Homework 4 : The General Theory of Relativity

(due at the start of class on the 24^th October 2002)

1. [5 points] What makes the Special Theory of Relativity less general than the General Theory of Relativity?

2. [5 points] Describe two aspects of the binary pulsar PSR1913+16 that require General Relativity to explain.

3. Many science fiction writers have imagined futuristic space-stations that produce “artificial gravity” via rotation. The figure below illustrates one possible design for such a space-station. It consists of a giant ring that is rotated about its center. Astronauts living in the ring will feel “artificial” gravity due to the centrifugal force that seems to push them onto the outer wall of the ring.
1. [5 points] Why is the centrifugal force experienced by the astronaut best described as a fictitious force?
2. [5 points] Suppose that there are two clocks on the space-station, one at the center of rotation (clock A on the diagram) and one on attached to the rotating ring (clock B on the diagram). Using Special Relativistic arguments, describe why clock B runs more slowly than clock A. Is there any step in your argument that makes you slightly uncomfortable?
3. [5 points] One of the astronauts is standing on the outer wall of the ring observing the two clocks A and B. He also observes the clock B running slower than clock A. To what would the astronaut attribute this effect? [Hint: your answer should invoke the Strong Equivalence Principle.]

4. [5 points extra credit] The full space-time diagram for a black hole can be very complicated, but it can be transformed (i.e. stretched, pulled, squeezed etc.) into the kind of space-time diagram that we’ve discussed. Below are the transformed space-time diagrams for two types of black hole (a “Schwarzschild” black hole which corresponds to a precisely non-rotating black hole, and a “Reissner-Nordstrom” black hole which corresponds to a charged black hole). As usual, the light rays follow straight line paths oriented at 45 degrees on these space-time diagrams. In which of these cases is it possible to avoid the singularity and, therefore, avoid destruction? What would happen to you in this case? Where would you end up? Explain your answers.