e-lecturer note

3+1 Formalism in General Relativity: Bases of Numerical Relativity

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The book starts with a chapter setting the mathematical background, which is
differential geometry, at a basic level (Chap. 2). This is followed by two purely
geometrical chapters devoted to the study of a single hypersurface embedded in
spacetime (Chap. 3) and to the foliation (or slicing) of spacetime by a family of
spacelike hypersurfaces (Chap. 4). The presentation is divided in two chapters to
distinguish between concepts which are meaningful for a single hypersurface and
those that rely on a foliation. The decomposition of the Einstein equation relative
to the foliation is given in Chap. 5, giving rise to the Cauchy problem with
constraints, which constitutes the core of the 3+1 formalism. The ADM Hamiltonian
formulation of general relativity is also introduced in this chapter. Chapter 6
is devoted to the decomposition of the matter and electromagnetic field equations,
focusing on the astrophysically relevant cases of a perfect fluid and a perfect
conductor (ideal MHD). An important technical chapter occurs then: Chap. 7
introduces some conformal transformation of the 3-metric on each hypersurface
and the corresponding rewriting of the 3+1 Einstein equations. As a by-product,
we also discuss the Isenberg-Wilson-Mathews (or conformally flat) approximation
to general relativity. Chapter 8 details the various global quantities associated with
asymptotic flatness (ADM mass, ADM linear momentum and angular momentum)
or with some symmetries (Komar mass and Komar angular momentum). In
Chap. 9, we study the initial data problem, presenting with some examples two
classical methods: the conformal transversetraceless method and the conformal
thin-sandwich one. Both methods rely on the conformal decomposition that has
been introduced in Chap. 7. The choice of spacetime coordinates within the 3+1
framework is discussed in Chap. 10, starting from the choice of foliation before
discussing the choice of the three coordinates in each leaf of the foliation. The
major coordinate families used in modern numerical relativity are reviewed.
Finally Chap. 11 presents various schemes for the time integration of the 3+1
Einstein equations, putting some emphasis on the most successful scheme to date,
the BSSN one. Appendix A is devoted to basic tools of the 3+1 formalism: the
conformal Killing operator and the related vector Laplacian, whereas Appendix B
provides some computer algebra codes based on the Sage system.

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Informasi Detail

Judul Seri

The Lecture Notes in Physics

No. Panggil

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Penerbit

New York : Springer-Verlag Berlin Heidelberg., 2012

Deskripsi Fisik

xv, 294 Hlm.

Bahasa

English

ISBN/ISSN

978-3-642-24525-1

Klasifikasi

NONE

Tipe Isi

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Tipe Media

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Tipe Pembawa

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Edisi

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Subjek

FISIKA
GEOMETRI

Info Detail Spesifik

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Pernyataan Tanggungjawab

agus

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