## J F M I977 Gate

Figure 4 Final results for Mars time delay experiments. Shapiro and colleagues increased the amplitude of the return signal greatly by using a Viking lander on the surface of Mars The radio signal from Earth triggered a return signal from the Viking lander. This figure shows the differences (technical term: residuais) between the observed data and the predictions of general relativity The dashed vertical line marks the time of closest approach of Mars to Sun on November 25, 1976 This figure looks much less dramatic than Figure 3 but "squeezes" general relativity much harder Note that the vertical scale is marked in tenths of a microsecond. See the reference at end of this project.

How lucky we are! We calculated 250 microseconds for the Earth-Mars round-trip delay, a value very close to the observed result. Our calculated result is also very close to the more precise value calculated by Shapiro using (1) the straight path that skims past Sun and (2) the Earth-Sun-Mars distances correct at the time of the experiment.

Figures 3 and 4 present observed results of Shapiro and colleagues for Venus and Mars respectively.

QUERY 8 Distance to Moon. Astronauts have left several corner reflectors on the surface of Moon. These are shaped like the corner of a room and have the property that they reverse the path of any laser pulse Incident on them. As described by Clifford Will (Chapter 7 in the reference at the end of this project), a round-trip laser pulse sent from Earth can be timed to measure the Earth-Moon distance between laser emitter and Moon reflector. Initially the uncertainty was about 15 centimeters; more recently the uncertainty has been reduced to the one-centimeter range. Does the Shapiro time delay have to be taken into account in doing this measurement? Will the distance measurement be wrong if no account is taken of the reduced speed of light in the curved spacetime near Earth and Moon? If so, what will be the approximate error in the measurement? In traveling to Moon, remember, the pulse moves outward through Earth's gravitational field and inward through Moon's gravitational field. Directions are reversed on the return trip, but the time delay adds for each direction. Hints: (1) Adapt equation [4] of the text separately for Earth and Moon. (2) The average distance between the centers of Earth and Moon is given inside the back cover. (3) The "equilibrium point" where (as Newton would say) the gravitational attraction is equal and opposite to Earth and Moon is at 0.90 the distance from Earth's center to Moon's center. (4) The ratio of masses is MMoon/MEart^ = 0.0123. (5) The ratio of Moon to Earth radii is rMoon/rEarth = 0.273.

QUERY 9 Delayed replay. Discussion question. You are in a stable circular free-float orbit around a black hole. You send yourself a message coded on a laser signal, firing your laser pulse inward so that its trajectory lies across your orbit, approaching nearer the black hole than you do. Can you use the light-retardation effect in such a way that you receive your own signal when you arrive somewhere else in your orbit? If not, why not? If so, what limits are there on the amount of delay you can experience in receiving your message? Hint: Look at Figure 6 on page 5-14.

### 6 References and Acknowledgments

Initial quote, The Quotable Einstein, edited by Alice Calaprice, Princeton University Press, 1996, page 182, and Albert Einstein by Albrecht Fôlsing, Penguin Books, New York, 1997, page 13.

Figure 3. "Fourth Test of General Relativity: New Radar Result," Irwin I. Shapiro, Michael E. Ash, Richard P. Ingills, William B. Smith, Donald B. Campbell, Rolf B. Dyce, Raymond F. Jurgens, and Gordon H. Pettengill, Physical Review Letters, Volume 26, Number 18, pages 1132-1135 (3 May 1971).

Figure 4. R. D. Reasenberg, 1.1. Shapiro, P. E. MacNeil, R. B. Goldstein, J. C Breidenthal, J. P. Brenkle, D. L. Cain, T. M. Kaufman, T. A. Komarek, and A. I. Zygielbaum, "Viking Relativity Experiment: Verification of Signal Retardation by Solar Gravity," Astrophysical Journal Letters, Volume 234, pages L219-L221 (15 December 1979).

The fascinating story of experiments verifying the time delay of radio signals passing near Sun is told by Clifford M. Will in his book Was Einstein Right? Putting General Relativity to the Test, Second Edition, Basic Books/ Perseus Group, New York, 1993, Chapter 6.

Irwin Shapiro read and commented on this project.

Query 8 was suggested by Glen Govertsen. Query 9 was suggested by Charles Holbrow.

Project F The Spinning Black Hole

1 Introduction F-1

2 Angular Momentum of the Black Hole F-2

3 The Kerr Metric in the Equatorial Plane F-2

4 The Kerr Metric for Extreme Angular Momentum F-5

5 The Static Limit F-7

6 Radial and Tangential Motion of Light F-10

7 Wholesale Results, Extreme Kerr Black Hole .. F-11

8 Plunging: The "Straight-ln Spiral" F-13

9 Ring Riders F-16

10 Negative Energy: The Penrose Process F-20

11 Quasar Power F-24

12 A "Practical" Penrose Process F-28

13 Challenges F-29

14 Basic References to the Spinning Black Hole .. F-31

15 Further References and Acknowledgments ... F-33

• How can I observe space dragging around a spinning black hole?

• Does a rapidly spinning black hole keep me from falling through the horizon?

• How many Sun masses per year does a quasar convert to light?

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