Log for exercise classes (Sofia)
Old log 08
- Exercise 1:
We discussed the meaning of symmetries and relativity. We also discussed Galileo transformations, some about Lorentz transformations and the invariance of the Minkowski length S^2 = x^2+y^2+z^2 - (ct)^2 under Lorentz transformations. We did [RQ] 4,6,1,2 in Chapter 1 and [C] 2.1,2.6.
- Exercise 2:
We did Review questions 3, 7, 9 of chapter 2 and problem 2.8 in Cheng. We also solved the problems 5, 6 and 7 from the problem collection Oe1.
- Exercise 3:
We discussed space time diagrams in Review questions 10 and 13 from chapter 2 in Cheng. We also solved problems 8 and 9 from Oe1 and discussed index notation for problem 2 from Oe1. We briefly started question 3a (second question 2) in Oe1 but saved most of it for next class due to lack of time.
- Exercise 4:
We did problem 3 in Oe1, started in previous exercise. We also did, from Cheng, problem 10.1 (basis and inverse basis vectors and metric) and problem 10.7 (velocity addition).
- Exercise 5:
The first half was dedicated to the mid term test (KS1) and in the second half we did, from Cheng, problem 10.8 (gravitational redshift), problem 3a (equivalence principle for the inclined plane) and discussed the twin paradox (problem 3.3).
- Exercise 6:
We made some more remarks om the twin paradox. The exercise was otherwise spent on variational calculus and also the metric of the two dimensional sphere. We did problems 1 and 3 in Oe2 and problems 4.1 and 4.2 from Cheng.
Some people discussed during the break the possibility of solving problem Oe2.1 using the Euler-Lagrange equation on L^2 instead of L. This is possible but not in the naive (simple) way. (Discussed in following lecture.)
- Exercise 7:
Most of the exercise was spent on problem 4.7 in Cheng, discussing the 3D positively (sphere) and negatively (pseudo-sphere) curved objects and their embedding in 4D flat spaces. We also did (from Cheng) 4.3 and 4.9 (how to detect curvature through measuring the relation between circumference and radius of a circle).
- Exercise 8:
We discussed the review questions 1 and 2 in Cheng, chapter 5. We also did problems (Cheng) 5.2 and 5.3 (discussing spatial distance between two neighboring points for curved spaces) and 5.6 (symmetry of the Christoffel symbol).
- Exercise 9:
We discussed the relation between proper time and coordinate time in problem 5.1 from Cheng. We also performed a rather long calculation in deriving light deflection around massive objects using the same procedure as for the perihelion precession of Mercury, for problem 6.2 in Cheng. Unfortunately there was not enough time to do the simpler problem 6.1.
- Exercise 10:
In the exercise class after the kontrollskrivning (mid term test) we discussed Christoffel symbols for given metrics and the connection with the geodesic equations, in problem 6a and 6b from Oe2.
- Exercise 11:
We did problem 1, 2, 3a and 4 from Oe3. We also did problems 11.1, 11.3 and 11.14 from Cheng.
- Exercise 12:
We discussed the Riemann and its properties, in problem 11.7, 11.8, 11.13 and 11.11 from Cheng.
- Exercise 13:
We discussed the behavior of electromagnetism under Lorentz transformations (special relativity) in problem 1 from Oe4. We also did half of problem 3 from Oe4, about how to find the Riemann tensor, before running out of time.
- Exercise 14:
In the hour after the final kontrollskrivning we finished problem 3 from Oe4. I mentioned the Ricci tensor for spherically symmetric metric from problem 6a (in Oe4) and we started solving Einstein equation for the Schwarzshild scenario in problem 6b.
- Exercise 15:
The plan is to, in the first hour, finish problem 6b and go through problem 10 from Oe4 about free fall towards a black hole. The second hour will be for you to ask questions before the exam.