This book offers an introduction to Newton–Cartan gravity, a reformulation of Newtonian gravity in differential-geometric language, which brings it closer to General Relativity (GR) than the standard formulation. The text systematically develops Newton–Cartan gravity from its geometric foundations, and provides a detailed discussion of the classical results on its relation to both standard Newtonian gravity and GR. Furthermore, it presents a global formulation of the gauge- theoretic perspective that underlies many modern developments in this field, and briefly discusses applications in Newtonian limits of string theory. This book is designed for researchers and graduate students in gravitational physics and related disciplines, providing a clear path toward understanding current research literature.
The book is written in a rigorous mathematical style. It includes an extensive set of illustrative exercises. A solid understanding of basic differential geometry is required; familiarity with principal bundles and associated vector bundles is necessary for the gauge-theoretic part.
Galilei manifolds.- Classical NewtonCartan gravity.- Galilei manifolds
via principal bundles.- Semidirect extensions of principal bundles.- Bargmann
structures.- Outlook.- Appendix A Semidirect products of Lie groups.
Philip K. Schwartz is a postdoctoral researcher at the Institute of Theoretical Physics at Leibniz University Hannover, Germany, where he earned his PhD in theoretical physics in 2020. His research focuses on exploring the relationship between (Poincaré-)'relativistic' gravity and Galilei-relativistic physics in terms of systematic post-Newtonian expansions, with a particular emphasis on theories of gravity based on modified geometry as well as quantum matter on gravitational backgrounds.
In recognition of his outstanding lecture course on NewtonCartan gravity, in 2022 he was honoured with a teaching prize.
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