Measuring the Value of r Inside the Horizon

Outside the horizon, r-values are determined by measuring the circumference of each spherical shell and dividing this circumference by 2jc (Section 4 of Chapter 2) Hence we call r the reduced circumference But no stationary concentric spherical shell exists inside the horizon Question How can the rain observer possibly determine the reduced circumference r of her location' Answer By measuring a piece of circumference

Two raindrops fall radially side by side past shell AA' outside the horizon, as shown schematically in the Schwarzschild map of Figure 2 One raindrop falls along the straight radial path ABCO and its companion along the nearby straight radial and converging path A'B'C'O Draw a circular segment connecting AA' and similar circular segments connecting BB' and CO Now the angle AOA' at the center is the same as the angle BOB' Hence each of the circular segments AA' and BB' represents the same fraction of the entire circumference of the circle of corresponding radius In equation form,

/ Length of ^circular segment AA

Circumference ofN

spherical shell passing through Ay

Length of circular segment BB',

''Circumference of^

spherical shell passing through B.

Call rA the reduced circumference at point A and the reduced circumference at point B Then the denominators of the two sides of the equation become 2jtrA and 2nra, respectively, and we can cancel the 2n from both sides. Now, if the angle at the center is small, the length of the circular segment is approximately equal to the straight-line distance between A and A' Call this distance AA'. And call

Horizon

Figure 2 Schwarzschild map of paths of two in-falling raindrops. By measuring their separations, such as AA' and CC', the in-falling observer can deduce her radius, even when she has passed inside the horizon.

BB' the corresponding straight-line distance between B and B' Then equation [5] becomes, for equal angles,

Suppose a diver reads the labeled radius rA of a spherical shell as she passes it and at that instant measures the distance AA' between the two raindrops Then at a later time she measures the raindrop separation to be BB' From this measurement she deduces the radius rB at that later time to be

Both points A and B lie outside the horizon We now assume that the same formula [7] holds for any radius, even for a radius rc less than that of the horizon In other words, the infalling observer can measure her radial position rc at point C inside the horizon using the equation

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