Short description:

This book provides a fresh perspective on the relationships between gravitation, electrodynamics, and quantum physics. Designed for graduate students and postdoctoral researchers with a background in mathematical physics, it organizes key ideas into a series of paradigms inspired by the history of scientific discoveries, from Aristotle and Euclid to modern physics. Framed within the language of modern differential geometry, these paradigms rely on essential concepts such as fiber bundles and manifolds, which are introduced in the text. Although the primary focus is on Einstein’s theory of gravitation, the discussion is set within a broader mathematical framework that includes arbitrary dimensional manifolds with linear connections, metric tensor fields (with any signature), torsion, and metric gradients. A chapter introduces the concept of Frenet-Serret frames along curves in various arbitrary-dimensional manifolds with metric tensor fields of arbitrary signature and provides examples relevant to spacetime physics. The book makes precise the concept of an “ideal spacetime observer” and a “standard clock in spacetime”,  highlighting the inevitable role of quantum wave-particle duality in interpreting local measurement processes.

The text offers a variational approach to deriving generalized theories of gravitation interacting with matter using the exterior calculus of differential forms. This provides an efficient calculus for deriving stress-energy-momentum tensors and leads to a detailed analysis of the Einstein-Maxwell paradigm in spacetime. Killing vector and Killing tensor fields are employed in analyzing the geodesics of Schwarzschild, Reissner-Nordstrom and Kerr spacetimes. Throughout the book emphasis is placed upon distinguishing between geometric and co-ordinate singularities, and is illustrated using charts constructed by Painleve-Gullstrand and Kruskal-Szekeres, leading to a discussion of the properties of black hole spacetimes. A geometrical framework is provided for analyzing the Tolman-Oppenheimer-Volkoff theory for stellar interiors and a chapter examines the “Oppenheimer-Schiff Debate” about the electromagnetic fields generated by rotating charged shells, clarifying key points in the literature. A chapter introduces chiral pulse models in Maxwell electrodynamics, Bopp-Lande-Podolsky electrodynamics and linearised Einstein gravitation. Spinor fields are introduced as sections of a Clifford algebra bundle and used to discuss spinor pulse fields in Minkowski spacetime. Several appendices complement the main text. They include a guide to notations, detailed proofs of mathematical identities, a table of physical dimensions for quantities discussed, and a primer on set and measure theory. For readers interested in further exploration, additional appendices outline the mathematical foundations of quantum mechanics, providing a stepping stone to future paradigms in modern physics.

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Long description:

This book provides a fresh perspective on the relationships between gravitation, electrodynamics, and quantum physics. Designed for graduate students and postdoctoral researchers with a background in mathematical physics, it organizes key ideas into a series of paradigms inspired by the history of scientific discoveries, from Aristotle and Euclid to modern physics. Framed within the language of modern differential geometry, these paradigms rely on essential concepts such as fiber bundles and manifolds, which are introduced in the text. Although the primary focus is on Einstein’s theory of gravitation, the discussion is set within a broader mathematical framework that includes arbitrary dimensional manifolds with linear connections, metric tensor fields (with any signature), torsion, and metric gradients. A chapter introduces the concept of Frenet-Serret frames along curves in various arbitrary-dimensional manifolds with metric tensor fields of arbitrary signature and provides examples relevant to spacetime physics. The book makes precise the concept of an “ideal spacetime observer” and a “standard clock in spacetime”,  highlighting the inevitable role of quantum wave-particle duality in interpreting local measurement processes.

The text offers a variational approach to deriving generalized theories of gravitation interacting with matter using the exterior calculus of differential forms. This provides an efficient calculus for deriving stress-energy-momentum tensors and leads to a detailed analysis of the Einstein-Maxwell paradigm in spacetime. Killing vector and Killing tensor fields are employed in analyzing the geodesics of Schwarzschild, Reissner-Nordstrom and Kerr spacetimes. Throughout the book emphasis is placed upon distinguishing between geometric and co-ordinate singularities, and is illustrated using charts constructed by Painleve-Gullstrand and Kruskal-Szekeres, leading to a discussion of the properties of black hole spacetimes. A geometrical framework is provided for analyzing the Tolman-Oppenheimer-Volkoff theory for stellar interiors and a chapter examines the “Oppenheimer-Schiff Debate” about the electromagnetic fields generated by rotating charged shells, clarifying key points in the literature. A chapter introduces chiral pulse models in Maxwell electrodynamics, Bopp-Lande-Podolsky electrodynamics and linearised Einstein gravitation. Spinor fields are introduced as sections of a Clifford algebra bundle and used to discuss spinor pulse fields in Minkowski spacetime. Several appendices complement the main text. They include a guide to notations, detailed proofs of mathematical identities, a table of physical dimensions for quantities discussed, and a primer on set and measure theory. For readers interested in further exploration, additional appendices outline the mathematical foundations of quantum mechanics, providing a stepping stone to future paradigms in modern physics.

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Table of Contents:

Newtonian Gravitation and Maxwell s Theory of Electromagnetism.- Vector Spaces.- Topological and Differentiable Manifolds.- Fibre Bundles.- Basic Differential Geometry.- eneral Linear Connections Curvature and Torsion Tensors.- Metric Compatible Connections Isometries and Killing Tensor Fields.- Intrinsic and Extrinsic Curvatures and Lagrangian Charts.- A Modern View of the Newtonian Paradigm.- Assigning Coherent Physical Dimensions and Units to Elements in Spacetime.- The Exponential Map and Geodesic Normal Coordinates.- Clocks and Observer Fields in Einsteinian Spacetimes.- Spacetime Projectors and Relativistic Energy Momentum.- Spacetime Reference Frame Fields and Their Neighbours.- Gibbs Differential Calculus via Exterior Differential Forms.- Split Structures for Electromagnetic Fields and Gauge Covariances.- The General Frenet Serret Equations on a Manifold.- Spacetime Chronology and Radar Coordinates.- Variational Methods on n Dimensional Manifolds.- Properties of the Maxwell Stress Energy Momentum Tensor in Spacetime.- Killing Conservation Laws and Killing Drive Forms.- The Ideal Perfect Thermodynamic Fluid in a Background Spacetime Metric.- The Non Thermodynamic Self Gravitating Perfect Fluid in a Dynamic Einstein de Sitter Metric.- Source-Free Maxwell Fields in Minkowski and Einstein de Sitter Spacetime and Einstein de Sitter Timelike Geodesics.- Displaying Spacetime Curves.- Tracking Particles and the Ubiquitous Photon.- The Historic Development of the Einstein Minkowski Spacetime Paradigm.- Instantaneous Multi Particle Collisions in Spacetimes.- Spherically Symmetric Black Holes in Einstein s Theory with the Levi Civita Connection.- Axially Symmetric Black Holes in Einsteins Theory with the Levi Civita Connection.- An Interpretation of Tidal Strains from the Einstein Minkowski Paradigm.- The Oppenheimer Schiff Debate.- Applications of Chiral Pulse Models in Physics.- An Introduction to Classical Analytical Dynamics.- Spectral Shifts.

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