## G6    Problems and Suggestions

1. Slowly pour colorless lemon syrup into a glass, which is already half filled with water. Aim the beam of a laser pointer through the glass and watch as the locally different indices of refraction provoke varying local light speeds and thus direction changes!
2. Correct and clarify the statement in [11] on page 124! It deals with the contraction of yardsticks in the vicinity of large masses.
3. Visit the website www.zarm.uni-bremen.de/index.htm to the drop tower in Bremen.
4. Calculate the Schwarzschild radius of the moon, the earth and the sun. Compare them with the effective radius of the spherical body, ie determine the ratio Rs / R.
5. Calculate the ratio Rs / R for an atom and a nucleus. Must the effects of GTR be taken into account in nuclear physics?
6. Read the Einstein biography [31] from Thomas Bührke !
7. Look for websites or documents describing the experiments of Eötvös and Dicke on the equality of inertial and gravitational mass, and study the basic idea of these experiments.
8. The American physicist Richard P. Feynman proposed the following to illustrate length contraction: Consider a huge stove, which is heated so that it is cool in the middle and gets warmer the further you go from the center. If done correctly then yardsticks in the radial direction will have the correct length due to thermal expansion! What is the power of this idea - and what are its flaws ?
9. Devise appropriate clocks for Feynman's hotplate version of the Schwarzschild metric.
10. How many seconds does a lifetime of 80 years have, if one lives in the Maldives or if one lives in the High Andes at an altitude of 4000 m ?
11. A whole new kind of matter has been found in a meteorite in which inertial and gravitational mass differ. How can we ever ascertain that? What would be the consequences for the gravitational theory of     a) Newton     b) Einstein
12. What would be ‘rectilinear’ in a gravitational field in which light rays are bent?
13. Show that a clock on board a satellite orbiting in a circle with radius r in a Schwarzschild field of mass M runs slower by the factor (1 - 3 • α / (2 • r)) than an identical clock at OFF position. You need both STR and GTR !
14. Calculate the quotient of the speed(s) of time of two clocks, which are near earth’s surface at locations with a height difference of Δh, by replacing gravity with a rocket of length Δh which is being accelerated with value g in a gravitation-free field. Set the initial speed to 0 and take into account the Doppler effect! When the signal arrives at the top, the clock at the top is already moving a bit faster than the clock at the bottom was when the signal was emitted ...

Some of the most beautiful ‘Einstein rings’ from the collection of the Hubble Space Telescope. They arise because the gravitational field of the central bright orange galaxy acts like a lens on the beam of blue light of a far distant source (usually a quasar). We then see the same object all around the edge of the foreground galaxy. Also the intensity of the light of the background object is massively increased. Einstein theoretically predicted such gravitational lenses in 1936, but he was rather pessimistic about the possibility of this effect in fact ever being observed. It did take 60 years ...

-> http://hubblesite.org/newscenter/archive , key words: ‘gravitational lens'