Relativity and Geometry: Foundations and Philosophy of Science and Technology Series 1st Edition by Roberto Torretti (PDF)

Description

Relativity and Geometry aims to elucidate the motivation and significance of the changes in physical geometry brought about by Einstein, in both the first and the second phases of relativity. The book contains seven chapters and a mathematical appendix. The first two chapters review a historical background of relativity. Chapter 3 centers on Einstein’s first Relativity paper of 1905. Subsequent chapter presents the Minkowskian formulation of special relativity. Chapters 5 and 6 deal with Einstein’s search for general relativity from 1907 to 1915, as well as some aspects and subsequent developments of the theory. The last chapter explores the concept of simultaneity, geometric conventionalism, and a few other questions concerning space time structure, causality, and time.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Many years ago I took a course on Kant’s Critique of Pure Reason with Robert Paul Woolf, a renowned Kantian. I loved the course. Here was a Philosopher not given to inscrutable mush about nothing (Hegel, for example). Kant’s early work was Scientific and he was interested in all things epistemological: what we know, how we know it and is that knowledge valid? The Critique represents the apex of his “Copernican Revolution”, in which he was forcibly awakened from “Dogmatic Slumber” by Hume’s Treatise of Human Nature. Kant’s Critique is a difficult read. He was inventing a new language for describing cognition and the processing of data presented to us by the Space-Time Manifold. His use of the word Manifold in this context is prescient. Kant also asked whether inferences extracted from that Manifold, and the chain of deductions inevitably attached to those inferences, held validity. These are the synthetic a priori judgements : non-tautological (in which the predicate of a sentence is not semantically contained in the subject-term) judgements held independently of sense experience. Since mathematical truths (and the sciences whose truths are written in mathematics) are dependent upon such judgements, Hume’s fierce attack on our notion of causality essentially forced Kant to verify the possiblity of Mathematics and Science. Roberto Torretti’s book is a wonderfully deep study of Relativity. He offers what he calls a “historico-critical” exposition in the spirit of Mach. His emphasis is on geometrical ideas but it is emblematic of the richness of this study that it begins with Kant and his analysis of geometry (as part of the structure of our minds) as a paradigm of our a priori knowlege of nature. Torretti returns to Kant throughout the book (as well as Philosophers like Reichenbach and Cassirer) because it is Einstein’s dismantling of Kant’s notion of geometry as well as the innate Categories of Time and Space that is the true revolution begun in 1905. But Philosophy is merely the background upon which events play out. For Torretti provides a clear, precise and well-documented analysis of the rise of Newtonian physics, the development of 19th Century electrodynamics and ideas about the aether, growth of non-Euclidean geometry and the experimental crisis at the end of the century. Einstein’s electrodynamics, Minkowski spacetime and Einstein’s revolutionary gravitational theory ease us into the 20th. All important concepts are discussed: simultaneity, causality, time, space, the nature of mass and motion. The very fabric of reality is what is at stake so the field of discussion is wide. An Appendix provides a splendid discussion of differentiable manifolds, fiber bundles, linear connections and various other topics. 85 pages of detailed notes at the end of the book are informative. The mathematics in this book are non-trivial. Maturity is assumed. You will not, however, see vast fields of tensor indices. You can refer to older volumes if you miss them. This book was originally published in 1983 and the geometric approach to Relativity holds sway. I found the discussion of various geometries to be endlessly fascinating, and there ARE many geometries mentioned here. For example, the chapter on Minkowski spacetime has an in-depth analysis of the Zeeman topology (1967) proposed by Zeeman as a substitute for the Minkowski geometry. Tensors are handled with relative delicacy (you knew there had to be some). The chapter on gravitational geometry is the densest mathematically, beginning with Ricci tensors, leading to Cartan’s geometric reformulation of Newton’s theory of gravity using bundles (enabling him to point out to Einstein that spacetime has a flat non-symmetric linear connection when essentially void of matter. This is Einstein’s final theory in essence). A. Friedmann’s brilliant analysis in 1922 of Einstein’s field equations pointing out the possibility of infinite solutions, any of which might entail the expansion or collapse (the solutions are indifferent to time reversal) of the galaxies is discussed here as well as Lie groups. I found this chapter the most difficult and re-read it several times. This is a book for physicists, mathematicians and philosophers of science with a strong background. It is not beyond the reach of the intelligent, motivated layman. This is a fascinating study even for old hands. I recommend it unreservedly.

⭐There are some folds. The print is okay (equations are hard to read).The book is not as comprehensive as I’d like though.

⭐This is a well-written exposition. It is not a popularization, it is not for the layman. It is an amalgamation of philosophy, physics, mathematics. Have handy these textbooks: Taylor & Wheeler’s Spacetime Physics, Misner, Thorne & Wheeler’s Gravitation and Pais’ Subtle is the Lord. A distinguishing hallmark is chapter five: Einstein’s Quest for a Theory of Gravity (55 pages). We are offered detailed exposition through lens of history of Einstein’s search for a theory of gravitation, culminating in the 1915 equations of general relativity. Now, Section #5.3 details other attempts (Abraham, Nordstrom, Mie) for a theory of gravitation (circa 1912). This section makes nice reading and complements what you will find in Pais (pages 231-236, Subtle is the Lord). Endnotes are copious (pages 283-350). There are appendices (essentially a survey of differential geometry in 20 pages–unfortunately of little utility for a novice). What else do we find ?(1) As customary, begin with a bit of “geometry” then proceed to Newton (absolute space, absolute time). Read: “Philosophical attitudes toward the concept of spacetime are often conditioned by the idea that it is unintuitive and…unnatural. However, you will surely acknowledge that the infinitely extended Euclidean space, with its infinitely articulated depth of depthless planes and lines, is not a whit more intuitive than Minkowski’s spacetime. (page 21).(2) Go from Newton (chapter one) to Maxwell’s electrodynamics (chapter two). Read: “Faraday’s views were scorned by the scientific establishment of the time because he was unable to formulate them in the standard language of mathematical analysis.” (page 36). From Faraday and Maxwell to Einstein in 1905.(3) Chapter three, that is, Einstein’s 1905 Special Relativity. Abolish action-at-a-distance. Maximum signal propagation is finite (not instantaneous). Simultaneity is abolished. Read: “Had more attention been paid to this fact (that relativistic clocks are hodometres of timelike worldlines), much of the effort spent on the so-called clock-paradox would have been spared.” (page 96). Supplement with Miller’s Special Relativity.(4) Chapter four (Minkowski) is preparation for chapter five: A lucid discussion. Read: “Following Penrose, we regard Minkowski spacetime, thus conceived, as a paradigm and a special case of a more general mathematical structure, called a causal space.” (page 123). Notice herein: no ‘worldline’ figures. I refer to Rindler or Dixon for elaborations of the 4-dimensional formalism of Special Relativity.(5) Scattered remarks about chapter five: Abraham Pais published Einstein’s scientific biography in 1982, Subtle is the Lord. Pais supplies pertinent details of Einstein’s search for a theory of gravitation. There are differences between Torretti and Pais. Here read: “before benefitting from Grossmann’s help, Einstein had already lighted on the key ideas of the new approach to gravity.” (page 143). Read: “Einstein must have almost made up his mind on this matter (building his theory on the moving sands of Riemann’s theory) before Grossmann introduced him to it.” (page 146). Now, proceed to Pais (chapter nine). Torretti recalls a quote from Steven Weinberg: “the theory of gravitational radiation provides a crucial link between general relativity and the microscopic frontier of physics.” (page 181).(6) The final two chapters (six and seven) touch upon Mach, cosmology, singularities. Simultaneity recurs chapter seven. Read: “Einstein’s theory of gravity is not easily expounded in terms of force.” (page 237). Read: “Before Einstein, however, nobody appears to have seriously disputed that any two events might be causally related to each other, regardless of their spatial and temporal distance.” (page 247). Go forth and study Hawking and Ellis (The Large Scale Structure of Space-Time).(7) Concluding: A useful monograph with a focus on Einstein’s geometrization of gravitation. Couched in the language of history, philosophy, and physics. The 30-page list of references is a rich source to peruse.

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