Here are my solutions to various problems in Thomas A. Moore’s textbook *A General Relativity Workbook*. As always, no guarantees that the answers are correct, but if you spot any errors, comments are always welcome.

After some consideration, I’ve decided to repost this index to the solutions. I understand that some folks may be concerned that I am providing ‘free’ answers to problems that some students have been assigned as homework, but there are several points that should be made:

- There are already many web resources (including physics blogs, tutorial sites and other pages where solutions are provided) available that provide answers to problems, so I’m not providing solutions that aren’t available elsewhere. What I
*am*providing, hopefully, are background explanations of the theory that expands on, or describes in a different way, material found in the textbook. Judging by comments I have received, many people find these explanations helpful, and that is my main goal. - If a teacher is concerned about students copying answers from the internet, s/he should consider making up their own problems. I did this for the courses that I taught in my 25 years as a university lecturer. For a field as rich as physics, it shouldn’t be too difficult to come up with original problems.
- If you’re a student seeking to copy solutions, you should realize that you won’t learn much unless you make a genuine effort to solve the problem on your own first. Remember that in most universities, the majority of the marks for a course are obtained from exams, and if you sit an exam without having worked out problems on your own beforehand, your chances of passing are pretty low. In my experience as a teacher myself, I found that most students realize this and do make a genuine effort to learn the material on their own.

**These solutions are the only ones that I’ve worked out so far, so please don’t ask me to post “the rest of the chapters” as I haven’t worked on those yet.
I will get to them eventually.**

There is an official site listing errata in the textbook. If you’re confused by something in the text itself, it’s worth having a look here to see if there is a typo on that page.

**Chapter 3 – Four-vectors**

Boxes

3.4.1, 3.4.2

Problems

P3.1, P3.2, P3.3, P3.4, P3.5, P3.6, P3.7, P3.8, P3.9, P3.10

**Chapter 4 – Index notation**

Problems

P4.1, P4.2, P4.3, P4.4, P4.5, P4.6, P4.7, P4.8, P4.9, P4.10, P4.11

**Chapter 5 – Arbitrary coordinates**

Problems

P5.1, P5.2, P5.3, P5.4, P5.5, P5.6, P5.7

**Chapter 6 – Tensor equations**

Problems

P6.1, P6.2, P6.3, P6.4, P6.5, P6.6, P6.7, P6.8, P6.9, P6.10, P6.11

**Chapter 7 – Maxwell’s equations**

Problems

P7.1, P7.2, P7.3, P7.5, P7.6, P7.7, P7.8

**Chapter 8 – Geodesics**

Problems

P8.1, P8.2, P8.3, P8.4, P8.5, P8.6, P8.7

**Chapter 9 – The Schwarzschild metric**

Exercises

9.1

Problems

P9.1, P9.2, P9.3, P9.4, P9.5, P9.6, P9.7

**Chapter 10 – Particle Orbits**

Exercises

10.1, 10.2, 10.3, 10.4

Problems

P10.2, P10.3, P10.4, P10.5, P10.6, P10.7, P10.8, P10.9, P10.10, P10.11, P10.12, P10.13, P10.14, P10.15

**Chapter 11 – Precession of the Perihelion**

Exercises

11.6.1

Problems

P11.1, P11.2, P11.3, P11.5, P11.6, P11.7, P11.8, P11.9, P11.11

**Chapter 12 – Photon Orbits**

Problems

P12.1, P12.2, P12.3, P12.4, P12.5, P12.6, P12.7,

P12.8, P12.9

**Chapter 13 – Deflection of Light**

Problems

P13.1, P13.2, P13.3, P13.4, P13.5, P13.6, P13.7, P13.8, P13.9

**Chapter 14 – Event Horizon**

Problems

P14.1, P14.2, P14.3, P14.4, P14.5, P14.7, P14.8

**Chapter 15 – Alternative Coordinates**

Exercises

15.1, 15.2, 15.3, 15.4

Problems

15.1, 15.3, 15.4, 15.5, 15.6, 15.7, 15.8, 15.9

**Chapter 16 – Black Hole Thermodynamics**

Exercises

16.1, 16.2, 16.3, 16.4

Problems

P16.1, P16.2, P16.3, P16.4, P16.5, P16.6, P16.7, P16.8, P16.9

**Chapter 17 – The Absolute Gradient**

Exercises

17.1, 17.2, 17.3, 17.4, 17.5, 17.6, 17.7

Problems

17.2, 17.3, 17.5, 17.6, 17.7, 17.8, 17.9, 17.10

**Chapter 18 – Geodesic Deviation**

Exercises

18.1, 18.2, 18.3, 18.4

Problems

18.1, 18.2, 18.3, 18.4, 18.5, 18.6, 18.7, 18.8

**Chapter 19 – The Riemann Tensor**

Exercises

19.1, 19.2, 19.3, 19.4, 19.5, 19.6

Problems

19.1, 19.2, 19.3, 19.4, 19.5, 19.6, 19.7, 19.8, 19.9

**Chapter 20 – The Stress-Energy Tensor**

Exercises

20.1,

20.2, 20.3, 20.4, 20.5

Problems

20.1, 20.2, 20.3, 20.4, 20.5, 20.6, 20.7, 20.8, 20.9, 20.10

**Chapter 21 – The Einstein Equation**

Problems

21.1, 21.2, 21.3, 21.5, 21.6, 21.7, 21.8, 21.9

**Chapter 22 – Interpreting the Equation**

Exercises

22.1, 22.2, 22.3, 22.4, 22.5, 22.6

Problems

22.1, 22.2, 22.3, 22.4, 22.5, 22.6, 22.7

**Chapter 23 – The Schwarzschild Solution**

Exercises

23.1, 23.2, 23.3, 23.4, 23.5

JERRYThanks Glenn for this series of solutions to this workbook. They have been extremely helpful in understanding this very difficult subject. Hope you continue with the remaining chapters

MihiThank you very much

Asher WeinermanThank you! Just a comment on p12.7: Use the + sign in the denominator for r3GM. Then the curve is smooth and continuous through 4GM to 3GM to 2Gm even to 1GM and even 0. Note that at 2GM the angle is not 0 because both the numerator and the denominator go to zero. The limit as r approaches 2GM is some angle (I can’t remember) between 35 and 45 degrees. Even at 1GM the angle is somewhere around 50 degrees (I can’t quite remember).

growescienceThat’s actually what I say after the first plot. I guess what I’m unsure of is why we’re allowed to do this.

Chris KranenbergHi Glenn,

If you are interested, I have a solution for Exercise 18.5. Again, your posts are of great value to the independent learner. Thanks again for taking the time posting solutions.

Best Regards,

Chris

gwrowePost authorThanks but I’d like to give it a try first. I’m currently wrestling with trying to get into quantum field theory. There are a lot of books out there but I haven’t found one yet that gives a gentle introduction. There’s definitely a gap in the market for a QFT book along the lines of Moore’s book on GR.

Chris KranenbergUnderstood. Best of luck in your search of a QFT text to your liking.

Christian FortinI find your comments around each problem very interesting and also reassuring.

The problems in the book by Moore are not difficult when we follow the instructions (practically algorithmic) of the author. It requires more thoughts (or readings of other chapters or books) to understand the eventual usefulness of the idea or model behind the problems.

Several problems seemed to me a little too long to develop, maybe because I do them by hand and not with the help of an algebraic software. It was reassuring to see that you do them in a similar way or length and often in a more detailed and pedagogic / andragogic way.