Here are some answers I have found or completed myself. I would love to hear about more.

******** SPOILER ALERT *************

Please don't use these for cheating in tests. Do use them for checking your answer or getting a hint. And remember they may be wrong! You don't want to end up like the well known cheat pictured, do you?

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**Chapter 1**

Exercise 7 Tensors and vectors: My Answer

Exercise 8 Un-conserved dust energy momentum tensor: My Answer

Exercise 9 Energy momentum tensor of point particles: My Answer | bonus

Exercise 8 Un-conserved dust energy momentum tensor: My Answer

Exercise 9 Energy momentum tensor of point particles: My Answer | bonus

Exercise 10 Electric and magnetic field 3-vectors: My Answer | bonus

Exercise 11 Maxwell in Relativity: My Answer

Exercise 12 Energy momentum tensors of two field theories: My Answer

#### Chapter 2

Exercise 1 Infinite cylinder: My Answer | Philip Saad (2.4) | bonus | PhysicsF

Exercise 3 2D Torus is a manifold: My Answer

Exercise 3 2D Torus is a manifold: My Answer

Exercise 4 Commutators: My Answer | bonus x 2

Exercise 6 Helix and tangent vector: My Answer

Exercise 7 Prolate spheroidal coordinates & Kepler problem: My Answer | bonus

Exercise 6 Helix and tangent vector: My Answer

Exercise 7 Prolate spheroidal coordinates & Kepler problem: My Answer | bonus

Exercise 8 Exterior derivative: My wrong answer | bonus

Exercise 10 Maxwells equations in 2-dimensional spacetime: My Answer

Exercise 10 Maxwells equations in 2-dimensional spacetime: My Answer

#### Chapter 3

Exercise 1: Consequences of metric compatibility: My answer

Exercise 2: Spherical gradient divergence curl as covariant derivatives: My answer

Exercise 3: Christoffel symbols with a diagonal metric: My Answer

Exercise 4: Paraboloidal coordinates: My Answer

Exercise 2: Spherical gradient divergence curl as covariant derivatives: My answer

Exercise 3: Christoffel symbols with a diagonal metric: My Answer

Exercise 4: Paraboloidal coordinates: My Answer

Exercise 4a: as 4 for simpler Spherical coordinates: My Answer

Exercise 5: 2-sphere: My Answer | bonus

Exercise 6: Metric outside Earth: My Answer | bonus

Exercise 8: Vital statistics of a 3-sphere: My Answer | bonus

Exercise 10: *Utah (4.2)

Exercise 12: Derivatives of Killing vectors My Answer | bonus x 2

Exercise 13: *Utah (4.3) | Guth

Exercise 13: *Utah (4.3) | Guth

Exercise 14: Killing vectors on two-sphere My Answer | bonus

#### Chapter 4

Exercise 6 Killing vector: *UCSB (6.1)

#### Chapter 5

Exercise 3: Inside the event horizon My Answer | bonus x 2

Exercise 4: *Utah (5.3)

Exercise 5: Observer and beacon outside black hole My Answer | bonus x 1

Exercise 5: Observer and beacon outside black hole My Answer | bonus x 1

#### Chapter 7

Exercise 2 Thin spherical shell: *UCSB (7.1)

Exercise 4 Harmonic gauge: *UCSB (7.2)

Exercise 6 Head-on collision: *UCSB (7.3)

Exercise 7: *Utah (6.3)

Exercise 7: *Utah (6.3)

#### Chapter 8

Exercise 1: N+n+1-dimensional spacetime: My answer

#### Appendix G

Exercise 1: Conformal Null Geodesic:s My Answer

#### Self imposed exercises

And here are some bonus exercises for masochists!

Exercise SI.01 Simple Lagrangians: My Answer

Inverses, determinants, Levi-Civita symbol and Laplacian: My Answer

### Notes

**My Answers**: On this blog

**Axolotl**: Petra Axolotl's blog

**PhysicsF**: Physics Forums

**bonus**: There is bonus material in my answer from other answers I have found.

**I have copied all the solutions below to my Google drive here before they were deleted.**

__University of California, Santa Barbara (UCSB)__
From a set of 7 assignments from October 15 to December 10 2014. 13 answers.

There are many other questions and answers there.

__University of Utah (Utah)__From a course by Pearl Sandick in Spring 2018. 10 answers. There were three typos and one extraordinary claim in the first one I looked at, Exercise 7.

__Guth__2 solutions from Semantic Scholar by Professor Alan Guth

Firstly, Thank u for your answer, I think there's something wrong with equation 42 in your chapter 3 exercise 4(b) answer, the basis of a vector should be the transformation of the down index, instead of the up index, and then the basis of the dual vector is also wrong.I am a beginner of GR. If there is any mistake, I hope you can correct me. Thank you.

ReplyDeleteThanks Chun! I think you are right. I have added your comment as not in the document.

ReplyDeletehello dear.

ReplyDeleteI need the answer of exercise 4 for chapter6.

So thanks

After a 10-year break, I decided to pick up GR again, using Carrol's book this time. Do you intend to make your solutions complete? If so, we can work together.

ReplyDeleteI see that you have not provided an answer to Ex. 13, Chapter 1. The gist of the exercise is that the extra term is a total derivative and therefore won't affect equations of motion. In fact, it is a so-called Chern-Simons term. For some other more interesting non-Abelian Gauge theories, such a term can have global topological consequences (i.e. instantons). For EM in 3+1 dimensions, it has no consequence.

ReplyDeleteIf you are still interested in 2.9. https://www.physicsforums.com/threads/differential-forms-integration-exercise.378255/

ReplyDeleteFor posterity, Exercise 2.11 is quite easy for those who have studied QFT. Here is a brief sketch. (a) The action has to have 0 mass-dimension. $A^{(3)}$ as a 3-form needs to be integrated over a 3-volume, of which one dimension is time. So the object itself must have two spatial dimensions. (b) The dual of a now 4-form gauge field is a 7-form field. So We need to integrate over a seven-sphere. The conservation part is a direct consqeuence of Stokes's theorem for forms. (c) Following the same reasoning as (a), the answer is 5 since tilde-A is a 6-form. (d) Only F and *F are allowed to appear in the Lagrangian. The reasoning is exactly the same as in vanilla EM.

ReplyDeleteI picked up Carroll to refresh some of my skills in GR (I'm working on my own solution set just for my own sake). These have been very helpful for checking and giving a nudge here and there. Just wanted to say thanks for posting them!

ReplyDelete