M

Moreover, shell observer clocks tick off not far-away time dt but shell time ¿'shell- This wristwatch time, dx, is defined as the time recorded in a frame in which two events occur at the same place; in the shell frame the shell clock does not change spatial coordinates between ticks. The relation between shell time d£shell aRd far-away time dt is given by equation [C] in Selected Formulas at the back of the book:

dt shell

The last two equations and equation [21] tell us that the radial velocity of the free-fall observer as measured by the shell observer has the value dr shell dt shell

This expression for radial velocity is the same as the result of the Newtonian analysis (box on page 2-22). However, that Newtonian expression failed to distinguish between shell coordinates and bookkeeper coordinates. Equation [24] makes clear that the expression refers to shell coordinates and shell measurements. (It is all right to leave the reduced circumference r in the right-hand side of this equation, since each spherical shell is stamped with its individual radius. By looking at this stamp, any observer can determine which shell he is talking about.)

At the reduced circumference r = 8M, or four times the Schwarzschild radius (2M) at the horizon, the particle falling from rest at infinity is moving inward at half the speed of light, as witnessed by shell observers. As the particle crosses the event horizon at r = 2M, nearby shell observers record it as moving at the speed of light. [Shells—and shell observers— cannot exist inside the horizon (see the following section) or even at the horizon, where the spherical shell experiences infinite stresses. The prediction of equation [24] at the horizon must be taken as a limiting case, an extrapolation.]

Shell observer measures a different speed of in-falling stone

Shell observer clocks stone at speed of light at horizon

Radical difference between "shell speed" and " bookkeeper speed " of stone at horizon

Figure 5 Computer plot of the two velocity values for a plunging stone (treated here as positive) A stone falling radially from rest at infinity has speed cfrsheii^sheiias measured by observers on shells through which the stone plunges and speed dr/dt as derived from the records of the Schwarzschild bookkeeper. At the horizon, the shell speed rises to the speed of light (equation 124]), while the bookkeeper speed drops to zero (equation [21])

What a contrast between these nearby measurements and the bookkeeper speed dr/dt, a speed that goes to zero at the event horizon! (See Figure 5.) There the in-falling stone moves with the speed of light as recorded directly by one observer (equation [24]); it moves with zero speed as reckoned by another observer (equation [21]). Nothing demonstrates more dramatically how far we have come from the phenomena that take place in flat spacetime as described by special relativity!

How can you use the time transformation [23] to describe an in-falling stone plunging from one spherical shell to another shell of different reduced circumference? Let a firecracker explode at each shell as the stone passes Clocks on these different shells "run at different rates" according to that very equation [23] How can you possibly combine readings from these two different-rate clocks to meter the shell time the stone takes between the two flash emissions?

For once we are caught by the sloppy way physicists do calculus' In writing the Schwarzschild metric, we use differentials dr, dt, and so forth to remind ourselves that event pairs so separated must be near enough to one another in spacetime to justify using a single value of r. In many cases this criterion can be met for events separated by many meters of distance and time When we measure velocity, however, we employ calculus in the conventional sense of a limiting process, with dr and dt and drsheii and cffShe|| all tending to zero. Then the firecrackers go off right next to each other; the two clocks are—in the limit—on the same shell, so they run at the same rate at this limit.

In summary, for a radially plunging object that starts from rest at infinity, the principle of constancy of energy tells us that the inward speed measured by shell observers increases steadily for smaller values of r, rising to the speed of light at the horizon. In contrast, the inward speed of the object drops to zero at the horizon when reckoned from the accounts of the Schwarzschild bookkeeper.

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