Modern introductory general relativity books at the advanced undergraduate level tend to cover mostly the same material: special relativity, tensor calculus, curvature and the Riemann tensor, Einstein's field equations, Schwarzschild solution, gravity waves, and cosmology. But they do not necessarily cover fluid mechanics, the material of this chapter. This chapter has come in lieu of one on electricity and magnetism. Either of these topics would give us practice working with tensors and developing frame-invariant equations as required to understand the development of the Einstein field equations. Schutz's choice here is especially useful because fluid mechanics is of more general relevance to astrophysical applications of general relativity. The last exercise of Schutz §4.10, Exercise 4.25, is on electricity and magnetism. Several texts (Rindler, 2006; Lawden, 2002; Hobson et al., 2006) have a full chapter on electricity and magnetism with the aim of preparing the student for the development of the Einstein field equations (but without a fluids chapter).

Exercises

4.1 Comment on whether the continuum approximation is likely to apply to the following physical systems:

(a) Planetary motions in the solar system.

(b) A lava flow from a volcano.

General relativity is a beautiful theory, our standard theory of gravity, and an essential component of the working knowledge of the theoretical physicist, cosmologist, and astrophysicist. It has the reputation of being difficult but Bernard Schutz, with his groundbreaking textbook, A First Course in General Relativity (first edition published in 1984, current edition in 2009), demonstrated that GR is actually quite accessible to the undergraduate physics student. With this solution manual I hope that GR, using Schutz's textbook as a main resource and perhaps one or two complementary texts (see recommendations at the end of this preface), is accessible to all “technically minded self-learners” e.g. the retired engineer with some time to devote to a dormant interest, a philosopher of physics with a serious interest in deep understanding of the subject, the mathematics undergraduate who wants to become comfortable with the language of the physicist, etc.

You can do it too!

I'm speaking with some experience when I say that an engineer can learn GR and in particular starting with Schutz's textbook. My bachelor's and master's degrees are in engineering and I started learning GR on my own when my academic career had gained enough momentum that I could afford a bit of time to study a new area in my free time. I must admit it wasn't always easy. I personally found the explanations of mathematics in the excellent textbook by Misner, Thorne and Wheeler (1973) more confusing then helpful. (In retrospect I'm at a loss to explain why; in no way do I blame the authors.) Soon two children arrived miraculously in our household, free time became an oxymoron, but with the constant reward I found from beavering away at Schutz's exercises I continued to learn GR, albeit slowly and with screaming (not always my own) interruptions. In his autobiography John A. Wheeler explains that he started learning GR in the 1940s when he finally got the chance to teach the subject. Similarly the real breakthrough for me came when I was offered the possibility to teach the subject to third-year undergraduate students at the Université de Bretagné Occidentale in Brest, France. Suddenly my hobby became my day job, fear of humiliation became my motivation, and most significantly I was forced to view the subject from the student's point of view.