Contents24 Fundamental Concepts of General Relativity 124.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Local Lorentz Frames, the Principle of Relativity, and Einstein’s EquivalencePrinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3 The Spacetime Metric, and Gravity as a Curvature of Spacetime . . . . . . . 624.4 Free-fall Motion and Geodesics of Spacetime . . . . . . . . . . . . . . . . . . 924.5 Relative Acceleration, Tidal Gravity, and Spacetime Curvature ........ 1424.5.1 Newtonian Description of Tidal Gravity . . . . . . . . . . . . . . . . 1424.5.2 Relativistic Description of Tidal G ravity . . . . . . . . . . . . . . . . 1624.5.3 Comparison of Newtonian and Relativistic D escriptions . . . . . . . . 1724.6 Properties of the Riemann Curvature Tensor . . . . . . . . . . . . . . . . . . 1924.7 Curvature Coupling Delicacies in the Equivalence Principle, and Some Non-gravitational Laws of Physics in Curved Spacetime1.............. 2324.8 The Einstein Field Equation2.......................... 2624.9 Weak Gravitational Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2924.9.1 Newtonian Limit of General Relativity . . . . . . . . . . . . . . . . . 3024.9.2 Linearized Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3124.9.3 Gravitational Field Outside a Stationary, Linearized Source . . . . . 3324.9.4 Conservation Laws for Mass, Momentum and Angular Momentum . . 351See MTW Chap. 16.2See MTW Chap. 17.0Chapter 24Fundamental Concepts of GeneralRelativityVersion 1024.1.K.p df, 12 October 2009Please send comments, suggestions, and errata via email to [email protected] or on paper toKip Thorne, 350-17 Caltech, Pasadena CA 91125Box 24.1Reader’s Guide• This chapter relies significantly on– The special relativity portions of Chap. 1.– Chapter 22, on the transition from special relativity to general relativity.• This chapter is a foundation for the applications of general relativity theory inChaps. 24–26.24.1 OverviewNewton’s theory of gravity is logically incompatible with the special theory of relativity:Newtonian gravity presumes the existence of a universal, frame-independent 3-dimensionalspace in which lives the Newtonian potential Φ, and a universal, frame-independent time twith respect to whic h the propagation of Φ is instantaneous. Special relativit y, by contrast,insists that the concepts of time and o f 3-dimensional space are frame-dependent, so thatinstantaneous propagation of Φ in one frame would mean non-instantaneous propagation inanother.The most straightforward way to remedy this incompatibility is to retain the assumptionthat gravity is described by a scalar field Φ, but modify Newton’s instantaneous, action-at-12a-distance field equation!∂2∂x2+∂2∂y2+∂2∂z2"Φ=4πGρ (24.1)(where G is Newton’s gravitation constant and ρ is the mass density) to read$∇2Φ ≡ gαβΦ;αβ= −4πGTµµ, (24.2)where$∇2is the squared gradient, or d’Alembertian in Minkowski spacetime and Tµµis thetrace (contraction on its slots) of the stress-energy tensor. This modified field equation atfirst sight is attractive and satisfactory (but see Ex. 24.1, below): (i)ItsatisfiesEinstein’sPrinciple of Relativity in that it is expressed asageometric,frame-independentrelationshipbetween geometric objects; and (ii)inanyLorentzframeittakestheform[withfactorsofc =(speedoflight)restored]!−1c2∂2∂t2+∂2∂x2+∂2∂y2+∂2∂z2"Φ=4πGc2(T00− Txx− Tyy− Tzz) , (24.3)which, in the kinds of situation contemplatedbyNewton[energydensitypredominantlydueto rest mass, T00∼=ρc2;stressnegligiblecomparedtorestmass-energy,|Tjk|%ρc2;and1/c × (time rate of change of Φ) negligible compared to spatial gradient of Φ], reduces tothe Newtonian field equation (24.1).Not surprisingly, most theoretical physicists inthedecadefollowingEinstein’sformulationof special relativity (1905–1915) presumed that gravity would be correctly describable, withinthe framework of special relativity, by this type of modification of Newton’s theory, orsomething resembling it. For a brief historical account see Chap. 13 of Pais (1982). ToEinstein, by contrast, it seemed clear as early as 1907 that the correct description of gravityshould involve a generalization of special relativit y rather than an incorporation into specialrelativity: Since an observer in a local, freelyfallingreferenceframeneartheearthshouldnot feel any gravitational accelerationatall,localfreelyfallingframes(local inertial frames)should in some sense be the domain of special relativity, and gravity should somehow bedescribed by the relative acceleration of such frames.Although the seeds of this idea were in Einstein’s mind as early as 1907 (see the discussionof the equivalence principle in Einstein, 1907), it required eight years for him to bring themto fruition. A first crucial step, which took half the eight years, was for Einstein to conquerhis initial aversion to Minkowski’s (1908) geometric formulation of special relativity, and torealize that a curvature of Minkowski’s 4-dimensional spacetime is the key to understandingthe relative acceleration of freely falling frames. The second crucial step was to master themathematics of differential geometry, which describes spacetime curvature, and using thatmathematics to formulate a logically self-consistent theory of gravit y. This second step tookan additional four years and culminated in Einstein’s (1915, 1916) general theory of relativity.For a historical account of Einstein’s eight-year struggle toward general relativity see, e.g.,Part IV of Pais (1982); and for selected quotations from Einstein’s technical papers duringthis eight-year period, which tell the story of his struggle in his own words, see Sec. 17.7 ofMTW.3It is remarkable that Einstein was led, not by experiment, but by philosophical and aes-thetic arguments, to reject the incorporation of gravity into special relativity [Eqs. (24.2)and (24.3) above], and insist instead on describing gravity by curved spacetime. Only af-ter the full formulation of his general relativity did experiments begin to confirm that hewas right and that the advocates of special-relativistic gravity were wrong, and only morethan 50 years after general relativity was formulated did the experimental evidence becomeextensiv e and strong. For detailed discussions see, e.g., Will (1981, 1986), and Part
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