Time Travel Using the Black Hole Stable Circular Orbits

You are on a panel of experts called together to evaluate a proposal to travel forward in time using the difference in rates between a clock in a stable circular orbit around a black hole and our far-away clocks remote from the black hole. Give your advice about the feasibility of the scheme, based on the following analysis or some other that you devise.

A. Consider two sequential ticks of the clock of a satellite in a stable circular orbit around a black hole. We want to find the ratio dx/dt. The numerator in this fraction is equal to the wrist-watch time dx between the ticks in the frame of the satellite; the denominator is the far-away time lapse dt recorded by the Schwarzschild bookkeeper. Use the expression for angular momentum to eliminate dty from the Schwarzschild metric in this case to obtain

Substitute from equation [47] and simplify to show that s-Hr-o-2?)"2 [s9'

B. What is the value of the ratio dx/dt for the stable circular orbit of smallest possible radius, r = 6M?

C. What rocket speed in flat spacetime gives the same ratio of rocket clock time to "laboratory" time"?

D. Based on this analysis, do you recommend in favor of—or against—the proposal for time travel using stable circular orbits around a black hole?

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