Time is precious — optimize your approach to learning general relativity
Written by: Robert Scott
Robert B. Scott, author of A Student's Manual for A First Course in General Relativity, 2016 considers the personal and professional reasons for writing his book and the best aspects of solutions manuals, offering advice on solving problems and getting the best out of the many resources available for learning general relativity
Why did you write A Student’s Manual ?
My goal was to make general relativity accessible to a wider audience, including self-learners who don’t have access to a professor or classmates to ask questions or verify their answers to exercises. I had in mind perhaps a retired engineer who studied basic calculus and linear algebra years ago and who is teaching himself general relativity just for the pure enjoyment of learning Einstein’s beautiful theory. But the book should be useful to others: physics students in an introductory general relativity course, the serious philosopher who needs a deeper understanding of general relativity for her analysis on the nature and meaning of spacetime, and so on.
Your book is 370 pages, that seems rather long for a solution manual.
My goal in A Student’s Manual was to be as clear as possible
Solutions that are given in physics textbooks are usually very brief. There is a sort of aesthetic in math and physics that says the more succinct an argument is the more beautiful it is. And instructors often seem to strive for beauty over clarity or transparency especially in solutions. One is often left wondering “But how do we know that? Why can we assume this?” My goal in A Student’s Manual was to be as clear as possible, even if that meant being more wordy. I did my best to explain my thinking and to anticipate the questions the student might have. Of course I cannot get in the minds of all readers and know what they might ask, but I tried to become aware of my own solution-finding process and put it down in words. When the detail is not needed, the student can simply read more quickly.
Why did you write a solution manual instead of writing your own textbook on general relativity?
There are many authors writing their own textbooks trying to improve upon the presentation of other books, succeeding perhaps in some ways but less so in other respects. I wanted to try a different approach, one that focused on engaging the reader in doing exercises because this is such a crucial part of the learning process. This led to the solution manual idea. I also had a more philosophical motivation. I feel there is such a glut of information available in general, and this is becoming true in physics publications as well. I wanted to find a way to work with the existing resources, to help the self-learner make use of existing books. I chose Schutz’s textbook as the source of exercises, but soon I realised I could use my solutions and new exercises as a way of pointing the reader to the benefits of some of my other favourite GR resources. By integrating the benefits of several textbooks, I hope I’ve created a resource that is more helpful than any single textbook.
How did you choose Schutz’s exercises as the basis of your solution manual? Why not one of the many other introductory GR textbooks?
With a firm footing in the basic mathematics the student has the language and tools to learn the subtle physics.
Well, Schutz is one of the best introductory textbooks on GR. It’s a tried and true classic, since 1985, and the new edition released in 2009 brings it up-to-date, especially in cosmology. Schutz has a lot of exercises, 338 in total. That said there are a lot of introductory books now, and several of them are quite good. Schutz is especially thorough in explaining the new mathematics the student will need to understand GR. I also prefer the style of teaching the math first, as Schutz does for example in the familiar setting of first polar coordinates, then curvilinear coordinates. With a firm footing in the basic mathematics the student has the language and tools to learn the subtle physics. This also makes Schutz’s book at better foundation for continued study at a more advanced level.
What are the weaknesses of Schutz’s book?
Study any textbook closely and you will find something you would prefer explained differently. I only found two things I did not like in Schutz’s book. In my opinion the derivation of geodesic deviation is not as clear as the textbook by Hobson et al. So I address this with a supplementary exercise with solution based on the Hobson et al. notation and approach. The other area is perhaps more pedagogically contentious, but personally I find it confusing to say that “time really does slow down” as a physical interpretation of the gravitational redshift effect. Many introductory GR textbooks avoid a physical interpretation of the gravitational time dilation, which is a shame. The textbook by Rindler is very strong on physical interpretations, so again I use my solution manual to point the reader to Rindler when appropriate, in this case his Chapter 9.
Do you have any advice for the intended readers of your book?
I’m convinced that anyone who makes the effort and doesn’t give up will succeed even if they are working on their own.
I’d like to encourage everyone who wants to learn GR to keep at it. The personal rewards of learning one of the most profound physical theories ever discovered make it worth the effort. Of course it is not easy. But I’m convinced that anyone who makes the effort and doesn’t give up will succeed even if they are working on their own. And of course you have to work the exercises on your own, before you’ve looked at my solution manual. Use my solutions as a way of verifying your answers or when you are stuck. Students sometimes ask me how long they should struggle on their own before consulting my solutions. It’s a personal decision and many factors are at play. If you have the luxury of no immediate time constraints like an exam next week, then the key question to ask yourself is: am I still learning things while I struggle with this problem? If not, if you have really ground to a halt, then it is time to look at the solution.
Do you have any specific advice for problem solving?
Try writing down exactly what you are stuck on and what you don’t understand. Then try re-reading the appropriate section in Schutz’s textbook. If that does not help, try consulting multiple books. My solution manual might point you to appropriate ones. Finally if something still does not make sense to you then you are most likely missing some essential background, often in calculus or linear algebra, that you will have to catch up on. Don’t give up!
Don’t give up!