### List of Answers to Exercises in Spacetime and Geometry : An Introduction to General Relativity – by Sean M Carroll

Here are some answers I have found or completed myself. I would love to hear about more.
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 1 Bouncing Ball: My AnswerAxolotl
Exercise 2 Three Torus: My AnswerAxolotl
Exercise 3 Events ABC: My AnswerAxolotl
Exercise 4 Super-luminal effects: My Answer  | Axolotl  | bonus
Exercise 5 Particle accelerator: My AnswerAxolotl
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 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 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 8 Exterior derivative: My wrong answer | bonus
Exercise 10 Maxwells equations in 2-dimensional spacetime: My Answer

#### Chapter 3

Exercise 1: Consequences of metric compatibility: 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 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

#### 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)

#### 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

Axolotl: Petra Axolotl's blog
PhysicsFPhysics 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

1. 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.

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

3. hello dear.
I need the answer of exercise 4 for chapter6.
So thanks