## A First Course in General Relativity

This morning I received delivery of a brand new copy of the Third Edition (left) of *A First Course in General Relativity* by Bernard Schutz. I bought the First edition (right) way back in 1985 when I started out as a graduate student. Not surprisingly there is a lot of additional material in the 3rd edition about gravitational waves, which had not been discovered when the first edition was published. I notice also that Bernard has lost his “F”…

In fact I have known Bernard for quite a long time, most recently as colleagues in the Data Innovation Research Institute in Cardiff. Before that he chaired the Panel that awarded me an SERC Advanced Fellowship in the days before STFC, and even before PPARC, way back in 1993. It just goes to show that even the most eminent scientists do occasionally make mistakes…

Anyway, the arrival of this book is a double coincidence because I’ve been thinking over the last couple of days about starting to organize teaching for next academic year. This isn’t easy as we still don’t know who is going to be available. We’re interviewing tomorrow for one of our vacant positions, actually. Yesterday also the University Bookshop sent out a request for textbooks to stock ahead of next academic year.

I was reflecting on the fact that I’ve been doing research in cosmology and theoretical astrophysics since 1985 and teaching undergraduate and postgraduate students since 1990 but I’ve never taught a course on general relativity. This may or may not change next year when teaching is allocated. There are many textbooks out there but, prompted by the arrival of Bernard’s new book, I was wondering if anyone reading this blog has any other recommendations, suitable for final-year undergraduate theoretical physics students, that might complement it on the reading list for my first course in general relativity, should I happen to give one?

Suggestions, please, through the comments box below!

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June 30, 2022 at 5:07 pm

For the sake of throwing in a name, Carroll was the book used in our TP postgrad course on GR. I think it is may be a bit technical for a final-year bachelor course, especially the second chapter on manifolds. However, the book is written really well and can be as entertaining as it is educating in my opinion. Maybe some sections could be adapted for a bachelor course.

Schutz was indeed the book recommended for experimental postgrad students learning GR here. I think this level is comparable to final year bachelor TP students so you are possibly in safe hands.

A final name would be Hartle, which is recommended for the final-year bachelor course here on gravitational waves. If it works for bachelor students in the Netherlands it may work in Ireland. Unfortunately I can say nothing about the content of the book however.

These are by no means exotic choices in the world of GR textbooks but I thought I would throw in the names I know 🙂

June 30, 2022 at 6:08 pm

Mike Hobson, George Efstathiou and Anthony Lasenby’s book (General Relativity: An Introduction for Physicists) is a worthy competitor.

July 1, 2022 at 7:22 am

I’ve heard of those guys…

June 30, 2022 at 9:13 pm

I took an undergrad course based on Schutz and was TA for an undergrad course based on Hartle. I got the impression that students liked Hartle better because Schutz sometimes is weeks of math and no physics. For instance, as soon as Hartle introduces the connection and geodesics it spends a lot of time discussing physics in curved space, going so far as just telling that Schwarzschild is the metric for spherical stars and seeing what the physics is like, whereas Schutz will wait until he has developed the math of curvature to derive the Schwarzschild metric and see what kind of physics it has.

Anyway, a good complement might be Schutz’s other book, “Geometrical Methods of Mathematical Physics”. It has a better introduction to differential geometry than the GR book, so students struggling through the math might like the companion. It’s also great because it comes with other applications of differential geometry to physics, in particular, Schutz’s discussion of thermodynamics is briliant and students might appreciate that the math goes beyond GR in terms of utility.

As for a physics complement, I would suggest Norbert Dragon’s “Geometry of Special Relativity”. The first two chapters do SR in the same fashion as one learns Euclidean Geometry in school, no coordinates, just diagrams, now suitably interpreted to be clocks in addition to rulers. It’s a great way to seeing the solution to various paradoxes, instead of just calculating them away.

The GR version of having a geometric centered discussion would be Geroch’s Lecture notes on General Relativity, recently printed in book form. Lots of diagrams and clear exposition of the underlying geometry of curved spacetimes.

Lastly, the best discussion of the physical interpretation of the Einstein equations is arguably John Baez’s “The Meaning of Einstein Equations”, a short pdf you can find on his website. The idea here is to show that Ricci curvature has a very straightforward interpretation as the second derivative of a volume element, divided by that same volume element (to first order), something that surprisingly rarely appears in GR textbooks.

I hope this is helpful

July 2, 2022 at 12:47 am

As an aside, Kempf’s General Relativity & Cosmology video course is outstanding…

July 2, 2022 at 7:52 pm

Though I wasn’t taught from it, I also recommend Hartle’s book for the fourth year undergraduates. As you (very probably!) know, there is a spectrum in GR textbooks that ranges from those like Weinberg – more emphasis on physics – to those like Wald – more mathematical.

After going through a graduate degree, my preference is for the latter as the math you learn is applicable in lots of other areas, i.e. things like differential geometry, real analysis, functional analysis. But for an undergraduate degree, you probably want something that builds on physical reasoning. Hartle mostly takes that approach, but also includes more mathematical material that you could expand on. Plus a nice treatment of gravitational waves.