## I10    Problems and Suggestions

1. Check the numbers in the last column of the table of section I1 using the values from Einstein's formula on the same page.
2. Check the numerical values claimed in the last few lines of text on the Hipparcos satellite at the end of section I2 !
3. In the Schwarzschild metric the diameter of the circle multiplied by Pi, is greater than the corresponding circumference. Epstein's bump yields an additional precession of the perihelion in the direction in which the planet orbits (I1). How could you realize with paper a geometry in which the circumference is greater than the diameter times Pi? How would that impact the additional precession of the perihelion?
4. Show that the value of α = G • M/c2 for the sun in units of ‘lig ht seconds’ is about 4.9261 • 10-6. What is the value of the gravitational constant G in these units?
5. We draw the cross section of the rolled up space-time as described in section H5. On the inside of the mass we have a spherical sector - and on the outside? For weak fields we can assume as a good approximation that y decreases there as a function of type y = a + b / x. Determine that solution f(x), where the x-position of the inflection point is (3 / 4)! What is the physical meaning of the x-Coordinate 3, and how big is RS? Is it still a weak field? Compare clocks on the surface of the body with those in the OFF!
6. When photons rise in a gravitational field they lose energy. Will they be slower or faster for an observer in OFF?
7. How many extra seconds elapse in OFF when in Amsterdam precisely a century has elapsed? Consider the GTR (gravitation of the sun and the earth) and the STR (orbital speed of the earth).
8. Determine based on the middle diagram of section I6 the average speed of the airplane during the 15-hour flight!
9. (Section I4) Determine the relative velocity v which belongs to a Doppler frequency shift of  Δf / f  of 2.22 • 10-15.
10. The nearly circular orbits of the NAVSTAR GPS satellites are so chosen that each satellite orbits the earth exactly twice during a sidereal day.
a) At what altitude above sea level do these satellites orbit?
b) By how many nanoseconds per orbit do the satellite clocks run fast compared to a clock at sea level at the equator, when the orbit passes over the poles?
c) Same as b) but for an orbit in the equatorial plane with the same rotational direction as the earth?
d) Same as c) but in the opposite rotation direction?
11. Visit the web-site http://homepage.univie.ac.at/Franz.Embacher/Rel/ . You can find there programs for STR and GTR. Work through some of them - you should now be well prepared to do so!
12. Search for "Michael Kramer" and "PSR J0737-3039 A/B" and read the web infos about this binary pulsar system that has turned out to be a perfect 'laboratory' to test GTR in the case of strong gravitational fields.

An Einstein poster in the background viewed through one of the high-precision quartz spheres
used in the  Gravity Probe B experiment as a gyroscope, depicted similarly to a convex lens.