Motion is always relative to some thing. Is this thing a concrete body like the earth, is it an abstract space, or is it an imagined frame? Do the laws of physics depend on the choice of reference? It there a choice for which the laws are simplest? Is this choice unique? Is there a physical cause for the choice made? These questions traverse the history of modern physics from Galileo to Einstein. The answers involved Galilean relativity, Newton's absolute space, the purely relational concepts of Descartes, Leibniz, and Mach, and many forgotten uses of relativity principles in mechanics, optics, and electrodynamics - until the relativity theories of Poincare, Einstein, Minkowksi, and Laue radically redefined space and time to satisfy universal kinds of relativity. Accordingly, this book retraces the emergence of relativity principles in early modern mechanics, documents their constructive use in eighteenth- and nineteenth-century mechanics, optics, and electrodynamics, and gives a well-rooted account of the genesis of special and general relativity in the early twentieth century. As an exercise in long-term history, it demonstrates the connectivity of issues and approaches across several centuries, despite enormous changes in context and culture. As an account of the genesis of relativity theories, it brings unprecedented clarity and fullness by broadening the spectrum of resources on which the principal actors drew.

The book aims to give a mathematical presentation of the theory of general relativity (that is, spacetime-geometry-based gravitation theory) to advanced undergraduate mathematics students. Mathematicians will find spacetime physics presented in the definition-theorem-proof format familiar to them. The given precise mathematical definitions of physical notions help avoiding pitfalls, especially in the context of spacetime physics describing phenomena that are counter-intuitive to everyday experiences.In the first part, the differential geometry of smooth manifolds, which is needed to present the spacetime-based gravitation theory, is developed from scratch. Here, many of the illustrating examples are the Lorentzian manifolds which later serve as spacetime models. This has the twofold purpose of making the physics forthcoming in the second part relatable, and the mathematics learnt in the first part less dry. The book uses the modern coordinate-free language of semi-Riemannian geometry. Nevertheless, to familiarise the reader with the useful tool of coordinates for computations, and to bridge the gap with the physics literature, the link to coordinates is made through exercises, and via frequent remarks on how the two languages are related.In the second part, the focus is on physics, covering essential material of the 20th century spacetime-based view of gravity: energy-momentum tensor field of matter, field equation, spacetime examples, Newtonian approximation, geodesics, tests of the theory, black holes, and cosmological models of the universe.Prior knowledge of differential geometry or physics is not assumed. The book is intended for self-study, and the solutions to the (over 200) exercises are included.

Die vorliegende Schrift behandelt die Kalenderlehre unter bewuBter Ausscheidung aller historischen Betrachtungen. Liegt darin einerseits e: ne Beschrankung, so glaubt der Ver fasser andererseits durch die Loslosung von der vielverschlungenen und daher mit mancherlei Ballast behafteten Entwickelungs geschichte des Kalenderwesens dessen mathematischen Kern sozusagen reinlicher herausschalen zu konnen, als es bei einer historischen Darstellung moglich ware. Die Schrift wendet sich an einen sehr allgemeinen Leser kreis. Sie mochte jedem Gebildeten zuganglich sein, der mathe matischen Oberlegungen einfachster Art Interesse entgegen bringt. Da und dort mag vieHeicht der Historiker bei chrono logischen Untersuchungen aus ihr Nutzen ziehen konnen. Eine besondere Freude ware es dem V erfasser, wenn zuweilen ein Feldgrauer nach dem Schriftchen griffe; weiB er doch aus eigener Erfahrung, wie es Zeiten gibt, in denen der Soldat nach irgendwelcher geistigen Anregung hungert - ganz ab gesehen davon, daB es im Felde von praktischem Interesse sein kann, nach kurzer Dberlegung zu wissen, welche Mondphase einem gegebenen Datum zukommt. Aber dariiber hinaus mochte das Biichlein auch dem an gehenden Mathematiker etwas zu sagen haben, und vor aHem dem Unterricht an hoheren Schulen dienen. Sollte die Schrift einem so verschiedenartigen Leserkreise gerecht werden, so war dies nur moglich durch eine besondere Behandlungsweise des Stoffes: die Darstellung zerfallt gewisser maBen in mehrere "Kreise," deren jeder in sich geschlossen und verstandlich ist. Der engste erfordert kaum mehr, als die Kennt nis der vier Grundrechnungsarten. Der nachste setzt schon Vorwort."

Gravity is one of the four fundamental interactions that exist in nature. It also has the distinction of being the oldest, weakest, and most difficult force to quantize. Understanding gravity is not only essential for understanding the motion of objects on Earth, but also the motion of all celestial objects, and even the expansion of the Universe itself. It was the study of gravity that led Einstein to his profound realisations about the nature of space and time. Gravity is not only universal, it is also essential for understanding the behaviour of the Universe, and all astrophysical bodies within it. In this Very Short Introduction Timothy Clifton looks at the development of our understanding of gravity since the early observations of Kepler and Newtonian theory. He discusses Einstein's theory of gravity, which now supplants Newton's, showing how it allows us to understand why the frequency of light changes as it passes through a gravitational field, why GPS satellites need their clocks corrected as they orbit the Earth, and why the orbits of distant neutron stars speed up. Today, almost 100 years after Einstein published his theory of gravity, we have even detected the waves of gravitational radiation that he predicted. Clifton concludes by considering the testing and application of general relativity in astrophysics and cosmology, and looks at dark energy and efforts such as string theory to combine gravity with quantum mechanics. ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.