Figure 3 - uploaded by Yvo Petkov
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Computer plot: Kerr map (Kerr bookkeeper plot) of the trajectory in space of a stone dropped from rest far from a black hole (therefore with zero angular momentum). According to the far-away bookkeeper, the stone spirals in to the horizon at r = M and circulates there forever.

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... use the Principle of Extremal Aging and other methods of Chapters 2 through 5 to derive expressions similar to results in those chapters and enter them in the right hand column of Table 1. Notes: (1) We limit ourselves to the equatorial plane. (2) Outside the static limit we can still set up stationary spherical shells (which we have limited to stationary rings in the equatorial plane). However, equation [21] with d φ /d τ = 0 tells us that a stationary ring has negative angular momentum. So during construction we need to provide an initial tangential rocket blast to give negative angular momentum to the ring structure in order to make it stationary. (See box page F-20.) Near the nonrotating black hole, the simplest motion was radial plunge (Chapter 3). What is the simplest motion near a spinning black hole? By analogy, examine the motion of a stone dropped from rest at a great distance which thereafter falls inward, maintaining zero angular momentum. Equation [22] gives the remarkable result that a particle with zero angular momentum nevertheless circulates around the black hole! This result is evidence for our interpretation that the black hole drags nearby spacetime around with it. Figure 3 shows the trajectory of an inward plunger with zero angular momentum, as calculated in what follows. Let’s see if we can set up the equations to describe a stone that starts at rest far from a rotating black hole and moves inward with zero angular momentum. At remote distance, in flat spacetime, the stone has energy E/m = 1. It keeps the same energy as it falls inward. From equation [19] in Table ...
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... computer has no difficulty integrating and plotting this equation, as shown in Figure 3. Since we used the Kerr bookkeeper angular velocity [22], the resulting picture is that of the Kerr bookkeeper. For her, the zero- angular-momentum stone spirals around the black hole and settles down in a tight circular path at r = M , there to circle forever. Remember that for the nonspinning black hole an object plunging inward slows down as it approaches the horizon, according to the records of the Schwarzschild bookkeeper. For both spinning and nonspinning black holes, the in-falling stone with L = 0 never crosses the horizon when clocked in far-away time. The observer who has fallen from rest at infinity has quite a different per- ception of the trip inward! For her there is no pause at the horizon; she has a quick, smooth trip to the center (assuming that the Kerr metric holds all the way to the center!). An algebra orgy similar to the previous one gives a relation between dr and d τ , where d τ is the wristwatch time increment of the ...
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... equations [25] and [26] show bookkeeper radial components of speed greater than unity in the region of small radius. The resulting speed is even more impressive when one adds the tangential component of motion forced on the diver descending into the spinning black hole ( Figure 3). Does such motion violate the “cosmic speed limit” of unity for light? A similar question is debated for the Schwarzschild black hole in Section 3 of Project B, Inside the Black Hole, pages B-6–12. Research note: When applied inside the horizon, equation [25] assumes that the Kerr metric correctly describes spacetime all the way to the center of the extreme Kerr black hole. This may not be the case. See the box Egg- beater Spacetime? on page ...