Probing the Universe: A Geometrical View for Observers of Spacetime Physics

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.