Question #1 (20%) - non-math
You observe an astronomical object to vary (change its brightness)
on a time-scale of 0.001 seconds.
Is there a way to estimate the maximum possible size of this object?
Question #2 (20%) - non-math
A space shuttle orbits the Earth and thus is an inertial reference
frame. It passes by a satellite that
also orbits the Earth but in the opposite direction.
From the point of view of the shuttle crew, the satellite is not
moving in one direction with the uniform speed, yet, it is
also an inertial reference frame. Explain this contradiction.
Question #3 (20%) - non-math
The conformal diagram below shows the space-time of the non-rotating
black hole. The black hole exists forever since the singularity and the
horizon touch the point of the infinite future. This is not realistic -
black holes do evaporate with time. Change this conformal diagram
so that the black hole only exists for a finite time and then
disappears. You can simply add your lines and cross over the wrong ones
on this plot, or supply your own drawing on a separate sheet of
paper. (Hint: since the black hole has to disappear at some finite
moment in time, the point of infinite future must lie above the
singularity.)
Question #4 (20%) - non-math
Imagine a sphere with a given size, say, a soccer ball (or any other
size). If we want to make the most massive object of that size, what
would this object be? Please give a full explanation for your answer.
Question #5 (20%) - non-math
Consider the following experiment (which you can actually perform): Obtain a
spring scale (e.g., a typical bathroom scale), place it in an elevator, and
stand on it. Note the exact value when the elevator is at rest. Now
ride up several floors. As the elevator starts up, there is an acceleration
upward. Note how the reading on the spring scale changes. Next ride down.
When the elevator starts down, note how the reading changes. Once the
elevator reaches a constant velocity up or down, note the reading of the
scale. What do you predict these three readings would be (compared to the
reading in the elevator at rest)?
Question #6 (20%) - math
A cosmic ray proton, moving in the frame of the Earth with the boost factor
of 20, hit the Earth's atmosphere and eventually came to rest (as measured
on Earth). During the
interaction it emitted three muons (proton mass is 9 times larger than
the muon mass) and no other particles. The first muon had a boost factor
of 100, the second muon had a boost factor of 50. What was the boost factor
of the third muon? (Hint: before you start, think what physical law you
will need to use in order to solve this problem.)