(ignoring the fact that acceleration is in the negative radial direction). In these units, what is the value of gE, the acceleration of gravity at the surface of Earth? LetgEconv be the acceleration of gravity in conventional units. Show that in geometric units has the units meter-1 and the approximate value gE = ^co"v = 10 16 meter 1 [51. Newton]
B. What is the corresponding prediction of general relativity? First we need to decide which dr and dt we are talking about. The statement of the exercise specifies that it is the shell worker whose measurements we are to predict. Therefore we want d^sheu/^shell- Start with the result of Exercise 7:
shell dt shell
Take the derivative of this expression with respect to dfshell' remembering that r0 is fixed, a constant. On the right side of the result you will have a factor dr/dtsj^j, where dr is the change in reduced circumference r, not shell coordinate. Use equation [D] in Selected Formulas at the end of this text to eliminate dr from your derivative. Then substitute for drsheii/dfsheii from equation . Evaluate the result at r = r0 to obtain the simple expression (again ignoring the minus sign)
d r shell dt shell
What are the limiting cases (1) as r0 approaches 2M and (2) as r0 becomes very large (but not infinite)?
C. The robot worker stands on a shell of radius r0 = 4M near a black hole of mass 5000 meters. How many "gee"—that is, how many times the value of gE at Earth's surface—is the initial acceleration of his dropped tool? What is the Newtonian prediction? (A fighter pilot risks blacking out when she makes her plane turn or rise at an acceleration of 7gE or more.)
D. What is the acceleration of gravity at the surface of the typical neutron star described in Exercise 3?
E. Optional. You want to hover, rockets blasting, just above the horizon of a black hole. Call your reduced circumference rD = 2M + dr, where dr « 2M. Find an approximate expression for gshell under these circumstances. Now, in order to stay conscious, you wantgsheu in your spaceship to be 7gE. If dr = 1 kilometer, what is the mass M of the black hole you should choose for this maneuver? Express your answer as a multiple of the mass of our Sun.
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