## Solution

A We want 1 - ~ = 1 - ltf6, which yields r « = 2 x 1 5 x 103 x 106 meters

= 3 x 109meters This radius is approximately four times the radius of Sun

B. This time we want 1 - — » 1 - 10~8, so r r = —^s = 2 x 1 5 x 103 x 108 meters

= 3 x lO11 meters which is approximately twice the radius of Earth's orbit

The embedding diagrams, Figures 6 and 7, represent one cut through the spatial part of the Schwarzschild geometry. Time does not enter, since dt = 0. There being no place on this surface for changing time, it depicts nothing moving. Therefore this representation has nothing to tell us directly about the motion of particles and light flashes through the space-time of Schwarzschild geometry (in spite of all the steel balls you have seen rolling on such surfaces in science museums!). In Chapters 3 and 4 we describe trajectories near a black hole, including trajectories that plunge through the Schwarzschild surface at r = 2M "into" the black hole. But first, Section 11 describes the meaning of "far-away time" t in the Schwarzschild metric.

Curvature of spaceTIME is needed to describe orbits

11 Far-AwayTime

Freeze space; examine curved spacetime.

It is not enough to know the geometry of space alone. To know the grip of spacetime that tells planets how to move requires knowing the geometry of spacetime. We have to know not merely the distance between two nearby points, P, Q, in space but the interval between two nearby events, A, B, in spacetime.

The Schwarzschild metric uses what we call far-away time t. There can be many remote clocks recording far-away time t. These remote clocks form a latticework that extends in all directions from the isolated black hole. Far from the influence of the black hole, these clocks are in a region of flat spacetime, so they can be synchronized with one another using light flashes similar to the synchronization pulse for free-float frames described in Chapter 1 (Section 9). However, in the present case the synchronizing

Far-away time t measured at large r.

SAMPLE PROBLEM 2 Sample of "Radial Stretching'

Verify the statement at the end of Section 4 that for a black hole of one solar mass, the directly measured radial distance calculates as 1 723 kilometers between a shell at r = 4 kilometers and a shell at r = 5 kilometers In Euclidean geometry, this measured distance would be 1 kilometer

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