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

A well known cheater and liar
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?
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 6 Differentiation: 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 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 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 


My Answers: On this blog
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.

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

  4. 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.

  5. I 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.

  6. If you are still interested in 2.9.

  7. For 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.

  8. I 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!

    1. Hey Matt, I'm a rookie and working on GR. If by any chance you could share your solutions, I would be extremely grateful as I am really struggling to solve some of these