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 developing the mathematical heritage (differential geometry, hypersurfaces embedded in space-time, foliation of space-time through 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. eventually, the decomposition of the problem and electromagnetic box equations is gifted, targeting the astrophysically proper instances of an ideal fluid and an ideal conductor (ideal magnetohydrodynamics). the second one a part of the ebook introduces extra complex themes: 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 common 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 facts 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 normal relativity direction with calculations and derivations provided intimately, making this article whole and self-contained. Numerical suggestions aren't coated during this book.
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Extra resources for 3+1 Formalism in General Relativity: Bases of Numerical Relativity
70) leads to ∇ u u = −K (u, u)n. If L were a geodesic of (M , g), one should have ∇ u u = κ u, for some non-affinity parameter κ. Since u is never parallel to n, we conclude that the extrinsic curvature tensor K measures the failure of a geodesic of (Σ, γ ) to be a geodesic of (M , g). Only in the case where K vanishes, the two notions of geodesics coincide. For this reason, hypersurfaces for which K = 0 are called totally geodesic hypersurfaces. 1 in Sect. 5 is a totally geodesic hypersurface: K = 0.
The latter is then compared with the actual value of λv at the point q, the difference (divided by λε) defining the Lie derivative Lu v so that (qq )α = x α ( p ) − x α ( p) = λvα ( p). 85) result in 1 α v (q) − vα ( p) ε 1 α 0 v (x + ε, x 1 , . . , x n−1 ) − vα (x 0 , x 1 , . . , x n−1 ) . 86) where we have used the standard notation for the components of a Lie derivative: Lu vα := (Lu v)α. Besides, using the fact that the components of u are u α = (1, 0, . . 25), are [u, v]α = ∂vα . 86): [u, v]α = Lu vα.
54), we get for v ∈ T (M ), ∇ · v = ∇μ vμ = ∂vμ + Γ μσ μ vσ . 63) where g := det(gαβ ) [Eq. 45)]. 64) where δ denotes any variation (derivative) that fulfills the Leibniz rule, tr stands for the trace and × for the matrix product. We conclude that ∂ 1 ∇·v= √ |g| ∂ x μ |g| vμ . 65) Similarly, for an antisymmetric tensor field of type (2, 0), ∇μ Aαμ = ∂ ∂ Aαμ ∂ Aαμ 1 α σμ μ ασ + Γ A +Γ A = +√ σ μ σ μ ∂xμ ∂xμ ∂ |g| x σ |g| Aασ , 0 where we have used the fact that Γσαμ is symmetric in (σ, μ), whereas Aσ μ is antisymmetric.