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.25.5: (a) Radial light rays, and light cones, for the Schwarzschild spacetime as depictedin Eddington-Finklestein coordinates [Eq.(25.66)].(b) These same light rays and light cones asdepicted in Schwarzschild coordinates [cf.Fig.25.3].t tM M6644(b)(a)2 2r M r M/ /0 02 4 6 8 2 4 6 8Fig.25.6: World line of an observer on the surface of an imploding star, as depicted (a) inan Eddington-Finklestein spacetime diagram, and (b) in a Schwarzschild spacetime diagram; seeExercise 25.5.Return, now, to the implosion of a star.The world line of the star s surface, which becameasymptotically frozen at the gravitational radius when studied in Schwarzschild coordinates,plunges unimpeded through r = 2M and into r = 0 when studied in Eddington-Finklesteincoordinates; see Exercise 25.5 and compare Figs.25.6(b) and 25.6(a).Thus, in order tounderstand the star s ultimate fate, we must study the region r = 0.As with r = 2M there are two possibilities: Either the tidal forces as measured on thestar s surface remain finite there, in which case something must be going wrong with thecoordinate system; or else the tidal forces diverge, destroying the star.The tidal forces arecomputed in Exercise 25.6, with a remarkable result: They diverge.Thus, the region r = 0is a spacetime singularity; a region where tidal gravity becomes infinitely large, destroyingeverything that falls into it.This, of course, is a very unsatisfying conclusion.It is hard to believe that the correctlaws of physics will predict such total destruction.In fact, they probably do not.As weshall discuss in Chap.27, when the radius of curvature of spacetime becomes as small as12lPW a" (G /c3) = 10-33 centimeters, space and time must cease to exist as classical entities;27tM6singularity4stellarsurfacenonsingular2stellarmatter, r =0r M0 /0 2 4 6 8Fig.25.7: Spacetime diagram depicting the formation and evolution of the horizon of a black hole.The coordinates outside the surface of the imploding star are those of Eddington and Finklestein;those inside are a smooth continuation of Eddington and Finklestein.Note that the horizon is theboundary of the region that is unable to send outgoing null geodesics to radial infinity.they, and the spacetime geometry must then become quantized; and, correspondingly, generalrelativity must then break down and be replaced by a quantum theory of the structure ofspacetime, i.e., a quantum theory of gravity.That quantum theory will describe and governthe classically singular region at the center of a black hole.Since, however, only rough hintsof the structure of that quantum theory are in hand at this time, it is not known what thattheory will say about the endpoint of stellar implosion.Unfortunately, the singularity and its quantum mechanical structure are totally invisibleto observers in the external universe: The only way the singularity can possibly be seen isby means of light rays, or other signals, that emerge from its vicinity.However, because thefuture light cones are all directed into it (Fig.25.6), no light-speed or sub-light-speed signalscan ever emerge from it.In fact, because the outer edge of the light cone is tilted inward atevery event inside the gravitational radius (Figs.25.5 and 25.6), no signal can emerge frominside the gravitational radius to tell external observers what is going on there.In effect, thegravitational radius is an absolute event horizon for our universe, a horizon beyond which wecannot see except by plunging through it, and paying the ultimate price for our momentaryexploration of the hole s interior.As most readers are aware, the region of strong, vacuum gravity left behind by theimplosion of the star is called a black hole.The horizon, r = 2M, is the surface of the hole,and the region r [ Pobierz całość w formacie PDF ]