reflect the latest advances in their respective fields. The volume is mainly intended for PhD students and young researchers getting started in this area, but also for scientists from other fields who are interested in the latest developments. The content will also appeal to radiobiologists and medical physicists, as it includes contributions on potential applications of laser-based particle accelerators.

Presenting the history of space-time physics, from Newton to Einstein, as a philosophical development DiSalle reflects our increasing understanding of the connections between ideas of space and time and our physical knowledge. He suggests that philosophy's greatest impact on physics has come about, less by the influence of philosophical hypotheses, than by the philosophical analysis of concepts of space, time and motion, and the roles they play in our assumptions about physical objects and physical measurements. This way of thinking leads to interpretations of the work of Newton and Einstein and the connections between them. It also offers ways of looking at old questions about a priori knowledge, the physical interpretation of mathematics, and the nature of conceptual change. Understanding Space-Time will interest readers in philosophy, history and philosophy of science, and physics, as well as readers interested in the relations between physics and philosophy.

This book contains a systematic exposition of the theory of spinors in finite-dimensional Euclidean and Riemannian spaces. The applications of spinors in field theory and relativistic mechanics of continuous media are considered. The main mathematical part is connected with the study of invariant algebraic and geometric relations between spinors and tensors. The theory of spinors and the methods of the tensor representation of spinors and spinor equations are thoroughly expounded in four-dimensional and three-dimensional spaces. Very useful and important relations are derived that express the derivatives of the spinor fields in terms of the derivatives of various tensor fields. The problems associated with an invariant description of spinors as objects that do not depend on the choice of a coordinate system are addressed in detail. As an application, the author considers an invariant tensor formulation of certain classes of differential spinor equations containing, in particular, the most important spinor equations of field theory and quantum mechanics. Exact solutions of the Einstein-Dirac equations, nonlinear Heisenberg's spinor equations, and equations for relativistic spin fluids are given. The book presents a large body of factual material and is suited for use as a handbook. It is intended for specialists in theoretical physics, as well as for students and post-graduate students of physical and mathematical specialties.

After an extensive introduction to the asymptotic safety approach to quantum gravity, this thesis explains recent key advances reported in four influential papers. Firstly, two exact solutions to the reconstruction problem (how to recover a bare action from the effective average action) are provided. Secondly, the fundamental requirement of background independence in quantum gravity is successfully implemented. Working within the derivative expansion of conformally reduced gravity, the notion of compatibility is developed, uncovering the underlying reasons for background dependence generically forbidding fixed points in such models. Thirdly, in order to understand the true nature of fixed-point solutions, one needs to study their asymptotic behaviour. The author carefully explains how to find the asymptotic form of fixed point solutions within the f(R) approximation. Finally, the key findings are summarised and useful extensions of the work are identified. The thesis finishes by considering the need to incorporate matter into the formalism in a compatible way and touches upon potential opportunities to test asymptotic safety in the future.

This book offers a detailed and stimulating account of the Lagrangian, or variational, approach to general relativity and beyond. The approach more usually adopted when describing general relativity is to introduce the required concepts of differential geometry and derive the field and geodesic equations from purely geometrical properties. Demonstration of the physical meaning then requires the weak field approximation of these equations to recover their Newtonian counterparts. The potential downside of this approach is that it tends to suit the mathematical mind and requires the physicist to study and work in a completely unfamiliar environment. In contrast, the approach to general relativity described in this book will be especially suited to physics students. After an introduction to field theories and the variational approach, individual sections focus on the variational approach in relation to special relativity, general relativity, and alternative theories of gravity. Throughout the text, solved exercises and examples are presented. The book will meet the needs of both students specializing in theoretical physics and those seeking a better understanding of particular aspects of the subject.

Describes the branch of astronomy in which processes in the universe are investigated with experimental methods employed in particle-physics experiments. After a historical introduction the basics of elementary particles, Explains particle interactions and the relevant detection techniques, while modern aspects of astroparticle physics are described in a chapter on cosmology. Provides an orientation in the field of astroparticle physics that many beginners might seek and appreciate because the underlying physics fundamentals are presented with little mathematics, and the results are illustrated by many diagrams. Readers have a chance to enter this field of astronomy with a book that closes the gap between expert and popular level.

Unlike most traditional introductory textbooks on relativity and cosmology that answer questions like "Does accelerated expansion pull our bodies apart?", "Does the presence of dark matter affect the classical tests of general relativity?" in a qualitative manner, the present text is intended as a foundation, enabling students to read and understand the textbooks and many of the scientific papers on the subject. And, above all, the readers are taught and encouraged to do their own calculations, check the numbers and answer the above and other questions regarding the most exciting discoveries and theoretical developments in general relativistic cosmology, which have occurred since the early 1980s. In comparison to these intellectual benefits the text is short. In fact, its brevity without neglect of scope or mathematical accessibility of key points is rather unique. The authors connect the necessary mathematical concepts and their reward, i.e. the understanding of an important piece of modern physics, along the shortest path. The unavoidable mathematical concepts and tools are presented in as straightforward manner as possible. Even though the mathematics is not very difficult, it certainly is beneficial to know some statistical thermodynamics as well as some quantum mechanics. Thus the text is suitable for the upper undergraduate curriculum.

This book presents a novel mathematical formalism, based on the tetrad formulation of differential geometry, for describing cosmological observables exactly and conveniently. It covers all the standard observables, i.e. distances, weak lensing, number counts and cosmic microwave background, and also includes a detailed derivation of general-relativistic matrix kinetic theory. All the fully nonlinear equations are derived in detail and the mathematical content is self-contained, so that readers require only a basic knowledge of general relativity. Moreover, the authors discuss several subtle issues that are usually overlooked in the literature and, in particular, issues that distinguish this formalism from the more approximative standard practice.

The goal of this essay is to discuss the future of discovery in particle physics. Its primary motivation is the 2019 European Strategy update, which aims to determine the future experimental and theoretical priorities for particle physics. A key question is to understand what the standard theory (Standard Model) of particle physics really is, which the author argues has been a foggy notion for several decades which he clarifies. It then is to decide what motivated beyond the Standard Model theories are to be targeted by experiment. This book brightly exposes these theories, and puts current particle physics research into its historical context and points the way toward future work.

This primer is a collection of notes based on lectures that were originally given at IIT Madras (India) and at IFT Madrid (Spain). It is a concise and pragmatic course on applied holography focusing on the basic analytic and numerical techniques involved. The presented lectures are not intended to provide all the fundamental theoretical background, which can be found in the available literature, but they concentrate on concrete applications of AdS/CFT to hydrodynamics, quantum chromodynamics and condensed matter. The idea is to accompany the reader step by step through the various benchmark examples with a classmate attitude, providing details for the computations and open-source numerical codes in Mathematica, and sharing simple tricks and warnings collected during the author's research experience. At the end of this path, the reader will be in possess of all the fundamental skills and tools to learn by him/herself more advanced techniques and to produce independent and novel research in the field.

The full inside story of the detection of gravitational waves at LIGO, one of the most ambitious feats in scientific history *Selected as a Book of the Year 2016 in the Sunday Times* 'This is empirical poetry. A fascinating tale of human curiosity beautifully told, and with black holes and lasers too' Robin Ince In 1916 Albert Einstein predicted the existence of gravitational waves: miniscule ripples in the very fabric of spacetime generated by unfathomably powerful events. If such vibrations could somehow be recorded, we could observe our universe for the first time through sound: the hissing of the Big Bang, the low tones of merging galaxies, the drumbeat of two black holes collapsing into one... In 2016 a team of hundreds of scientists at work on a billion-dollar experiment made history when they announced the first ever detection of a gravitational wave, confirming Einstein's prediction a century ago. Based on complete access to LIGO (Laser Interferometer Gravitational-Wave Observatory) and the scientists who created it, Black Hole Blues offers a first-hand account of this astonishing achievement: an intimate story of cutting-edge science at its most awe-inspiring and ambitious.

Ryan Wasserman explores a range of fascinating questions raised by the possibility of time travel. This volume explores a wide-range of puzzles such as the grandfather paradox, the bootstrapping paradox, and the twin paradox of special relativity. Ryan Wasserman draws out their implications for our understanding of time, tense, freedom, fatalism, causation, counterfactuals, laws of nature, persistence, change, and mereology. Paradoxes of Time Travel is written in an accessible style, and filled with entertaining examples from physics, science fiction, and popular culture.

This book provides an introduction to the mathematics and physics of general relativity, its basic physical concepts, its observational implications, and the new insights obtained into the nature of space-time and the structure of the universe. It introduces some of the most striking aspects of Einstein's theory of gravitation: black holes, gravitational waves, stellar models, and cosmology. It contains a self-contained introduction to tensor calculus and Riemannian geometry, using in parallel the language of modern differential geometry and the coordinate notation, more familiar to physicists. The author has strived to achieve mathematical rigour, with all notions given careful mathematical meaning, while trying to maintain the formalism to the minimum fit-for-purpose. Familiarity with special relativity is assumed. The overall aim is to convey some of the main physical and geometrical properties of Einstein's theory of gravitation, providing a solid entry point to further studies of the mathematics and physics of Einstein equations.

Special relativity is the basis of many fields in modern physics: particle physics, quantum field theory, high-energy astrophysics, etc. This theory is presented here by adopting a four-dimensional point of view from the start. An outstanding feature of the book is that it doesn't restrict itself to inertial frames but considers accelerated and rotating observers. It is thus possible to treat physical effects such as the Thomas precession or the Sagnac effect in a simple yet precise manner. In the final chapters, more advanced topics like tensorial fields in spacetime, exterior calculus and relativistic hydrodynamics are addressed. In the last, brief chapter the author gives a preview of gravity and shows where it becomes incompatible with Minkowsky spacetime. Well illustrated and enriched by many historical notes, this book also presents many applications of special relativity, ranging from particle physics (accelerators, particle collisions, quark-gluon plasma) to astrophysics (relativistic jets, active galactic nuclei), and including practical applications (Sagnac gyrometers, synchrotron radiation, GPS). In addition, the book provides some mathematical developments, such as the detailed analysis of the Lorentz group and its Lie algebra. The book is suitable for students in the third year of a physics degree or on a masters course, as well as researchers and any reader interested in relativity. Thanks to the geometric approach adopted, this book should also be beneficial for the study of general relativity. "A modern presentation of special relativity must put forward its essential structures, before illustrating them using concrete applications to specific dynamical problems. Such is the challenge (so successfully met!) of the beautiful book by Eric Gourgoulhon." (excerpt from the Foreword by Thibault Damour)

This book serves two main purposes: firstly, it shows, in a simple way, how the possible existence of an extra-spatial dimension would affect the predictions of four-dimensional General Relativity, a model known as the Brane world; secondly, it explains, step-by-step, a new technique called Minimal Geometric Deformation, which was introduced for the purpose of solving the correspondingly modified Einstein field equations. This method gave rise to the Gravitational Decoupling in General Relativity, which is widely used to solve the Einstein field equations in various contexts.

At its most basic, philosophy is about learning how to think about the world around us. It should come as no surprise, then, that children make excellent philosophers! Naturally inquisitive, pint-size scholars need little prompting before being willing to consider life's "big questions," however strange or impractical. Plato & Co. introduces children and curious grown-ups to the lives and work of famous philosophers, from Socrates to Descartes, Einstein, Marx, and Wittgenstein. Each book in the series features an engaging and often funny story that presents basic tenets of philosophical thought alongside vibrant color illustrations. In Albert Einstein's Flashes of Inspiration, the young Albert Einstein has a very important job: he must deliver electricity to the big Oktoberfest celebration in Munich. As he hurries from one merry-go-round to another, nothing seems to be going as planned. With his sister, Maja, Heinrich the dog, and Niels Bohr, a qualified dwarf-thrower, can he win a battle against the laws of the universe? The key just may lie in the question of whether a dumpling can fly faster than light?

This book derives and analyzes all solutions to the Kepler problem with dark energy (DE), presenting significant results such as: (a) all radial infinite motions obey Hubble's law at large times; (b) all orbital infinite motions are asymptotically radial and obey Hubble's law; (c) infinite orbital motions strongly dominate the finite ones. This clearly shows the effect of repulsive DE: In the classical Kepler problem, all orbital motions are finite for negative energies and infinite in the opposite case. Another DE effect is spatial localization of bounded orbits: mostly, they are within the equilibrium sphere, where the attractive Newtonian force outbalances the repulsive force of DE. This problem is of particular current interest due to recent studies of the local flows of galaxies showing domination of DE in their dynamics; the book discusses this observation in detail.

This fascinating book provides an accessible and up-to-date overview of modern cosmology. In particular, the book discusses the formation of the Cosmic Microwave Background and the evolution of large scale structures in the universe, the distribution of galaxies and clusters of galaxies on very large distance scales. Following a brief introduction, the authors describe the scientific method - how science is done. They then discuss observational cosmology, the instruments and what observations can be done with them, and what is derived from those observations. After discussing the constituents of the universe, including dark matter and dark energy, the authors provide an outline of the forces that shape the universe, with particular emphasis on gravitation. Following this, the reader is taken on a journey in time from the present day back to the very beginning of the universe, a period called inflation, which sets the initial conditions for the subsequent evolution of the universe. The book ends with a brief chapter on what lies beyond. Written by two experts, the book is aimed at the interested lay-person with little or no physics background, but an interest in modern cosmology.

A readable and entertaining look at how Einstein's special theory of relativity gives us a new understanding of the nature of time Relativity ought to be an important part of everyone's education. Its subject is time, with which we all think we are familiar. Einstein's special theory of relativity reveals that some of our most intuitive notions about time are shockingly wrong. This clear, lively, and informal exposition of special relativity takes a highly original approach to introduce readers to the true nature of time. It is accessible to anyone who remembers a little high school algebra and elementary geometry. It's About Time offers deep insights to curious readers who have no technical scientific background.