By Éric Gourgoulhon
This graduate-level, course-based textual content is dedicated to the 3+1 formalism of common relativity, which additionally constitutes the theoretical foundations of numerical relativity. The ebook begins via setting up the mathematical heritage (differential geometry, hypersurfaces embedded in space-time, foliation of space-time through a kin of space-like hypersurfaces), after which turns to the 3+1 decomposition of the Einstein equations, giving upward thrust to the Cauchy challenge with constraints, which constitutes the middle of 3+1 formalism. The ADM Hamiltonian formula of normal relativity can also be brought at this level. eventually, the decomposition of the problem and electromagnetic box equations is gifted, concentrating on the astrophysically proper instances of an ideal fluid and an ideal conductor (ideal magnetohydrodynamics). the second one a part of the booklet introduces extra complex subject matters: the conformal transformation of the 3-metric on each one hypersurface and the corresponding rewriting of the 3+1 Einstein equations, the Isenberg-Wilson-Mathews approximation to normal relativity, international amounts linked to asymptotic flatness (ADM mass, linear and angular momentum) and with symmetries (Komar mass and angular momentum). within the final half, the preliminary information challenge is studied, the alternative of spacetime coordinates in the 3+1 framework is mentioned and numerous schemes for the time integration of the 3+1 Einstein equations are reviewed. the must haves are these of a easy basic relativity direction with calculations and derivations provided intimately, making this article entire and self-contained. Numerical options usually are not lined during this book.
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Extra info for 3+1 Formalism in General Relativity: Bases of Numerical Relativity
60) → → → Notice that for any multilinear form A on Σ, − γ ∗ (− γ ∗M A) = − γ ∗M A, for a vector − → − → − → → ∗ ∗ γ , and v ∈ T p (M ), γ v = γ (v), for a linear form ω ∈ T p (M ), γ ∗ ω = ω ◦ − − → − → ∗ ∗ for any tensor T , γ T is tangent toΣ, in the sense that γ T results in zero if one . 2 Relation Between K and ∇n A priori the unit vector n normal to Σ is defined only at points belonging to Σ. Let us consider some extension of n in an open neighbourhood of Σ. If Σ is a level surface of some scalar field t, such a natural extension is provided by the gradient are well defined of t, according to Eq.
2 Generalization to Any Tensor Field The Lie derivative is extended to any tensor field by (i) demanding that for a scalar field f, Lu f = ∇ f, u and (ii) using the Leibniz rule. αk β1 ... β + ↑ i=1 i i k ↓ T α1 ... β i=1 ∂u αi ∂xσ ∂u σ . 90) In particular, for a 1-form, Lu ωα = u μ ∂ωα ∂u μ + ω . 91) As for the vector case [Eq. 88)], the partial derivatives in Eq. β − T i ↓ α1 ... αk β1 ... β ∇βi u σ . 16 Both the covariant derivative (affine connection) and the Lie derivative act on any kind of tensor field.
E. we shall treat tensor fields defined on Σ as if they were defined on M . For covariant tensors (multilinear forms), if not mentioned explicitly, the four→ dimensional extension is performed via the − γ ∗M operator, as above for γ and K . For contravariant tensors, the identification is provided by the push-forward mapping Φ∗ discussed in Sect. 1. This four-dimensional point of view has been advocated by Carter [8–10] and results in easier manipulations of tensors defined in Σ, by treating them as ordinary tensors on M .
3+1 Formalism in General Relativity: Bases of Numerical Relativity by Éric Gourgoulhon