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Introduction to \(3+1\) numerical relativity. (English) Zbl 1140.83002
International Series of Monographs on Physics 140. Oxford: Oxford University Press (ISBN 978-0-19-920567-7/hbk). xiv, 444 p. (2008).
This very carefully developed book gives much more than the title seems to promise: it is a well-prepared introduction to the mathematics of general relativity, and has among many other topics, also a large part about possible numerical solutions of the Einstein field equation. The preface shortly deals with the history of numerical relativity.
The 10 main chapters are entitled as follows: 1. Brief review of general relativity, 2. The \(3+1\) formalism, 3. Initial data, 4. Gauge conditions, 5. Hyperbolic reductions of the field equations, 6. Evolving black hole spacetimes, 7. Relativistic hydrodynamics, 8. Gravitational wave extraction, 9. Numerical methods, 10. Examples of numerical spacetimes.
Multipole expansion, Newman Penrose formalism, apparent horizons, and many other objects are discussed in detail in the first 8 chapters, whereas chapters 9 and 10 concentrate on the methods for numerical solutions like Runge-Kutta etc.
The four appendices deal with A: total mass and momentum, B: Christoffel symbols, C: Conformal rescalings, and D: Spin-weighted spherical harmonics. This last appendix D contains a clear deduction of these harmonic functions not easily to be found elsewhere. The book ends with an extended reference list of 309 items and a short subject index.

83-02 Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory
83-04 Software, source code, etc. for problems pertaining to relativity and gravitational theory
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C75 Space-time singularities, cosmic censorship, etc.
83C57 Black holes
83C35 Gravitational waves
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