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?

This is discussed on page 251 in the textbook H&H. Imagine a round object that is dark. Now the center flares, and light starts to propagate from the center toward the edge. The object gets all lit up after the light reaches its edge. Thus, if we know the time it takes the object to brighten up, i.e. it takes the light to go from the center to the edge, we also know the size: it is simply the distance the light travels during the time the objects brightens up. This is the maximum possible size - the object can be smaller: if we switch the light in the room on and off every second, it does not mean that the room is one light-second (300 thousand kilometers or 186 thousand miles) across.

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.

There is actually no contradiction. According to General Relativity, a massive body (like the Earth) curves the space-time, and makes straight lines curved. Thus the statement that all inertial reference frames move with constant velocity (i.e. with constant speed along a straight line) is only true in the flat space-time. In the curved space-time there may not be straight lines at all, like on a surface of a sphere: no matter what kind of line you draw on a ball, it will not be straight. In the curved space-time inertial reference frames move along special lines called geodesics, and an orbit of a space shuttle or a satellite is just such a line.

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.)

Here is our solution. The dotted black line shows the boundary that needs to be removed, and red lines show new borders. Now observer O (who is outside of a black hole) will see the last image of F falling into the black hole (thick blue line), and at later times O will see no black holes, just the empty space. This is not the only solution, so if your sketch is different, it does not mean that it is wrong!

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.

Let us imagine that we take a soccer ball and start pumping matter in it. We also prevent it from bursting by some super-mechanism. As we pump more and more matter inside, it becomes more and more massive, and its gravity gets stronger and stronger. So, sooner or later it turns into a black hole. But this is only one half of the answer. What if we continue pumping matter into that black hole? The mass of the black hole will grow, but so will its size, since the radius of the black hole (Schwarzschild radius) is proportional to the mass. Thus, the second half of the answer is that we cannot make that black hole more massive and keep its size the same, so the black hole is indeed the most massive object of this size.

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)?

The weak equivalence principle states that the upward accelerated reference frame is equivalent to the gravity force. Thus, there is more gravity in the elevator accelerating upward, so you weigh more. Conversely, there is less gravity in the elevator accelerating downward, so you weigh less. At at constant speed the weight is the same - this is an inertial reference frame in the absence of gravity.

Question #6 (20%) - math
A cosmic ray proton, moving in the frame of the Earth with the boost factor of 20, hit the Earth 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.)

A physical law that you need to use is the law of conservation of energy. The energy does not disappear or appear from nothing, it can only move from one form into another. A proton was initially moving very fast, with the boost factor of 20, and it had a lot of energy. Precisely, it had 20 Mc² where M is the mass of a proton. When it stopped, it had only one Mc², its rest energy. Thus by interacting with the Earth atmosphere, it lost 19Mc² of energy. This energy did not disappear, but went into creating muons. Now, the mass of the proton is 9 times larger than the mass of a muon. Let m means the mass of a muon. Them M = 9m. Thus, the proton lost 19 * 9mc² = 171 muon mc². The energy of the first muon is 100mc² (boost factor of 100), the energy of the second one is 50mc². Thus the energy of the third one is 21mc² to make it up to the total of 171mc².