ContentsVI GENERAL RELATIVITY 223 From Special to General Relativity 123.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Special Relativity Once Again . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.1 Geometric, Frame-Independent Formulation . . . . . . . . . . . . . . 323.2.2 Inertial Fra mes and Components of Vectors, Tensors and Physical Laws 523.2.3 Light Speed, the Interval, and Spacetime Diagrams . . . . . . . . . . 723.3 Differential Geometry in G eneral Bases andin Curved Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.3.1 Non-Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . . . 923.3.2 Vectors as Differential Operators; Tangent Space; Commutators . . . 1323.3.3 Differentiation of Vectors and Tensors; Connection Coefficients . . . . 1523.3.4 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2023.4 The Stress-Energy Tensor Revisited . . . . . . . . . . . . . . . . . . . . . . . 2523.5 The Proper Reference Frame of an Accelerated Observer [MTW pp. 163–176,327–332] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301Part VIGENERAL RELATIVITY2Chapter 23From Special to General RelativityVersion 0623.1.K.pdf, 25 April 2007.Please send comm e nts, suggestions, and errata v ia email to [email protected] or on paperto Kip Thorne, 130-33 Caltech, Pasadena CA 911 25Box 23.1Reader’s Guide• This chapter relies significantly on– The sp ecial relativity portions of Chap. 1.– The discussion of connection coefficients in Sec. 10.3.• This chapter is a foundation for the presentation of general relativity theory inChaps. 24–27 .23.1 Ove rviewWe have reached the final Part of this book, in which we present an introduction to the basicconcepts of general relativity and its most important applications. This subject, althougha little more challenging than the material t hat we have covered so far, is nowhere near asformidable as its reputation. Indeed, if you have mastered the techniques developed in thefirst five Parts, the path to the Einstein Field Equations should be short and direct.The G eneral Theory of Relativity is the crowning achievement of classical physics, thelast 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-venture undertaken by Newton 250 years earlier. It was created after many wrong turnsand with little experimental guidance, almost by pure thought. Unlike the special theory,whose physical foundations and logical consequences were clearly appreciated by physicists12soon a fter Einstein’s 1905 formulation, the unique a nd distinctive character o f the generaltheory only came to b e widely appreciated long after its creation. Ultimately, in hindsight,rival classical theories of gravitation came to seem unnatural, inelegant and ar bitrar y bycomparison.1Experimental tests of Einstein’s theory also were slow to come; only in thelast three decades have there been striking tests of high enough precision t o convince mostempiricists that, in all probability, and in its domain of applicability, general relativity isessentially correct. Despite this, it is still very poorly tested compared with, for example,quantum electrodynamics.We begin our discussion of general relativity in this chapter with a review and elabora-tion of relevant material already covered in earlier chapters. In Sec. 23.2, we give a briefencapsulation of the special theory drawn largely f r om Chap. 1, emphasizing those aspectsthat we must generalize to deal with non-inertial frames of reference. Then in Sec. 23.3 wecollect, review and extend the fundamental ideas of differential geometry that have beenscattered throughout the book and which we shall need as foundations for the mathematicsof spacetime curvature (Chap. 24); most importantly, we generalize differential geometry toencompass coordinate systems and bases that are not orthogonal. Einstein’s field equationsare a relationship between the curvature of spacetime and the matter that generates it, akinto the Maxwell equations’ relationship between the electromagnetic field and electric cur-rents and charges. The matter is described using the stress-energy tensor that we introducedin Sec. 1.12. We revisit the stress-energy tensor in Sec. 23 .4 and develop a deeper under-standing of its properties. In general relativity one often wishes to describe t he outcome ofmeasurements made by observers who refuse to fall freely—e.g., an observer who hovers in aspaceship just above the horizon of a black hole, or a gravitational-wave experimenter in anearth-bound laboratory. As a foundation for treating such observers, in Sec. 23.5 we examinemeasurements made by a ccelerated observers in the flat spacetime of special relativity.This chapter will leave us well prepared to develop, in Chap. 24, the basic concepts ofgeneral relativity, including spacetime curva ture, the Einstein Field Equation, and the lawsof physics in curved spacetime. In Chaps. 25–27 we shall explore the major applications ofgeneral relativity: to stars, black holes, gravitational waves, and cosmology. We begin inChap 25 by studying the spacetime curvature around and inside highly compact stars (such asneutron stars) and showing how, in the weak field limit, non-Newtonian effects are predictedin our own solar system and in binary neutron star systems and how these predictions havebeen verified. We also discuss the implosion of massive stars and describe the circumstancesunder which the implosion inevitably produces a black hole, and we explore the surprisingand, initially, counter-intuitive properties of black holes. In Chap. 26 we study gravitationalwaves, i.e. ripples in the curvature of spacetime that propagate with the speed of light, andwe explore their close analogy with the electromagnetic waves that were first predicted byMaxwell’s equations. We explore the properties of these waves, their production by binarystars and merging black holes, projects to detect them, both on earth and in space, and theprospects for using them to explore observationally the dark side of the universe and thenature of ultrastrong spacetime curvature. Finally, in Chap. 27 we draw once more uponall the previous Parts of this book, combining them with general relativity t o describe theuniverse on the largest of
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