By Éric Gourgoulhon
This graduate-level, course-based textual content is dedicated to the 3+1 formalism of normal relativity, which additionally constitutes the theoretical foundations of numerical relativity. The booklet begins through setting up the mathematical heritage (differential geometry, hypersurfaces embedded in space-time, foliation of space-time via a kinfolk of space-like hypersurfaces), after which turns to the 3+1 decomposition of the Einstein equations, giving upward push to the Cauchy challenge with constraints, which constitutes the middle of 3+1 formalism. The ADM Hamiltonian formula of common relativity can be brought at this level. ultimately, 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 publication introduces extra complicated subject matters: the conformal transformation of the 3-metric on every 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 info challenge is studied, the alternative of spacetime coordinates in the 3+1 framework is mentioned and diverse schemes for the time integration of the 3+1 Einstein equations are reviewed. the must haves are these of a uncomplicated basic relativity path with calculations and derivations offered intimately, making this article whole and self-contained. Numerical innovations will not be lined during this book.
Keywords » 3+1 formalism and decomposition - ADM Hamiltonian - Cauchy challenge with constraints - Computational relativity and gravitation - Foliation and cutting of spacetime - Numerical relativity textbook
Related matters » Astronomy - Computational technology & Engineering - Theoretical, Mathematical & Computational Physics
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Extra info for 3+1 Formalism in General Relativity - Bases of Numerical Relativity
One can deduce from Eq. 67) an interesting formula about the derivative of a vector field v along another vector field u, when both vectors are tangent to Σ. Indeed, from Eq. 67), ( D u v)α = u σ Dσ vα = u σ γ v σ γ α μ ∇v vμ = u v δ α μ + n α n μ ∇v vμ α uv α v = u ∇v v + n u n μ ∇v v μ = u v ∇v v α − n α u v v μ ∇v n μ , v −vμ ∇v n μ where we have used n μ vμ = 0 (v being tangent to Σ) to write n μ ∇v vμ = −vμ ∇v n μ . Now, from Eq. 20) and the symmetry of K, u v vμ ∇v n μ = −K (v, u) = −K (u, v), so that the above formula becomes ∀(u, v) ∈ T (Σ) × T (Σ), D u v = ∇ u v + K (u, v)n .
Vn ). 1 By itself, the embedding Φ induces a mapping from vectors on Σ to vectors on M (push-forward mapping Φ∗ ) and a mapping from 1-forms on M to 1-forms on Σ (pull-back mapping Φ ∗ ), but not in the reverse way. For instance, one may define “naively” a reverse mapping F : T p (M ) −→ T p (Σ) by v = (vt , v x , v y , vz ) −→ Fv = (v x , v y , vz ), but it would then depend on the choice of coordinates (t, x, y, z), which is not the case of the push-forward mapping defined by Eq. 4). As we shall see below, if Σ is a spacelike hypersurface, a coordinateindependent reverse mapping is provided by the orthogonal projector (with respect to the ambient metric g) onto Σ.
18) where the κi are the three eigenvalues of χ . 3 The curvatures defined above are not to be confused with the Gaussian curvature introduced in Sect. 3. The latter is an intrinsic quantity, independent of the way the manifold (Σ, γ ) is embedded in (M , g). On the contrary the principal curvatures and mean curvature depend on the embedding. For this reason, they are qualified of extrinsic. 19) is symmetric. It is called the second fundamental form of the hypersurface Σ. It is also called the extrinsic curvature tensor of Σ (cf.
3+1 Formalism in General Relativity - Bases of Numerical Relativity by Éric Gourgoulhon