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Spinors and Space-Time: Volume 2, Spinor and Twistor Methods in Space-Time Geometry (Cambridge Monographs on Mathematical Physics)

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The first volume of this set provided a through development of two-spinor calculus. This book, the second volume, uses two-spinor calculus to analyze some problems in physics. It examines situations where spinor methods unquestionably provide value, such as Witten's proof of the positive energy theorem. It also introduces twistors (only briefly mentioned in volume I). The original goal of twistor theory was very ambitious, to essentially quantize space-time by making it a concept derived from twistor space. Obviously that hasn't be achieved, but that goal isn't a primary focus of this book.

The first chapter is a quick recap of volume I, including most of the formulas from it that are used in this book. After this, twistors are introduced. The algebra is easy to follow, but there is a lot of it. A geometric description of null twistors is given in terms of the spinors it's composed of. Later it's shown how to construct a null twistor, up to a phase factor, from the momentum and spin of a massless particle. I thought this provided a nice physical model. One of the interesting things is that conformal invariance has a fairly central role, as it does in string theory. Overall the contact with physics in this chapter is fairly light.

From this point on application to physics are more central, of course there is still plenty of math. The first topic considered is congruences of null geodesics. A couple of things stand out. One is the coverage is more complete than usual. The other is that the authors describe the relation of twistors to shear free ray congrucences, normally it's only spinors that are used to describe the congruences.

The next chapter covers on of the more widely used applications of spinor methods to general relativity, the classification of the Weyl tensor. While this is perhaps the most widely discussed application of spinors in general relativity, the depth of the discussion here is much greater than the usual. Instead of just showing how much more transparent the analysis is with spinors, this book also adds a twist that cannot be done with tensors, it considers changing the phase and magnitude of the Weyl spinor, i.e. it considers more than the principal null directions. There is also material on the classification of the Ricci curvature, which is uncommon.

Following this spinors are applied to asymptotic questions in general relativity. The chapter begins with a review of causal structure and compactification (compactification in the sense of conformal compactification, not in the sense of wrapping extra dimensions in a small torus). The view that space-time points are entities derived from twistor space is further developed here, but still not in great detail. For me the best parts of this, rather long, chapter were the discussions of peeling properties of gravitational radiation the use of spinors and twistors to analyze energy-momentum and angular momentum. Regarding the latter, it also includes the use of spinors to prove the positive energy theorem.

I liked this book a lot, even more than volume I. One thing that surprised me is that I expected more development of twistor theory, there was a fair amount, but I would have liked to have seen more. A lot of the material in this book is clearly relevant to physics, however there is also a substantial amount that is more speculative.