In their 1279-page book "Gravitation," Misner, Thorne and Wheeler emphasize that the student of general relativity must master differential geometry on three different levels: (1) a pictorial level that reflects deep geometric intuition; (2) a conceptual level where equations may be expressed in coordinate-free or frame-independent notation; and (3) a computational level in local coordinates, which involves acquiring skill with the "debauch of indices" computations that are so characteristic of the subject, especially in the physics literature.
Barrett O'Neill is a highly accomplished differential geometer who worked in Riemannian geometry for some time before he began writing books on Lorentzian geometry and general relativity. All of his work is characterized by a mathematician's primary emphasis on the coordinate-free level (2), as mentioned in the preceding paragraph, before turning to local coordinate expressions. Mathematicians who are approaching general relativity as "outsiders," in particular, will find O'Neill's works extremely accessible---a welcome relief from the physics texts that are often written almost exclusively in index-based notation. O'Neill's book "Semi-Riemannian Geometry with Applications to Relativity" was written in 1983, and in my opinion it still remains the best introduction to Lorentzian geometry and general relativity for the well-prepared student who wants to see the mathematics "done right."
The book under review was first published in 1995, and it offered the first book-length treatment of Kerr spacetime written in a modern mathematical style, stressing both coordinate-free and coordinate-based computations. A casual comparison of O'Neill's book with Chandrasekhar's classic "The Mathematical Theory of Black Holes" (1983) will immediately reveal profound differences in the mathematical style of the two books. As Misner, Thorne and Wheeler said, the student who would master relativity theory must learn to read both styles of text with comfort.
The great virtue of O'Neill's books, however, is that they first provide profound conceptual insights into the more elusive concepts of general relativity through their elegant, coordinate-free expression. Once one has understood a particular concept, it is then relatively easy to explore its local coordinate expression; moving in the reverse direction can be quite difficult, however, especially for those who have not developed the intimate familiarity with complex index manipulations that comes from years of practice.
The fifth chapter of O'Neill's "The Geometry of Kerr Black Holes" contains an amazingly lucid discussion of the Weyl curvature tensor and its use in assigning Petrov Types to spacetimes. This chapter alone is worth the price of the entire book. Again, a comparison of this chapter with the related Chapter 4 in Stephani, Kramer, et. al.'s well-known "Exact Solutions to Einstein's Field Equations, 2nd Ed.," will
reveal sharp contrasts. O'Neill begins with a discussion of how the Hodge star operator * provides a complex vector space structure on the second exterior product of each tangent space; the Weyl curvature tensor induces an operator on this vector space that commutes with * and hence may be viewed as a complex linear operator. The Petrov classification emerges through a consideration of the complex eigenvalues of this linear operator. One might struggle for some time to apprehend these fundamental mathematical facts from the coordinate-based approach, which begins with ponderous eigenvalue equations written out in index notation and little or no discussion of the basic linear-algebraic concepts that underlie those equations.
In summary, O'Neill's book is highly recommended to the mathematician who is interested in general relativity, and to the physicist who desires to see the mathematics of GR expressed in both coordinate-free and coordinate-based formulas. The book stands in good company with related works by authors such as Theodore Frankel, Norbert Straumann, Rainer Sachs, and Jerrold Marsden, all of whom have written wonderful books on mathematical physics that emphasize the modern approach to differential geometry. It is regrettable that at the time of this review, O'Neill's book appears to have gone out of print.