### Preface

This book is an introduction to general relativity, intended for advanced undergraduates or beginning graduate students in either mathematics or physics. The goal is to describe some of the surprising implications of relativity without introducing more formalism than necessary. “Necessary” is of course in the eye of the beholder, and this book takes a nonstandard path, using differential forms rather than tensor calculus, and trying to minimize the use of “index gymnastics” as much as possible. 1)

As with most of my colleagues in relativity, I learned the necessary differential geometry the way mathematicians teach it, in a coordinate basis. It was not until years later, when trying to solve two challenging problems (determining when two given metrics are equivalent, and studying changes of signature) that I became convinced of the advantages of working in an orthonormal basis. This epiphany has since influenced my teaching at all levels, from vector calculus to differential geometry to relativity. The use of orthonormal bases is routine in physics, and was at one time the standard approach to the study of surfaces in three dimensions. Yet no modern text on general relativity makes fundamental use of orthonormal bases; at best, they calculate in a coordinate basis, then reinterpret the results using a more physical orthonormal basis.

This book attempts to fill that gap.

The standard basis vectors used by mathematicians in vector analysis possess several useful properties: They point in the direction in which the (standard, rectangular) coordinates increase, they are orthonormal, and they are the same at every point. No other bases have all of these properties; whether working in curvilinear coordinates in ordinary, Euclidean geometry, or on the curved, Lorentzian manifolds of general relativity, some of these properties must be sacrificed.

The traditional approach to differential geometry, and as a consequence to general relativity, is to abandon orthonormality. In this approach, one uses a coordinate basis, in which, say, the basis vector in the $\theta$ direction corresponds to the differential operator that takes $\theta$-derivatives. In other words, one defines the basis vector $\ev_\theta$ by an equation of the form $\ev_\theta \cdot \grad{f} = \Partial{f}{\theta}$

Physics, however, is concerned with measurement, and the physically relevant components of vector (and tensor) quantities are those with respect to an orthonormal basis. The fact that angular velocity is singular along the axis of symmetry is a statement about the use of angles to measure “distance”, rather than an indication of a physical singularity. In relativity, where we don't always have a reliable intuition to fall back on, this distinction is especially important. We therefore work exclusively with orthonormal bases.

In both approaches, however, one must abandon the constancy of the basis vectors. Understanding how the basis vectors change from point to point leads to the introduction of a connection, and ultimately to curvature. These topics were discussed extensively in a companion volume on differential forms, The Geometry of Differential Forms, and are briefly reviewed in the first chapter.

We also follow an “examples first” approach, beginning with an analysis of the Schwarzschild geometry based on geodesics and symmetry, and only later discuss Einstein's equation and cosmological models. This allows the reader an opportunity to master the geometric reasoning essential to relativity before being asked to follow the more sophisticated arguments leading to Einstein's equation.

No prior knowledge of physics is assumed in this book, although the reader will benefit from familiarity with Newtonian mechanics and with special relativity. This book does however assume familiarity with differential forms, at the level of the companion volume cited above, which in turn requires familiarity with vector calculus and linear algebra. For the reader in a hurry, the essentials of both special relativity and differential forms are reviewed in the first chapter.

1) For the expert, the only rank-2 tensor objects that appear in the book are the metric tensor, the energy-momentum tensor, and the Einstein tensor, all of which are instead described as vector-valued 1-forms; the Ricci tensor is only mentioned to permit comparison with more traditional approaches.