It is a challenging but rewarding task to teach general relativity to undergraduates. Time and experience are in short supply. One can rely neither on the
undivided attention of students who are studying many other exciting topics
in the final years of their course, nor on easy familiarity with the classical
tools of applied mathematics and geometry. Not only are the ideas themselves
difficult, but the calculations needed to solve even quite simple problems are
themselves technically challenging for students who have only recently learned
about multivariable calculus and partial differential equations.
For those with a strong background in pure mathematics, there is the temptation to present the theory as an application of differential geometry without
conveying a clear understanding of its detailed connection with physical observation. At the other extreme, one can focus too exclusively on physical prediction, and ask the audience to take too much of the mathematical argument on
trust.
This book is based on a course given at the Mathematical Institute in Oxford over many years to final-year mathematics students. It is in the tradition
of physical applied mathematics as it is taught in this country, and may, I hope,
be of use elsewhere. It is coloured by the mathematical leaning of our students,
but does not present general relativity as a branch of differential geometry. The
geometric ideas, which are of course central to the understanding of the nature
of gravity, are introduced in parallel with the development of the theory—the
emphasis being on laying bare how one is led to pseudo-Riemannian geometry
through a natural process of reconciliation of special relativity with the equivalence principle. At centre stage are the ‘local inertial coordinates’ set up by
an observer in free-fall, in which special relativity is valid over short times and
distances