ContentsVI GENERAL RELATIVITY ii22 From Special to General Relativity 122.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Special Relativity Once Again . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.1 Geometric, Frame-Independent Formulation . . . . . . . . . . . . . . 222.2.2 Inertial Frames and Components of Vectors, Tensors and Physical Laws 422.2.3 Light Speed, the Interval, and Spacetime Diagrams . . . . . . . . . . 622.3 Differential Geometry in General Bases andin Curved Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.3.1 Non-Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . . . 822.3.2 Vectors as Differential Operator s; Tangent Space; Commutators . . . 1222.3.3 Differentiation of Vectors and Tensors; Connection Coefficients . . . . 1522.3.4 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2022.4 The Stress-Energy Tensor Revisited . . . . . . . . . . . . . . . . . . . . . . . 2422.5 The Proper Reference Frame of an Accelerated Observer [MTW pp. 163–176,327–332] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29iPart VIGENERAL RELATIVITYiiGeneral RelativityVersion 0822.1.K.pdf, 22 April 2009We have reached the final Part of this book, in which we present an introduction tothe basic concepts of general relativity and its most important applications. This subject,although a little more challenging than the material that we have covered so far, is nowherenear as formidable as its reputation. Indeed, if you have mastered the techniques developedin the first five Parts, the path to the Einstein Field Equations should be short and direct.The General Theory of Relativity is the crowning achievement of classical physics, t helast great fundamental theory created prior to the discovery of quantum mechanics. Itsformulation by Albert Einstein in 1915 marks the culmination of the great intellectual ad-vent ure undertaken by Newton 250 years earlier. Einstein created it after many wrong turnsand with little experimental g uidance, almost by pure thought. Unlike the special theory,whose physical foundations and logical consequences were clearly appreciated by physicistssoon after Einstein’s 1905 formulation, the unique and distinctive character of the generaltheory only came to be widely appreciated long after its creation. Ultimately, in hindsight,rival classical theories of gravitation came to seem unnatural, inelegant and arbitrary bycomparison.1Exp erimental tests of Einstein’s theory also were slow to come; only since 1970 havethere been striking tests of high enough precision to convince most empiricists that, in allprobability, and in its domain of applicability, general relativity is essent ially correct. Despitethis, it is still very poorly tested compared with, for example, quantum electrodynamics.We begin our discussion of general relativity Chap. 22 with a r eview and elaborationof special relativity as d eveloped in Chap. 1, focusing on those t h at are crucial for thetransition to general relativity. Our elaboration includes: (i) an extension of differentialgeometry to curvilinear coordinate systems and general bases both in the flat spacetime ofspecial relativity and in the curved spacetime that is the venue for general relativity, (ii)an in-depth exploration o f the stress-energy tensor, which in general relativity generatesthe curvature of spacetime, and (iii) construction and exploration of the reference frames ofaccelerated observers, e.g. physicists who reside on the Earth’s surface. In Chap. 23, weturn to the basic concepts of general relativity, including spacetime curvature, the EinsteinField Equation that governs the g eneration of spacetime curvature, the laws of physics incurved spacetime, and weak-gravity limits of general relativity.1For a readable a ccount a t a popular level, see Will (1993); for a more detailed, scholarly account see,e.g. Pais (1982).iiiivIn the remaining chapters, we explore applications of general relativity to stars, blackholes, gravitational waves, experimental tests of the theory, and cosmology. We begin inChap 24 by studying the spacetime curvature around and inside highly compact stars (such asneutron stars). We then discuss the implosion of massive stars and describe the circumstancesunder which the implosion inevitably produces a black hole, we explore the surprising and,initially, counter-intuitive properties of black holes (both nonspinning holes and spinningholes), and we learn ab out the many-fingered nature of time in general relativity. In Chap.25 we study experimental tests of general relativity, and then turn attention to gravitationalwaves, i.e. ripples in the curvature of spacetime that propagate with the speed of light. Weexplore the properties of these waves, their close analog y with electromagnetic waves, theirproduction by binary stars and merging black holes, projects to detect them, bot h on earthand in space, and the prospects for using them to explore observationally the dark side of theuniverse and the nature of ultrastrong spacetime curvature. Finally, in Chap. 26 we drawupon all the previous Parts of this book, combining them with general relativity to describethe universe o n the largest of scales and longest of times: cosmology. It is here, more thananywhere else in classical physics, that we are conscious of reaching a frontier where thestill-promised land of quantum gravity beckons.Chapter 22From Special to General RelativityVersion 0822.1.K.pdf, 22 April 2009Pleas e send com ments, suggestions, and errata via email to [email protected] or on paperto Kip Thorne, 130-33 Caltech, Pas adena CA 91125Box 22.1Reader’s Guide• This chapter relies significantly on– The special relativity portions of Chap. 1.– The discussion of connection coefficients in Sec. 10.5.• This chapter is a foundation for the presentation of general relativity theory inChaps. 23–26.22.1 OverviewWe begin our discussion of general relativity in this chapter with a review and elab orationof releva nt material already covered in earlier chapters. In Sec. 22.2, we give a brief encap-sulation of the special theory drawn largely from Chap. 1, emphasizing those aspects thatunderpin the transition to general relativity. Then in Sec. 22.3 we collect, review and extendthe fundamental ideas of differential geometry that have been scattered throughout the bookand which we
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