Statistics – Applications
Scientific paper
Jun 2006
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2006cqgra..23.4011b&link_type=abstract
Classical and Quantum Gravity, Volume 23, Issue 11, pp. 4011-4012 (2006).
Statistics
Applications
Scientific paper
Null hypersurfaces are a mathematical consequence of the Lorentzian signature of general relativity; singularities in mathematical models usually indicate where the interesting physics takes place. This book discusses what happens when you combine these ideas.
Right from the preface, this is a no-nonsense book. There are two principal approaches to singular shells, one distributional and the other 'cut and paste'; both are treated in detail. A working knowledge of GR is assumed, including familiarity with null tetrads, differential forms, and 3 + 1 decompositions. Despite my own reasonably extensive, closely related knowledge, there was material unfamiliar to me already in chapter 3, although I was reunited with some old friends in later chapters.
The exposition is crisp, with a minimum of transition from chapter to chapter. In fact, my main criticism is that there is no clear statement of the organization of the book, nor is there an index. Everything is here, and the story is compelling if you know what to look for, although it is less easy to follow the story if you are not already familiar with it.
But this is really a book for experts, and the authors certainly qualify, having played a significant role in developing and extending the results they describe. It is also entirely appropriate that the book is dedicated to Werner Israel, who pioneered the thin-shell approach to (non-null) singular surfaces and later championed the use of similar methods for analysing null shells.
After an introductory chapter on impulsive signals, the authors show how the Bianchi identities can be used to classify spacetimes with singular null hypersurfaces. This approach, due to the authors, generalizes the framework originally proposed by Penrose [1]. While astrophysical applications are discussed only briefly, the authors point out that detailed physical characteristics of signals from isolated sources can be determined in this manner. In particular, they describe the behaviour of test particles in such a spacetime, in an initial attempt to outline a framework for the detection of impulsive gravitational waves.
Subsequent chapters describe the singular null hypersurfaces obtained by boosting isolated gravitational sources, building on the work of Aichelburg and Sexl [2], and by colliding impulsive waves, building on the work of Khan and Penrose [3]. In between, the special case of spherical symmetry is considered, both with and without collisions. There is also a short chapter discussing the effect of replacing GR by alternative theories of gravity, and an appendix which briefly summarizes the non-null case.
The references are reasonably complete, from Synge and Penrose to the recent work of the authors. However, there are a few relatively minor errors and omissions. For instance, the results in chapter 3 about shells of matter in both Schwarzschild and Reissner Nordström geometries are presented without reference or derivation. And I was disappointed to see that my own work with 't Hooft [4] on the horizon shift due to the impulsive wave of a massless particle at the horizon of a Schwarzschild black hole—a direct generalization of the work by Aichelburg and Sexl—is not mentioned.
But none of these minor complaints detracts from my appreciation of having a complete discussion of singular null hypersurfaces all in one place. The three fundamental papers [1 3] which started this area of research all appeared at essentially the same time, 35 years ago; it is high time there was a unified presentation of the entire field. This book fills that need admirably, and could serve as the core of a graduate seminar for students having already taken a course in general relativity, or as a reference. My copy will have a treasured place in my library.
References
Penrose R 1972 The geometry of impulsive gravitational waves General Relativity: Papers in Honour of J L Synge ed L Ó Raifeartaigh (Oxford: Clarendon) pp 101 30
Aichelburg P C and Sexl R U 1971 On the gravitational field of a massless particle Gen. Rel. Grav. 2 303 12
Khan K A and Penrose R 1971 Scattering of two impulsive gravitational plane waves Nature 229 185 6
Dray T and 't Hooft G 1985 The gravitational shock wave of a massless particle Nucl. Phys. B 253 173 88
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