Four-force

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A First Course in General Relativity (Second Edition)

Abstract: A few years ago, in my review of Sean Carroll's book in Classical and Quantum Gravity [1], I wrote that while the 1970s was the decade of Weinberg [2] and Misner, Thorne and Wheeler [3], and while the eighties was the decade of Schutz [4] and Wald [5], the 2000s was clearly the decade of Hartle [6] and Carroll [7] In my opinion, these books continue to stand out in the surprisingly dense crowd of introductory textbooks on general relativity At the dawn of this new decade I look forward to see what fresh pedagogical insights will be produced next, and who will be revealed as the winners of the 2010s It is, of course, much too early to tell, but Schutz is back, and he will set the standard just as he did back in 1985 This is the long-awaited second edition of his `First Course', a short, accessible, and very successful introduction to general relativity The changes from the first edition are modest: Schutz wisely refrained from bloating the text with new topics, and limited himself to updating his discussion of gravitational-wave sources and detectors, neutron-star and black-hole astrophysics, and suggestions for further reading Most importantly, he completely rewrote the chapter on cosmology, a topic that has evolved enormously since the first edition The book begins in chapter 1 with a beautiful review of special relativity that emphasizes spacetime geometry and stays away from an algebraic approach based on the Lorentz transformation, which appears only later in the chapter This is followed up in chapters 2 and 3 with an introduction to vector and tensor analysis in flat spacetime The point of view is modern (tensors are defined as linear mapping of vectors and one-forms into real numbers) but the presentation is very accessible and avoids an overload of mathematical fine print In chapter 4 the book introduces the spacetime description of fluids; it is here that the energy?momentum tensor makes its first appearance The move to curved spacetime is tackled next In chapter 5 the principle of equivalence is used to motivate the notion that gravity is a manifestation of spacetime curvature Tensor calculus in curved spacetime is approached gently, by first working through a generalization to curvilinear coordinates A systematic introduction to differential geometry is provided in chapter 6; here the reader is initiated in Riemannian manifolds, covariant differentiation, parallel transport, geodesics, the curvature tensors, and the Bianchi identities This is a formidable chapter, but the student is guided by a sure hand, and the presentation is both beautiful and accessible The next two chapters bring differential geometry to physics In chapter 7 the reader learns how to formulate the laws of physics in a curved spacetime, and in chapter 8 the Einstein field equations are finally formulated The chapter ends with a thorough treatment of the weak-field limit in the Lorenz gauge The following chapters present applications of the theory Chapter 9 is devoted to gravitational waves: propagation, detection, generation, energy balance, and astrophysical sources Here, as always, the discussion is accessible and fully up-to-date I could identify one weakness, which I have noted in many other textbooks (this is a pet peeve of mine, which seems to be turning into an obsession): the quadrupole formula for the gravitational-wave field is derived on the basis of the linearized theory, without warning the reader that the derivation does not apply to self-gravitating systems This is, however, compensated by a major strength: Schutz's derivation of the energy carried off by gravitational waves is based on a beautiful physical argument that bypasses the construction of an energy?momentum tensor for the gravitational-wave field; the complexities associated with such a construction are well known, and it is nice to see that Schutz has found a nice way around In chapter 10 the exact theory is applied to stellar structure, and in chapter 11 the student is introduced to black holes A large part of the chapter is devoted to the study of geodesic motion in Schwarzschild spacetime, and this allows Schutz to make contact with the classical tests of general relativity: perihelion advance and light deflection The singular behaviour of the Schwarzschild coordinates at the event horizon is described in detail This reveals another weakness of the book: the Kruskal coordinates are simply written down, with no derivation and little motivation; it is a pity that Schutz did not choose to introduce the Eddington--Finkelstein coordinates, or the Painlev??Gullstand coordinates, as easier alternatives The chapter ends with a general discussion of black holes (including their place in astrophysics and a description of the Hawking effect) and a detailed presentation of the Kerr solution The last chapter (chapter 12) is devoted to cosmology, and this is the part of the book that was the most thoroughly revised The presentation begins with the enunciation of the cosmological principle and a derivation of the Friedmann?Lemaitre models It continues with a discussion of cosmological dynamics in the presence of pressureless matter, radiation, and a cosmological constant (of which nobody wanted to be reminded at the time of the first edition) It concludes with an up-to-date review of cosmological measurements and a (very) brief history of the Universe, from the big bang to inflation, to recombination, to structure formation The presentation of general relativity and its applications contained in this book is suitable for undergraduate students who would prefer the standard `math-first approach' to Hartle's `physics-first approach' The student will learn the essentials of differential geometry in a gentle way, and will then apply these tools to physics in curved spacetime; all of this can be accomplished in a brisk one-semester course The book leaves out many topics than can be found in more advanced texts, such as Lie differentiation, differential forms, Killing vectors, the more abstract formulation of differential geometry (in terms of charts and diffeomorphisms), and the Lagrangian formulation of general relativity This limitation of scope is wise: Schutz masterly covers the essentials in an efficient and small package, and relegates all refinements to further reading in other textbooks; this is a sound learning strategy To conclude I will state that I just love this book I love it today as much as I did when I first came across it as an undergraduate student The revisions bring the book up-to-date, and they ensure that Schutz's text will remain in the pantheon of introductory general relativity books for many years to come References [1] Poisson E 2005 Review of Spacetime and Geometry: An Introduction to General Relativity, by S M Carroll Class Quantum Grav 22 4385?4386 [2] Weinberg S 1972 Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (New York: Wiley) [3] Misner C W, Thorne K S, and Wheeler J A 1973 Gravitation (San Francisco, CA: Freeman) [4] Schutz B F 1985 A First Course in General Relativity (Cambridge: Cambridge University Press) [5] Wald R M 1984 General Relativity (Chicago : Chicago University Press) [6] Hartle J B 2003 Gravity: An Introduction to Einstein's General Relativity (San Francisco, CA: Addison-Wesley) [7] Carroll S 2003 Spacetime and Geometry: An Introduction to General Relativity (San Francisco, CA: Benjamin Cummings)

Introduction to relativity

Abstract: Special relativity is a cornerstone of the structure of all fundamental theories, and general relativity has blossomed from Einstein's original theory into a cutting-edge applied science. Applications of Einstein's field equations describe such phenomena as supermassive black holes at the center of galaxies, the spiraling paths of binary pulsars, gravitational lensing caused by massive compact halo objects (Macho's), and the possibility of detecting gravitational waves emitted in cataclysmic cosmic events. In Introduction to Relativity, physics teacher and researcher Bill McGlinn explains the fundamental concepts of Einstein's special and general theories of relativity. He describes the basic consequences of special relativity-length contraction and time dilation-and the enigma of the twin paradox, as well as the Doppler shift of light. Relativistic dynamics is contrasted to Newtonian dynamics, followed by a discussion of relativistic tensor fields, including those of the electromagnetic field and the energy-momentum density of fluids. After a study of Einstein's early attempt at incorporating the equivalence principle into physics, McGlinn presents the general theory of relativity, discussing the three classic tests of relativity: the deflection of light by a gravitational field; the precession of perihelia; and the gravitational redshift of light. He also discusses other important applications, such as the dynamics of orbiting gyroscopes, the properties of stellar interiors, and black holes. The book ends with a chapter on cosmology, which includes discussions of kinematics and dynamics of the famed Robertson-Walker metric, Hubble's constant, cosmological constant, and cosmic microwave background radiation. For anyone seeking a brief, clear overview of modern general relativity which emphasizes physics over mathematics, McGlinn's Introduction to Relativity is indispensable.

New derivation of Lopez's source of the Kerr-Newman field

Abstract: Lopez's source of the Kerr-Newman field is constructed by means of Israel's formulation of the theory of surface layers in general relativity.

Applied general relativity

Abstract: Important relativistic effects and issues are discussed which must be considered in the interpretation of current measurements such as ranging measurements to LAGEOS and to the moon, in the implementation of the Global Positioning System, in the synchronization of clocks near the earth's surface, and in the adoption of appropriate scales of time and length for the communication of scientific results.