( 2 M\l/2 ¿'shell = I1 ~ —J dt t15- stationary]

drshell VS. dr

¿'-shell = {l~yndr 116]

¿''shell = (* " 7) dr I17- stationary]

Energy (constant of the motion)

I ™

E(x_ 2Myt + 2M% m \ r Jdi r dt 1 1

Angular momentum (constant of the motion)

t - -ldi ™

L=R2d*_2M2dt m dx r dx

you 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 last 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), but we must use continual tangential rocket blasts to keep these rings from rotating in the tangential direction.

QUERY 15 Energy and angular momentum as constants of the motion. Derive Table 1, entries [19] and [21] for energy and angular momentum of a free object moving in the equatorial plane of an extreme Kerr black hole.

8 Plunging: The "Straight-ln Spiral"

For the nonrotating black hole the simplest motion was radial plunge (Chapter 3). What is the simplest motion near a spinning black hole? By analogy, let us examine motion starting from infinity and proceeding with zero angular momentum.

Figure 3 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 atr-M and circulates there forever

QUERY 16 No angular momentum. But angular motion! Set angular momentum [21] equal to zero and verify the following equation:

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 follow 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 1,

Equations [22] and [23] are two equations in the four unknowns dr, dt, dx, and dty. A third equation is the metric [3] for the extreme-spin black hole. With these three independent equations, we can eliminate three of the four unknowns to find a relation between any two remaining differentials. We choose the quantities dr and dty, because we want to draw the trajectory, the Kerr map. Don't bother doing the algebra—it is a mess. After substituting equation [4] for R2 into the result, one obtains the relation between dr and di|>:

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