One Way Motion Inside the Horizon

Why can't we escape from a black hole? Why is travel inside the horizon inevitably a one-way trip to the center? This section treats these questions from the point of view of one special diver: a "raindrop," namely, a plunger that falls freely from rest at a great distance. The basic conclusion, however, is true for all travelers inside the horizon: Inside the horizon your radius r decreases inexorably.

The raindrop is a local inertial free-float frame. As in any inertial frame, no object or message can move forward or backward faster than light. The worldline of a passing particle must lie within the so-called forward light cone, moving no faster than light in either the inward or the outward radial direction. In the following analysis we trace out the local worldlines of light moving in the raindrop frame.

Preview: The rain observer launches light pulses toward the center and away from the center. We shall find that inside the horizon, light shot out the front and light shot out the back both move inward. Hence all particles, shot out the front or out the back (necessarily with speeds less than that of light), must also move inward. These facts are important because the only true proof that a horizon exists is the demonstration that world-lines can run through it only in the inward direction, not outward.

QUERY 9 Light cones in rain coordinates

A. Multiply out the following expression to show that it is equivalent to the rain-frame metric [15]:

B. For light (dx = 0) moving radially (d<|> = 0), show that equation [16] has two solutions, one for the "headlight flash" shot inward and one for the "taillight flash" shot out of the back of the raindrop.

C. Show that velocities for both the headlight and taillight flashes can be summarized in the equation

D. Show that equation [17] predicts that inside the horizon (for r < 2M) even the taillight flash, shot out of the back of the raindrop, moves inward.

E. Fill out the argument that, once the raindrop has crossed the horizon, anything launched from the raindrop in either radial direction, no matter at what speed relative to the raindrop, inevitably moves inward.

Figure 5 shows these initial light cones outside and inside the horizon. Note that these light cones display only the initial motion of inward and "outward" light flashes. In the following section we will trace these flashes for some time after their emissions.

QUERY 10 Initial motion of taillight light flash outside and at the horizon. As the raindrop falls inward, find the radial location r/M at which the taillight flash emitted from the back of the descending raindrop moves outward with the following proper velocities:

rain

rain

Figure 5 Computer plot Worldline of a raindrop emitting flashes as it passes inward through the horizon of a black hole Arrowheads show the direction of motion of the raindrop along its worldline Little cones represent light spreading out in all directions from flash emissions along the worldline The lower line segment leaving each dot represents the initial motion of the portion of the flash sent inward (minus sign chosen in equation [ 17]) The upper line segment represents the initial motion of the port/on of the flash aimed radially outward (plus sign chosen in equation [ 17]) Inside the horizon even the portion of the flash aimed radially outward moves inward, toward the center Note that the figure shows only the initial motion of these light flashes For an example of a full trajectory of light, see Figure 7, page B-19.

Figure 5 Computer plot Worldline of a raindrop emitting flashes as it passes inward through the horizon of a black hole Arrowheads show the direction of motion of the raindrop along its worldline Little cones represent light spreading out in all directions from flash emissions along the worldline The lower line segment leaving each dot represents the initial motion of the portion of the flash sent inward (minus sign chosen in equation [ 17]) The upper line segment represents the initial motion of the port/on of the flash aimed radially outward (plus sign chosen in equation [ 17]) Inside the horizon even the portion of the flash aimed radially outward moves inward, toward the center Note that the figure shows only the initial motion of these light flashes For an example of a full trajectory of light, see Figure 7, page B-19.

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