Radial Trajectories of Light

You are given the task of providing occasional radio news bulletins for the divers. Each of these bulletins, covering the latest news and scientific reports from outside the horizon, will be broadcast radially inward from a fixed station on a shell external to the horizon.

To prepare for this task, you want to know how long it takes light to move from radius r^ outside the horizon to radius inside the horizon. And since no clock can be stationary inside the horizon, you choose to measure "how long" for this trip in rain time frain. The following query is rather technical. You may choose (or may be instructed) to carry it out, or you may simply use the result. In this query and in much of the remainder of this project, we use dimensionless variables that not only simplify the analysis but also make the results independent of the mass M of the black hole being considered. For every domain of radius r and time t we define the dimensionless variables

QUERY 13 Rain time for light to move from one radius, r<\, to another, r2. Rewrite equation [17] of Query 9 to read, for inward-moving light or radio waves,

(For a flash sent "outward," the plus sign becomes a minus sign in both denominators in equations [19].) Make the substitution u = V2 + r*1/2 [20]

Integrate the resulting equation from u^ to u2, recall that

In A - lnfi = In (A/B), then resubstitute equation [20] for u-\ and u2 to show that for the signal moving inward from rto r*2, the integrated rain time is

'*r2 rain — rain = ('* 1 ~ r*2> " 2^(r*|/2 - r*^2) + 41n j2 + r*Y*

[21. headlight flash] Messy? Sure, but the computer doesn't care and easily plots the results.

QUERY 14 Horizon-to-crunch rain time for light. Verify that when r*2 = r% the elapsed rain time is zero. Why is it zero? Show that when r* 1 is at the horizon and r*2 = 0 (at the crunch point), the elapsed rain time t*rajn = 0.773 or train = 0.773 M. Compare this result with the horizon-to-crunch wrist-watch time x = (4/3)M for the rain diver herself, derived in the box on page 3-22. Why is the result using equation [21] less than the result for the rain diver herself?

Figure 6 is a computer plot of some curves derived from equation [21]. For each curve, select r*i and set f*rl rain = 0. Then choose a sequence of smaller values of r*2 and let the computer calculate t*r2 rain and plot the result. For comparison, we include curves showing the worldline of several raindrop divers. It is evident from Figure 6 that our news bulletins can be timed to catch up with the descending diver community.

What about the light flash that the in-falling diver launches radially outward? If this launch occurs before the diver falls through the horizon, then the outward flash indeed moves outward. Once the diver has passed inward across the horizon, however, even the "outward" flash moves inward. The little cones sprouting from the worldline in the earlier Figure 5 show the initial motions of inward and "outward" flashes emitted inside the horizon. More complete worldlines for emitted light are shown in Figure 7. You can trace the worldline of the "outward" flash by placing a minus sign in front of the second term in the denominator on the right side of equation [19] and carrying out the integration. Inside the horizon (where r*\ is always greater than r*2), a convenient form of the solution is

'*r2 ram "'*rl rain = " ('*! " r*2) " 2j2(r* ¡^ - r*^2) + 41n

[22. taillight flash]

A diver who has arrived at emission point A shown in Figure 7 can influence only those future events that lie within the shaded region in the diagram. The more time passes—the smaller the radius at which the diver arrives—the fewer events the diver can affect. With unlimited rocket power, the diver could herself be present at any of the events in this shaded region. Indeed, the worldline of the "outward" flash might be called a worldline of inevitability; for the raindrop traveler arriving at event A, no known power in the universe can defer extinction longer than the upper limit traced by this curve.

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