Relativity : The Special and the General Theory, The Masterpiece Science Edition,
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From the Commentary by Robert Geroch (The corresponding section of Einstein's text can be found below the comment. Please note that in the book, the Commentary is placed after the complete text of Relativity.)
Section 17. Space-Time
Minkowski's viewpoint represents a "geometrization" of relativity. These ideas have, over the years, come to the forefront: They reflect the perspective of the majority of physicists working in relativity today. Let us expand on this viewpoint. The fundamental notion is that of an event, which we think of as a physical occurrence having negligibly small extension in both space and time. That is, an event is "small and quick," such as the explosion of a firecracker or the snapping of your fingers. Now consider the collection of all possible events in the universe-all events that have ever happened, all that are happening now, and all that will ever happen; here and elsewhere. This collection is called space-time. It is the arena in which physics takes place in relativity. The idea is to recast all statements about goings-on in the physical world into geometrical structures within this space-time. In a similar vein, you might begin the study of plane geometry by introducing the notion of a point (analogous to an event) and assembling all possible points into the plane (analogous to space-time). This plane is the arena for plane geometry, and each statement that is part of plane geometry is to be cast as geometrical structure within this plane. This space-time is a once-and-for-all picture of the entire physical world. Nothing "happens" there; things just "are." A physical particle, for example, is described in the language of space-time by giving the locus of all events that occur "right at the particle." The result is a certain curve, or path, in space-time called the world-line of the particle. Don't think of the particle as "traversing" its world-line in the same sense that a train traverses its tracks. Rather, the world-line represents, once and for all, the entire life history of the particle, from its birth to its death. The collision of two particles, for example, would be represented geometrically by the intersection of their world-lines. The point of intersection-a point common to both curves; an event that is "right at" both particles-represents the event of their collision. In a similar way, more complicated physical goings-on-an experiment in particle physics, for example, or a football game-are incorporated into the fabric of space-time. One example of "physical goings-on" is the reference frame that Einstein uses in his discussion of special relativity. How is this incorporated into space-time? The individuals within a particular reference frame assign four numbers, labeled x, y, z, t, to each event in space-time. The first three give the spatial location of the event according to these observers, the last the time of the event. These numbers completely and uniquely characterize the event. In geometrical terms, a frame of reference gives rise to a coordinate system on space-time. In a similar vein, in plane geometry a coordinate system assigns two numbers, x and y, to each point of the plane. These numbers completely and uniquely characterize that point. The statement "the plane is two-dimensional" means nothing more and nothing less than that precisely two numbers are required to locate each point in the plane. Similarly, "space-time is four-dimensional" means nothing more and nothing less than that precisely four numbers are required to locate each event in space-time. That is all there is to it! You now understand "four-dimensional space-time" as well as any physicist. Note that the introduction of four-dimensional space-time does not say that space and time are "equivalent" or "indistinguishable." Clearly, space and time are subjectively different entities. But a rather subtle mixing of them occurs in special relativity, making it convenient to introduce this single entity, space-time. In plane geometry, we may change coordinates, i.e., relabel the points. It is the same plane described in a different way (in that a given point is now represented by different numbers), just as the land represented by a map stays the same whether you use latitude/longitude or GPS coordinates. We can now determine formulae expressing the new coordinate-values for each point of the plane in terms of the old coordinate-values. Similarly, we may change coordinates in space-time, i.e., change the reference frame therein. And, again, we can determine formulae relating the new coordinate-values for each space-time event to the old coordinate-values for that event. This, from Minkowski's geometrical viewpoint, is the substance of the Lorentz-transformation formulae in Section 11. A significant advantage of Minkowski's viewpoint is that it is particularly well-adapted also to the general theory of relativity. We shall return to this geometrical viewpoint in our discussion of Section 27.
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"""It is remarkable that Einstein should have produced such an account for the lay public so soon after his general theory was completed. This view has now superbly survived the tests of time.... In this new edition we are fortunate in having the advantage of an excellent exposition of Einstein's foundational ideas on relativity from a more modern perspective in Robert Geroch's commentary.""
¿From the Introduction by Roger Penrose, author of The Emperor's New Mind and The Road To Reality.
""One good way to get a feeling for what relativity theory is all about is to read, in these pages, what the originator of the subject had to say. I have provided comments, attached to various sections of Einstein's book. The key to understanding relativity is to think about it for yourself.""
¿From the Commentary by Robert Geroch, Professor of Physics, University of Chicago.
""Relativity was a highly technical new theory that gave new meanings to familiar concepts and even to the nature of theory itself. The general public looked upon relativity as indicative of the seemingly incomprehensible modern era, educated non-scientists despaired of ever understanding what Einstein had done, and political ideologues used the new theory to exploit public fears and anxieties¿all of which opened a rift between science and the broader culture that continues to expand today.""
¿From the Historical Essay by David C. Cassidy, author of J. Robert Oppenheimer and the American Century, and Einstein and Our World.
""All religions, arts and sciences are branches of the same tree. All these aspirations are directed toward ennobling man's life, lifting it from the sphere of mere physical existence and leading the individual toward freedom.""
¿Albert Einstein
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--This text refers to the
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Relativity : The Special and the General Theory, The Masterpiece Science Edition,
Relativity: The Special and the General Theory, The Masterpiece Science Edition,,Albert Einstein,Roger Penrose,Robert Geroch,David C. Cassidy,Pi Press,0131862618,Physics,Relativity,Relativity (Physics),Relativity Theory,Science,Science/Mathematics,Relativity physics,Science / Physics,Science / Relativity
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