**HW 1:** generally solved very well, with a class average of 3.62 (out of 4).

**Quiz 1:** The class average was 3.83 (quizzes are always out of 5).

- (5) Many didn't get the second part of this question right. The answer can be found here. Notice that this is a 'why'-question.

**HW 2:** generally solved very well, with a class average of 1.88 (out of 2).

**Quiz 2:** The class average was 3.67.

- (1) The left or rear light actually flashes first. Ask me and I'll draw a spacetime diagram on the board in class.
- (5) Surprisingly many didn't get this one right. But I actually explained the answer to this in the lecture before the quiz! You absolutely have to know how Einstein's simultaneity works!

**HW 3:** generally well solved very well, with a class average of 3.41 (out of 4, although no one scored 4 points). For the first problem, I would have expected that you don't simply state what is moving with respect to what, but also what electric currents are present.

**Quiz 3:** The class average was 3.46 (highest score was 4).

- (2) I told you in my comments after last time that you absolutely have to know how Einstein's simultaneity works, but many didn't...
- (3) It's exactly the same procedure, but the result will be different, as simultaneity is, after all, relative.
- (4) It doesn't suffice to just remark that the hypersurface is tilted so that when A reads '4', B's clock only reads '3'; it is that the actual time lapsed between equidistant hyperplanes of simultaneity is shorter on B's clock than on A's. Look it up here.

**Quiz 4:** The class average was 4.00.

- (1) It is important not to just state
*that* the discovery of non-Euclidean geometry threatened Kant's theory, but *how*.
- (4) Actually, I really wanted you to mark a circumference and radius
*in the drawing*.
- (5) Notice that there are
*two* parts to the question.

**HW 6:** The class average was 5.28.

- (5b) Many didn't really draw worldlines in a space-time sheet of masses aligned at the same altitude but just some group of converging geodesics.
- (6b) Stating that Minkowski spacetime is flat and
*therefore must have vanishing stress-energy tensor* (presumably by virtue of the Einstein equations) doesn't answer the question. Instead, one needs to first state both that it is flat and is a vacuum solution, and that therefore, the Einstein equations are satisfied.

**Quiz 5:** The class average was 3.36.

- (2) Almost everybody chose the bending of light.
- (3) Spatial geometry: in GR, space-space sheets are curved, which they are not in Newtonian theory; Big bang: in Newtonian theory, the big bang is the expansion of a nugget of matter into a pre-existing spacetime, whereas in GR, it's the explosion of space itself, as it were.

**HW 7:** The class average was 3.9. Some of you have now reached 25 points and need not submit any more homework. I will, however, still correct it if you do!

- (3c) Many of you didn't get this one. Perhaps we should go over it in class.
- (4)-(6) I only gave you a maximum of 2 points for these three exercises.

**Quiz 6:** The class average was only 2.58, with the highest score being 4.

- (1) Notice that this question, like a number of others in this quiz, contain several sub-questions!
- (2) Again, there are two things you're supposed to do. First, say that this is a conformal diagram of a Schwarzschild spacetime.
- (3) Bohr's theory contradicted classical electrodynamics because it assumed stable (and discrete) orbits on which the move without radiating off energy. (And there's a second sub-question)
- (4) momentum = h / wavelength (Plus another sub-question)
- (5) The answer can be found in the chapter on relativistic cosmology ("The cosmological constant (lambda)")

**Final exam:** The class average was 20.1 (out of 30), however, with a rather large standard deviation.

Have a happy spring break!