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General Relativity and Gravitation, Vol. 35, No. 12, December 2003 (C© 2003)

Book Review

The Relativistic Boltzmann Equation: Theory and Applications. By C.Cercignani and G. M. Kremer.396p., Birkhauser, Basel, 2002. EUR88.00SFr132.00, ISBN 3-7643-6693-1.

Starting with a self-contained summary of special relativity theory, the first part ofthis text gives a very comprehensive survey of the special-relativistic Boltzmannequation and its application to determining equations of state of gases (degen-erate and non-degenerate, relativistic and non-relativistic, neutral and ionised)and studying properties of gas mixtures in various ways (Chapman-Enskog, Grad,Marle, etc.). This part of the book is very careful and complete, including a compre-hensive account of equilibrium distribution functions, transport equations, trans-port coefficients, wave propagation, and so on, and for example giving a derivationof the Chandrasekhar limits on white dwarf masses and a discussion of relativisticBose-Einstein condensation.

The latter part of the book gives a self-contained summary of general relativitytheory and the Boltzmann equation in gravitational fields, with application tospherically symmetric geometries and proof of some formal existence results inthis case. This part is far less complete and satisfactory than the first part. Thepresentation of general relativity is very old-fashioned and does not emphasizegeometrical aspects of the theory, nor is it very complete. It does not for instancemention the important paper by Tauber and Weinberg [1] on equilibrium solutions,nor explain satisfactorily the implications of the requirement that existence ofequilibrium solutions of the Boltzmann equation for massive particles requiresa timelike Killing vector, nor comment on the general relativistic version of theH-theorem. Indeed, although the section on general relativity is structured as aself-contained introduction, it does not explain the significance of Killing vectors,or clearly explain the geometrical meaning of geodesics. The section on cosmologyis cursory, and does not get to either case of real interest – on the one hand theuse of the Boltzmann equation in studying the evolution of the hot early universe(as discussed for example in Bernstein’s brief book [2]), and on the other itsuse in studying structure formation in the evolving universe and the associated

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anisotropies of the cosmic background radiation (CBR). The latter has been a veryactive area in the past decade, with many important theoretical developments andobservational results coming in. However the authors do not give the Ehlers-Geren-Sachs theorem (exact or approximate versions), neither do they give the multipoleequations used to calculate CBR anisotropies nor discuss approximate methodsused for its solution, let alone discussing issues like the existence of peaks in theCBR anisotropy multipole structure that have caused excitement in recent years.

The book will be very helpful to those interested in special relativistic equa-tions of state and their basis in kinetic theory and more mathematical aspects of theBoltzmann equation, but those interested in important uses of kinetic theory andthe Boltzmann equation in present day cosmology will have to look elsewhere.

REFERENCES

[1] Tauber, G. E. and Weinberg, J. W. (1961).Phys. Rev.122, 1342.[2] Bernstein, J. (1988).Kinetic Theory in the Expanding Universe(Cambridge University Press,

Cambridge, UK).

George EllisMathematics DepartmentUniversity of Cape Town

Rondebosch 7700Cape Town, South Africa

E-mail: ellis@maths.uct.ac.za