Geometry of Kerr Black Holes

Description

Reader's review:
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.

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