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This book is based on lectures given at the first edition of the Domoschool, the International Alpine School in Mathematics and Physics, held in Domodossola, Italy, in July 2018. It is divided into two parts. Part I consists of four sets of lecture notes. These are extended versions of lectures given at the Domoschool, written by well-known experts in mathematics and physics related to General Relativity. Part II collects talks by selected participants, focusing on research related to General Relativity.

Einstein field equations --- Einstein's field equations --- Einstein's gravitational field equations --- Einstein's law of gravitation --- Field equations, Einstein --- Differential equations --- Field theory (Physics) --- General relativity (Physics) --- Gravitational fields --- Gravitation. --- Mathematical physics. --- Classical and Quantum Gravitation, Relativity Theory. --- Mathematical Physics. --- Physical mathematics --- Physics --- Matter --- Antigravity --- Centrifugal force --- Relativity (Physics) --- Mathematics --- Properties

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As the structures in our Universe are mapped out on ever larger scales, and with increasing detail, the use of inhomogeneous models is becoming an essential tool for analyzing and understanding them. This book reviews a number of important developments in the application of inhomogeneous solutions of Einstein's field equations to cosmology. It shows how inhomogeneous models can be employed to study the evolution of structures such as galaxy clusters and galaxies with central black holes, and to account for cosmological observations like supernovae dimming, the cosmic microwave background, baryon acoustic oscillations or the dependence of the Hubble parameter on redshift within classical general relativity. Whatever `dark matter' and `dark energy' turn out to be, inhomogeneities exist on many scales and need to be investigated with all appropriate methods. This book is of great value to all astrophysicists and researchers working in cosmology, from graduate students to academic researchers.

Cosmology --- Einstein field equations. --- Inhomogeneous materials. --- Heterogeneous materials --- Inhomogeneous media --- Media, Inhomogeneous --- Materials --- Matter --- Einstein's field equations --- Einstein's gravitational field equations --- Einstein's law of gravitation --- Field equations, Einstein --- Differential equations --- Field theory (Physics) --- General relativity (Physics) --- Gravitational fields --- Astronomy --- Deism --- Metaphysics --- Mathematics.

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A brief introduction to gravity through Einstein's general theory of relativityOf the four fundamental forces of nature, gravity might be the least understood and yet the one with which we are most intimate. From the months each of us spent suspended in the womb anticipating birth to the moments when we wait for sleep to transport us to other realities, we are always aware of gravity. In On Gravity, physicist A. Zee combines profound depth with incisive accessibility to take us on an original and compelling tour of Einstein's general theory of relativity. Inspired by Einstein's audacious suggestion that spacetime could ripple, Zee begins with the stunning discovery of gravity waves. He goes on to explain how gravity can be understood in comparison to other classical field theories, presents the idea of curved spacetime and the action principle, and explores cutting-edge topics, including black holes and Hawking radiation. Zee travels as far as the theory reaches, leaving us with tantalizing hints of the utterly unknown, from the intransigence of quantum gravity to the mysteries of dark matter and energy.Concise and precise, and infused with Zee's signature warmth and freshness of style, On Gravity opens a unique pathway to comprehending relativity and gaining deep insight into gravity, spacetime, and the workings of the universe.

Einstein field equations. --- General relativity (Physics) --- Gravity. --- Relativity (Physics) --- SCIENCE / Gravity. --- Gravitation --- Nonrelativistic quantum mechanics --- Space and time --- Geophysics --- Mechanics --- Pendulum --- Relativistic theory of gravitation --- Relativity theory, General --- Physics --- Einstein's field equations --- Einstein's gravitational field equations --- Einstein's law of gravitation --- Field equations, Einstein --- Differential equations --- Field theory (Physics) --- Gravitational fields

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This monograph presents a self contained mathematical treatment of the initial value problem for shock wave solutions of the Einstein equations in General Relativity. The first two chapters provide background for the introduction of a locally intertial Glimm Scheme, a non-dissipative numerical scheme for approximating shock wave solutions of the Einstein equations in spherically symmetric spacetimes. What follows is a careful analysis of this scheme providing a proof of the existence of (shock wave) solutions of the spherically symmetric Einstein equations for a perfect fluid, starting from initial density and velocity profiles that are only locally of bounded total variation. The book covers the initial value problems for Einstein's gravitational field equations with fluid sources and shock wave initial data. It has a clearly outlined goal: proving a certain local existence theorem. Concluding remarks are added and commentary is provided throughout. The book will be useful to graduate students and researchers in mathematics and physics.

Einstein field equations. --- Shock waves. --- Shock (Mechanics) --- Waves --- Einstein's field equations --- Einstein's gravitational field equations --- Einstein's law of gravitation --- Field equations, Einstein --- Differential equations --- Field theory (Physics) --- General relativity (Physics) --- Gravitational fields --- Mathematics. --- Applications of Mathematics. --- Classical and Quantum Gravitation, Relativity Theory. --- Math --- Science --- Applied mathematics. --- Engineering mathematics. --- Gravitation. --- Engineering --- Engineering analysis --- Mathematical analysis --- Matter --- Physics --- Antigravity --- Centrifugal force --- Relativity (Physics) --- Mathematics --- Properties

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First published in 1922 and based on lectures delivered in May 1921, Albert Einstein's The Meaning of Relativity offered an overview and explanation of the then new and controversial theory of relativity. The work would go on to become a monumental classic, printed in numerous editions and translations worldwide. Now, The Formative Years of Relativity introduces Einstein's masterpiece to new audiences. This beautiful volume contains Einstein's insightful text, accompanied by important historical materials and commentary looking at the origins and development of general relativity. Hanoch Gutfreund and Jürgen Renn provide fresh, original perspectives, placing Einstein's achievements into a broader context for all readers.In this book, Gutfreund and Renn tell the rich story behind the early reception, spread, and consequences of Einstein's ideas during the formative years of general relativity in the late 1910s and 1920s. They show that relativity's meaning changed radically throughout the nascent years of its development, and they describe in detail the transformation of Einstein's work from the esoteric pursuit of one individual communicating with a handful of colleagues into the preoccupation of a growing community of physicists, astronomers, mathematicians, and philosophers.This handsome edition "es extensively from Einstein's correspondence and reproduces historical documents such as newspaper articles and letters. Inserts are featured in the main text giving concise explanations of basic concepts, and short biographical notes and photographs of some of Einstein's contemporaries are included. The first-ever English translations of two of Einstein's popular Princeton lectures are featured at the book's end.

General relativity (Physics) --- History --- Einstein, Albert, --- Relativistic theory of gravitation --- Relativity theory, General --- Gravitation --- Physics --- Relativity (Physics) --- Albert Einstein. --- Generalization of Gravitation Theory. --- On the Cosmologic Problem. --- Princeton. --- The Meaning of Relativity. --- astronomers. --- biographical notes. --- experimental tests. --- formative years. --- general relativity. --- general theory of relativity. --- gravitational field equations. --- gravitational waves. --- lectures. --- mathematicians. --- modern physics. --- philosophers. --- physicists. --- relativistic cosmology. --- relativity theory. --- relativity. --- scientists. --- special relativity. --- special theory of relativity. --- theories of relativity. --- theory of relativity.