The computer has no difficulty integrating and plotting this equation, as shown in Figure 3. Since we used the Kerr bookkeeper angular velocity , 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.
QUERY 17 Final circle according to the bookkeeper. Verify that dr goes to zero (that is, r does not change) once this stone reaches the horizon.
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.
QUERY 18 Bookkeeper speed in the "final circle." At the horizon, what is the numerical value of the tangential speed Rdty/dt of the stone dropped from rest at infinity, as measured by the Kerr bookkeeper?
The observer who has fallen from rest at infinity has quite a different perception 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 di, where dx is the wristwatch time increment of the in-faller:
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